 

75313430
0

v 7.?
I: I...
A I

. It):

5:31:'
:I . . I: . €972?»
(.1! ”Parr .1 ..

11.).0n- '
Id It‘lgr'vr.'
gt.-
1 It I
‘tl‘l'i.
‘A‘AI'a I.)
Allbuvlraul I:
VA. I In!
l)|l..|lo.t
((5.!!7301 urn:
.v‘)?¢’.sr-|~IP
!.a.l.\¢’l.~

. A. .’I.
1?)? pi

17:62::
31.1.6.2: 5.13..
5.1311.-.»

4?! . :1!2..1!!...n¢n..

55129:}. - - .

 

THESIS

NIVERSITY LlBHARlE

Illliill‘l’ililllill I lilllUlllll Inuit

3 1293 00899 2822

 

 

 

 

 

 

 

 

 

 

This is to certify that the
dissertation entitled

Microwave Processing of Epoxy Resins and Graphite
Fiber/Epoxy Composites in a Cylindrical

Tunable Resonant Cavity
presented by

Jianghua Wei

has been accepted towards fulﬁllment
of the requirements for

Ph.D. Chemical

degree in
Engineering

 

 

t

Major professor

Date 11/6/92

 

MS U i: an Afﬁrmative Action/Equal Opportunity Institution 0-12771

 

LIBRARY

Michigan State ‘
University

 

PLACE IN RETURN BOX to remove thls checkout from your record
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

'll..q . 3.5.21}:-
égp"? in

(A:

 

 

 

 

 

 

 

 

 

NOV 282007
'0825 07

 

 

 

 

 

 

 

 

MSU Is An Affirmative ActiorVEqual Opportunity Institution
cMMpmS—M

Microwave Processing of Epoxy Resim and Graphite Fiber/Epoxy
Composites in a Cylindrical Tunable Resonant Cavity

By

J ianghua Wei

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirement

for the degree of

DOCTOR OF PHILOSOPHY

Department of Chemical Engineering
1992

ABSTRACT

Microwave Processing of Epoxy Resins and Graphite Fiber/Epoxy
Composites in a Cylindrical Tunable Resonant Cavity

By
J ianghua Wei

This dissertation presents a systematic study for the use of microwave energy as
an alternative to thermal energy in the processing of polymers and composites, including

the results of the neat resin study, various microwave processing technique, and
processing model.

In the cure study of neat epoxy resins, two stoichiometric epoxy/amine systems
were studied, diglycidyl ether of bisphenol A (DGEBA, Dow Chemical DER 332) with
diaminodiphenyl sulfone (DDS , Aldrich Chemical) and DGEBA with meta phenylene
diamine (MPDA, Aldrich Chemical). Comparative isothermal cures were conducted
using microwave and thermal energy. Compared to thermal cure, microwave cure
increased reaction rates and glass transition temperature of both DGEBA/DDS and
DGEBA/mPDA systems. The increase is much more signiﬁcant in the DGEBA/DDS
system than in the DGEBA/mPDA system. The microwave and thermal cure of both

systems were successfully described by second order autocatalytic reaction kinetics.

Batch, scale up of batch, and continuous processing techniques were also studied
using microwave energy. In the study of batch processing, 24-ply Hercules AS4/ 3501-6
graphite ﬁber/epoxy laminates were processed in a 17.78 cm diameter cylindrical tunable
resonant cavity using 2.45 GHz microwave radiation. The temperature uniformity across
the laminate was a strong function of the electromagnetic (EM) heating modes. With
proper choice of the heating mode, 3. 8 cm thick composites were uniformly heated. The
mechanical properties of microwave processed composites were higher than those of

thermally processed samples using similar processing temperature cycles. Better bonding
between the graphite ﬁbers and the matrix was observed in the microwave processed
composites than in the thermally processed samples. The scale up of batch processing
was studied using a 17.78 cm cavity with 2.45 GHz microwave radiation and a 45.72
cm cavity with 915 MHz microwave radiation. For low to medium loss materials, the
location of a given resonant mode can be scaled up accordingly, that is with the same
electromagnetic ﬁeld pattern and the heating characteristics, when every dimensions of
the sample was scaled by the seale-up factor of the cavities. A unique microwave
applicator was invented for continuous processing of conductor and nonconductor

reinforced composites using microwave energy without microwave leakage.

A ﬁve parameter microwave power absorption model and a one dimensional
composite processing model were developed. The process model was numerically coded
use FORTRAN and used in various simulations of microwave and thermal processing.
Simulation results show that: 1) the thickest Hercules AS4/3501-6 composite that can be
processed using the manufacture’s cure cycle is 4 cm, 2) good heat convection at the
composite surface is very important in reducing the temperature excursion during thermal
processing, 3) microwave processing is much better than thermal processing in process

controllability, process speed, and temperature uniformity inside the composite.

To my wife

Qianqian

iv

ACKNOWLEDGEMENT

I wish to thank Prof. Martin C. Hawley for his wisdom and guidance during the
course of this research. Thanks are also extended to Prof. Dennis Nyquist and Dr.
Jinder low for their many discussions and valuable suggestions, and to Mr. Brook
Thomas for his proofreading of the dissertation. Finally, I would like to thank my

wife for her understanding, caring, support, and endless love through the duration of
this work.

This research was supported in part by DARPA.

TABLE OF CONTENTS

CHAPTER 1 INTRODUCTION .............................. 1
1.1 Characteristics of Microwave and Thermal Heating ........... 1

1.2 Research Topics ................................. 5
CHAPTER 2 BACKGROUND FOR MICROWAVE PROCESSING ........ 6
2.1 Introduction ......................... . .......... 6

2.2 Interactions Between Electromagnetic Radiation and Materials ..... 6

2.2.1 Electron or Optical Polarization .................. 8

2.2.2 Atomic Polarization ......................... 10

2.2.3 Orientation or Dipole Polarization ................ 10

2.2.4 Interfacial or Space-charge Polarization ............. 11

2.3 Dielectric Properties of Polar Materials .................. 11

2.4 Microwave Heating Applications ...................... 14

2.5 Microwave Heating Systems .......................... 16

2.5.1 Cylindrical Tunable Resonant Cavity ............... 16

2.5.2 Q Factor of the Resonant Cavity ................. 23

2.5.3 Electromagnetic Field Inside the Resonated Cavity ...... 25

2.5.4 Microwave Processing System ................... 26

CHAPTER 3 MICROWAVE CURE OF EPOXY ................... 31
3.1 Introduction .................................... 31

3.2 Thermal Cure of Epoxy Resins ....................... 33

3.2.1. Cure Kinetics ............................ 33

3.2.1.1 The n“ order kinetics model .............. 33

3.2.1.2 Autocatalytic Kinetics Model .............. 35

3.2.2 Glass Transition Temperature ................... 42

3.3 Experiments ................................... 46

3.4 Results and Discussion ............................ 51

3.4.1 Reaction Kinetics .......................... 51

3.4.2 Glass transition temperature .................... 67
3.4.3 Master Curve and TIT diagrams ................. 71
3.5 Conclusion .................................... 77

CHAPTER 4 MICROWAVE PROCESSING OF UNIDIRECI'IONAL AND
CROSSPLY CONTINUOUS GRAPHITE FIBER/EPOXY

COMPOSITES ................................ 79
4.1 Introduction .................................... 79
4.2 Experiments ................................... 80
4.2.1 Resonant Heating Mode Selection and Maintenance ..... 82
4.2.2 Microwave Processing and Mechanical Properties Test . . . . 83
4.3 Results and Discussion ............................ 95
4.4 Conclusion ................................... 108

CHAPTER 5 FIBER ORIENTATION EFFECTS ON THE MICROWAVE
HEATING OF CONTINUOUS GRAPHITE FIBER/EPOXY

COMPOSITES ............................... 111
5.1. Introduction .................................. 111
5.2. Experiments .................................. 112
5.3. Results and Discussion ........................... 112
5.4. Conclusion .................................. 141

CHAPTER 6 SCALE-UP STUDY OF MICROWAVE HEATING IN

TUNABLE CAVITIES .......................... 143
6.1. Introduction .................................. 143
6.2. Scale-up Frequency .............................. 143
6.3. Experiments .................................. 144
6.4. Results and Discussion ........................... 146
6.4.1. Scale-up of Epoxy Loaded Cavities .............. 147

vii

6.4.2 Seale-up Study of Graphite Fiber Composite Loaded
Cavities ............................... 155
6.5. Conclusions .................................. 158

CHAPTER 7 CONTINUOUS PROCESSING OF GRAPHITE FIBER/EPOXY

TAPE IN A MICROWAVE APPLICATOR ............. 160
7.1 Introduction .................................. 160
7.2 Experiments .................................. 161
7.3 Results and Conclusion ............................ 164
7.4 Conclusion ................................... 168

CHAPTER 8 POWER ABSORPTION MODEL FOR MICROWAVE
PROCESSING OF COMPOSITES IN A TUNABLE

RESONANT CAVITY .......................... 170
8.1 Introduction ................................... 170
8.2 Problem Simpliﬁcation ........................... 170
8.3 Electromagnetic Model ............................ 173
8.4 Parameter Estimation ............................. 190
8.4.1 Energy Balance ........................... 191
8.4.2 Optimization of Parameters ................... 191

8.5 Measurement of Five Parameters for Microwave Power
Absorption Model ............................... 194
8.6 Conclusion ................................... 199

CHAPTER 9 PROCESSING MODEL FOR MICROWAVE AND THERMAL

PROCESSING OF COMPOSITES ................... 200
9.1 Introduction .................................. 200
9.2 Background ................................... 201
9.2.1 Viscosity ............................... 201
9.2.2 Resin Flow ............................. 204

9.2.3 Voids ................................
9.2.4 Composite Properties .......................

9.3 Microwave and Thermal Process Model .................
9.3.1 Reaction Kinetics ..........................
9.3.2 The Measurement of the Effective Heat Transfer

Coefﬁcient .............................

9.4 Simulation ....................................
9.4.1 Thermal Processing ........................
9.4.2 Microwave Processing ........... ' ...........
9.5 Conclusion ...................................

CHAPTER 10

CHAPTERll

APPENDIX I

APPENDIX II

APPENDIX III

SUMMARY OF RESULTS ..................

FUTURE WORK .........................

FORTRAN Code for Generating the 'I'I'T Diagram for
DGEBA/DDS System ......................

FORTRAN Code for Calculating the Parameters
for Microwave Power Absorption Model ..........

FORTRAN Code for Processing model ...........

000000000000000000000000000000000000000

210
211
214

216
218

218
224

235

237

Table 1.1

Table 3.1.
Table 3.2.

Table 3.3

Table 3.4
Table 3.5
Table 3.6
Table 3.7
Table 3.8

Table 3.9
Table 4.1

Table 4.2

Table 4.3

Table 5.1

Table 5.2

LIST OF TABLES

Comparison of Microwave and Thermal Heating Characteristics . . 2
Temperatures Used for Microwave and Thermal Cure ........ 49
The values of bl, b2, b3, b4, and e from the least-squares ﬁt . . . 52

Reaction Rate Constants for Both Thermal and Microwave Cure of
DGEBA Reacting with DDS and mPDA With Assumption of n =1,

L= ...................................... 58
Activation Energies and Pre-exponential constants for DGEBA Reacting
with DDS and mPDA With Assumption of n= 1, L=0 ....... 59
Calculations based 11" order assumption for DGEBA/mPDA . . . . 61
Reaction rate constants for DGEBA/DDS system ........... 64

Calculated reaction kinetics parameters for DGEBA/DDS system . 64
Reaction kinetic parameters for both DGEBA/DDS and DGEBA/mPDA
systems
The Parameters for DiBenedetto Model
Flexural Properties of Microwave Processed 24-crossply AS4/3501-6

Composite ................................... 98
Flexural Properties of Microwave Cured 24-p1y Unidirectional
AS4/3501-6 Composites at Various Orientations and Modes . . . 100

Comparison of Flexural Properties between Microwave and Thermally
Cured 24-p1y AS4/3501-6 Composites ................ 103
Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 0° Fiber Orientation ............. 128
Conditions and Results for Microwave Heating of 7 .8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 15 ° Fiber Orientation. ............ 129

Table 5.3

Table 5.4

Table 5.5

Table 6.1
Table 6.2
Table 6.3
Table 6.4

Table 6.5

Table 7.1

Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 45 ° Fiber Orientation. ............ 130
Conditions and Results for Microwave Heating of 7. 8 X 7 .8 X 3.8cm
Fully Thermally Cured Cressply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 75 ° Fiber Orientation. ............ 131
Conditions and Results for Microwave Heating of 7.8 X 7 .8 X 3. 8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm

Cylindrical Cavity for 90° Fiber Orientation. ............ 132
Heating Conditions for Scale-up Experiments ............ 146
Theoretical and Measured Results of Modes for Empty Cavity . . 147
Scale-up Results for Epoxy Loaded Cavity .............. 150
Heating Modes and Results for 60-ply Unidirectional Composite in a

45.72 cm Cavity .............................. 155
Heating Modes and Results for 24-ply Unidirectional Composite in a

17.78 cm Cavity .............................. 156

Heating Conditions and Results ..................... 164

xi

Figure 1.1

Figure 2.1
Figure 2.2
Figure 2.3
Figure 2.4
Figure 2.5
Figure 2.6
Figure 2.7

Figure 2.8
Figure 2.9

Figure 2.10
Figure 2.11

Figure 3.1

Figure 3.2

Figure 3.3
Figure 3.4

Figure 3.5

Figure 3.6

Figure 3.7
Figure 3.8

LIST OF FIGURES

Temperature Proﬁles During Microwave and Thermal Cure of Pure

Epoxy Resins ................................ 4
Four Types of Polarization Mechanisms ................ 9
Frequency Dependance of e' and e" ................... 12
Diagram of Cylindrical Tunable Resonant Cavity .......... 17
Theoretical EM Field of T'Mm in a Cylindrieal Empty Cavity . . . l9
Mode Chart for Circular Cylindrical Cavities ............. 21
Mode Chart for 15.24 cm Cylindrical Cavity ............. 21
Mode Chart for 17.78 cm Cylindrical Cavity ............. 22
Mode Chart for 45.72 cm Cylindrical Cavity ............. 23
Q—factor Determination From Power Absorption Curve ....... 25
Microwave Processing Systems ...................... 28
Deﬁnition of Pseudo-single, Controlled-hybrid, and Uncontrolled-
hybrid Modes ................................. 30
Comparison of Tg Determination for Thermally Cured

DGEBA/DDS . ................................ 44
Comparison of Tg Determination for Microwave Cured

DGEBA/DDS ................................ 45
Chemical Structure of DGEBA, DDS, and mPDA .......... 48
Electromagnetic Field Patterns for TEm Mode in the Cylindrieal

Cavity ....... ' .............................. 50
Comparison of FTTR Spectra of Microwave Cured, Thermal Cured,

and Fresh DGEBA/DDS samples ..................... 53
Comparison of FITR Spectra of Microwave Cured, Thermal Cured,

and Fresh DGEBA/mPDA samples ................... 54

Reaction Rates of Microwave and Thermally Cured DGEBA/DDS 56
Reaction Rates of Microwave and Thermally Cured DGEBA/mPDA 56

xii

Figure 3.9

Figure 3.10

Figure 3.11

Figure 3.12

Figure 3.13

Figure 3.14

Figure 3.15

Figure 3.16

Figure 4.1
Figure 4.2

Figure 4.3

Figure 4.4

Figure 4.5

Figure 4.6

Figure 4.7
Figure 4.8

Regeneration of Reaction Rates for Microwave and Thermally Cure of

DGEBA/mPDA From the Model Using n=1 and L=0 ....... 59
Regeneration of Reaction Rates for Microwave and Thermally Cure of
DGEBA/DDS From the Model Using n=1 and L=O ........ 62
Regeneration of Reaction Rates for Microwave and Thermally Cure of
DGEBA/DDS From the Model Using n¢1 and L¢0 ........ 66
Comparison of Tg of Microwave and Thermally Cured

DGEBA/DDS . . . - ............................. 68
Comparison of Tg of Microwave and Thermally Cured

DGEBA/mPDA ............................... 68
Master Curves for Microwave and Thermal Cure of DGEBA/mPDA at
Reference Temperature of 120°C ..................... 72
T'IT Diagrams for Both Microwave and Thermal Cure of
DGEBA/mPDA System .......................... 74
TIT Diagrams for Both Microwave and Thermal Cure of
DGEBA/DDS System ............................ 75
Composite Layup for Microwave Process ................ 81
Power Absorption Curve of Empty Cavity under Various Coupling
Situation .................................... 83
Typical Non-resonant Power Absorption Curve in the Composite
Loaded Cavity ................................ 84
The Locations of Four Temperature Probes and Meaning of Fiber
Orientations .................................. 85
Temperature/position/ time Proﬁle during Microwave Processing of 24
Crossply AS4/3501-6 Composites at Various Heating Modes. . . . 91
Typical Radial Electric Field Strength Distribution

Along Axial Position ............................ 91
Load Versus Deﬂection Curves for 3-point Bending Test ...... 92
Temperature/ time Proﬁles during Microwave Processing of 72-p1y
Unidirectional AS4/3501-6 Composite at a PM Mode. ........ 94

xiii

Figure 4.9

Figure 4.10

Figure 4.11

Figure 4.12

Figure 4.13

Figure 4.14

Figure 4.15

Figure 4.16

Figure 5.1

Figure 5 .2

Figure 5.3

Figure 5 .4

Figure 5.5

Spatial Distribution of the Extent of Cure in the Cross Section of the

Microwave Cured 72-ply Unidirectional Composite. ......... 94
Temperature/time Proﬁles during Thermal Processing of 72-ply
Unidirectional AS4/3501-6 Composite ................. 95

Temperature/time Proﬁle During Microwave Cure of 24 ply
Unidirectional AS4/3501-6 Composite at Fiber Orientation of 0° with
respect to the Coupling Probe ...................... 101
Temperature/time Proﬁle During Microwave Cure of 24 ply
Unidirectional AS4/3501-6 Composite at Fiber Orientation of 45 ° with
respect to the Coupling Probe ...................... 101
Temperature/time Proﬁle During Microwave Cure of 24 ply
Unidirectional AS4/3501-6 Composite at Fiber Orientation of 90° with

respect to the Coupling Probe ...................... 102
Comparison of the Temperature/time Proﬁles During Microwave Cure
of 24-ply AS4/3501-6 Composites ................... 102

Interlarninar Shear Strength of Microwave Processed 24-p1y
Unidirectional AS4/3501-6 Composites at Various Extent of Cure 105
SEM Pictures of Delaminated Surface from SBS Test for Microwave
and Autoclave Processed Unidirectional AS4/3501-6 Composites. 107
Temperature/position/time proﬁles during microwave heating of 3.8 cm
thick crossply AS4/3501-6 composite at 0° ﬁber orientation . . . 115
Temperature/position/time proﬁles during microwave heating of 3.8 cm
thick crossply AS4/3501-6 composite at 15° ﬁber orientation . . . 117
Temperature/position/ time proﬁles during microwave heating of 3.8 cm
thick crossply AS4/3501-6 composite at 45° ﬁber orientation . . . 120
Temperature/position/time proﬁles during microwave heating of 3.8 cm
thick crossply AS4/3501-6 composite at 75° ﬁber orientation . . . 122
Temperature/position/time proﬁles during microwave heating of 3.8 cm
thick crossply AS4/3501-6 composite at 90° ﬁber orientation . . . 125

xiv

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Figure 5.10

Figure 5.11
Figure 5.13

Figure 6.1
Figure 6.2

Figure 6.3

Figure 6.4

Figure 6.5

Figure 6.6

Figure 6.7

Figure 7 .1
Figure 7.2

' ﬁber orientation

Heating time for various resonant heating mode at various ﬁber
orientation .................................. 134
Temperature difference for various resonant heating mode at various
.............................. 135
Quality index for various resonant heating mode at various ﬁber
orientation .................................. 137
Qualitative description of penetration depth for individual ﬁbers . 139
Qualitative description of penetration depth for unidirectional
................................. 139
Qualitative description of penetration depth for crossply composites 140

composites

Temperature/position/time proﬁles during microwave processing of
ZOO-crossply fresh AS4/3501-6 composite. .............. 141
Loeation of the temperature probes .................. 146
Temperature Proﬁles and Radial Electric Field Pattern Along Axial
Direction during Microwave Heating of Epoxy Squares at

TMm Mode ................................ 151
Temperature Proﬁles and Radial Electric Field Pattern Along Axial
Direction during Microwave Heating of Epoxy Squares at

TE"; Mode ................................ 152
Shift of the Scale-up Factor during Microwave Heating of Epoxy
Squares ................................... 153
Shift of Cavity Length during Microwave Heating of Epoxy Square in a
45.72 cm Cavity .............................. 154
Shift of Cavity Length during Microwave Heating of Epoxy Square in a
17.78 cm Cavity .............................. 154
Heating Rate and Heating Uniformity Relationship for Microwave
Heating of Graphite Fiber/epoxy Composite . . . . h ......... 158
Modiﬁed 17.78 cm Tunable Cavity for Continuous Processing . . 162
Microwave System for Continuous Processing ............ 163

XV

Figure 7.3

Figure 7.4
Figure 7.5

Figure 7.6

Figure 8.1
Figure 8.2
Figure 8.3
Figure 8.4
Figure 8.5

Figure 8.6

Figure 8.7
Figure 8.8

Figure 8.9

Figure 8.10

Figure 8.11

Figure 8.12

Center Surface Temperature during Continuous Processing of Hercules

AS4/3501-6 Prepreg using Microwave Energy ........... 165
Extent of Cure at Various Resident Time in Three Modes . . . . 166
Extent of Cure Distribution Across the Tape as Function of Resident
Time for a CH mode at 12.45 cm ................... 167
Extent of Cure Distribution Across the Tape as Function of Resident
Time for a PS mode at 15.85 cm .................... 167
Incident Waves on the Composites During Processing ....... 172
One Dimensional Conﬁguration for Power Absorption Model . . 174
TEM Wave at Two Isotropic Medium a and b ............ 174
Base and Principal Coordinates ..................... 175
Waves at Interface with TEM Waves Propagating at Both
Directions .................................. 180
Electric Field Inside n"l Ply ....................... 182
TEM Waves at Top and Bottom Plies ................. 184

Temperature Distribution Across the Fully Cured 72-ply Unidirectional
Hercules AS4/3501-6 Composite during Microwave Heating at

L,= 16.03 cm with 70W Input Power ................. 196
Temperature Distribution Across the Fully Cured 72-ply Unidirectional
Hercules S4/3501-6 composite during Microwave Heating at L,=16.03
cm with 80W Input Power ........................ 196
Temperature Distribution Across the Fully Cured 72-p1y Unidirectional
Hercules 84/3501-6 Composite during Microwave Heating at L,= 16.03
cm with 150W Input Power ....................... 197
Input Power Effect on the Parameters in Microwave Power Absorption

Model during Microwave Heating of Composite at Resonant Mode with

L,=16.03 cm ................................ 197
Linear Relationship Between AT and P” During Microwave Heating of
AS4/3501-6 Composite at Resonant Mode with L,=16.03 cm . . 198

xvi

Figure 8.13

Figure 8.14

Figure 9.1
Figure 9.2

Figure 9.3
Figure 9.4

Figure 9.5

Figure 9.6
Figure 9.7
Figure 9.8
Figure 9.9

Figure 9.10

Figure 9.11

Linear Relationship Between A, and P” During Microwave Heating of

AS413501-6 Composite at Resonant Mode with L,=16.03 cm . . 198
Linear Relationship Between" P1 and P” During Microwave Heating of
AS4/3501-6 Composite at Resonant Mode with L,= 16.03 cm . . 199
Composite conﬁguration for processing ................ 212
Temperature/time Proﬁle During Cooling of 2-ply Hercules AS4/3501-
6 composite ................................. 217
Ln(T-T_) Versus Time Curve ...................... 218
Temperature/time Proﬁles for Thermal Processing of AS413501—6
laminate .................................. 220
Extent of Cure Proﬁles for Thermal Processing of AS4/3501-6
laminate .................................. 220
Temperature Proﬁles for Various laminate Thickness ....... 223
Extent of Cure Proﬁles for Various laminate Thickness ..... 223

Temperature Proﬁles for Various Effective Heat Transfer

Coefﬁcient ................................. 224
Temperature/ time Proﬁles during Microwave Processing of 72-ply
Unidirectional AS4/3501-6 Composite at L,=16.03 cm and 70 W 225
Extent of Cure Proﬁles during Microwave Processing of 72-p1y
Unidirectional AS4/3501-6 Composite at L,=16.03 cm and 70 W 225
Temperature! time Proﬁles during Microwave Processing of 72-p1y
Unidirectional AS4/3501-6 Composite at L,= 16.03 cm and 500 W 226

xvii

CHAPTER 1
INTRODUCTION

Because of their superior speciﬁc properties and high performance, graphite
ﬁber reinforced epoxy composites are widely used in aerospace, military, recreation,
transportation, and other industries 1". This type of composite is usually processed
in the autoclave by ﬁrst arranging uncured prepreg into a designed shape and then
curing at elevated temperatures and pressures using thermal energy. However,
autoclave curing requires long cure cycles, such as 4.5 to 8.5 hours for Hercules
AS4/3501-6 composites, and even this long cure cycle is only suitable for thin parts
because of the nature of thermal heating and large exotherm resulting from the
polymerization reaction. To shorten the processing time and to process thick-section

composites, new processing techniques need to be developed.

1.1 Characteristics of Microwave and Thermal Heating

Thermal heating is characterized by surface-driven, non-selective heating. The
heating efﬁciency is controlled by the heat transfer coefﬁcient at composite surface
and the thermal conductivity of the composite. During processing, the heat front
moves from the surface to the interior of the composite. For thermoset polymers, the
heat initiates the exothermic polymerization reaction. The thermoset undergoes
liquefaction, gelation and vitriﬁcation as the reaction proceeds. The direction of the
heat front represents, to a certain degree, the direction of the vitriﬁcation front. Once
the thermoset matrix reaches vitriﬁcation, the extra resin and trapped gases in the
interior will remain inside the composite. This processing drawback will reduce the
mechanical properties of the processed composites. Also as the polymerization of
thermoset is a strong exothermic reaction, the exotherm not only accelerates the

movement of the heat front, but also creates a large temperature excursion in the

1

2

interior of the composite. The temperature excursion in the interior may cause
thermal degradation of the matrix. To reduce the temperature excursion and remove
the extra resin, slow heating rates and long cure cycles with several relatively low
soak temperatures are required. The temperature excursion can be eliminated if the
heat source can be readily removed when the thermal excursion occurs, which is
impossible during thermal processing.

To overcome the disadvantages of thermal heating, microwave heating has
been studied as an alternative in the processing of polymer composites 3'”. Table
1.1 shows one-to-one comparison between microwave and thermal heating
characteristics.

Table 1.1 Comparison of Microwave and Thermal Heating Characteristics

     

Microwave Heating

 

Thermal Heating
Heating Rate SLOW, Controlled by FAST, Direct coupling of

   

 

    
    
 

 

        
 
 

heat transfer energy into molecules
‘ Selectivity NO, Heating is due to YES, Heating is proportional
the temperature gradient to material loss factor and 1
input power

  

 

, Heat Movement OUT TO IN, Surface IN TO OUT, Volumetric
driven heating ﬁom hot heating with boundary heat
to cold loss

      
     
     

    

 

   

YES, Microwave energy can 3

‘ Controllability NO, Heat source can’t be
‘ be readily removed

readily controlled.

 

 

    
   

The advantages of microwave heating over thermal heating in Table 1.1 can be
easily explained by microwave power absorption in the material. The power

absorption rate, P, (in W/m’) inside a homogeneous material is 7

3
= -;-e0e”to 1212 (H)

where E is the electric ﬁeld strength inside the material, V/m.
co is the frequency, rad/sec. w=2tf, f is the frequency Hz.
Go is the permittivity of free space, eo=l/(36r) x 10" F/m.
e’ is the effective relative loss factor, e'=e.’ + o/(eow),
e" is relative loss factor due to dipolar contribution,
a is the material conductivity, S/m.

Clearly, e" is responsible for selective heating. 3 is responsible for fast

heating and controllability. Because the electric ﬁeld can penetrate into the material
as far as its skin depth, the microwave heating is volumetric. The volumetric heating
and high controllability of microwave heating not only eliminate the temperature
excursion during processing but also allow higher curing temperatures. Figure 1.1
shows the comparison of temperature proﬁles during microwave and thermal curing
of pure epoxy resin, diglycidyl ether of bisphenol A (DGEBA)/ diaminodiphenyl
sulfone (DDS). The thermal curve was obtained when the epoxy sample was placed
in a thermal oven at 200°C. The two microwave curves were obtained using
continuous power input to heat the resin to the control temperature, 190°C or 230°C,
then using pulsed power input to maintain the resin temperature at the control
temperature. As shown in the Figure 1.1, the initial heating rate is much faster in the
microwave cure than in the thermal cure due to the different heating mechanism. No
temperature excursions were observed during microwave cure while a temperature
excursion of 40°C was observed during thermal cure. Because the temperature
excursion can be eliminated during microwave cure, higher cure temperatures can be
used to process thermoset composites. If the thermal degradation temperature is
240°C as shown in Figure 1.1, the maximum cure temperature that can be used in
microwave cure is 230°C while that in thermal cure is only 200°C.

4

The volumetric and inside-out characteristics of microwave heating cause an
inside out movement of the heat and vitriﬁcation fronts. This allows both excess
resin and trapped gases to be fully removed from the interior. A highly consolidated
composite with better mechanical properties can be expected from microwave
processing. The selective characteristic of microwave heating results in a higher
temperature in the graphite ﬁber than the surrounding matrix. This unique
temperature proﬁle improves the interfacial bonding between graphite ﬁber and epoxy
matrix ’-‘.

 

 

 

300
260
0's. .

I? 220 . ' 230°C microwave
V ' zoo-cw
“s, 180 l .0 IN'Ctaktmve
*5 -'
53 140 , '
a. : O
E .
,2 100 :

so .

20. . 1 . . r . . . . .

o 10 2'0 3'0 4'0 5'0 5'0 7'0 8'6 so

Time (minutes)

Figure 1.1 Temperature proﬁles during microwave and thermal cure

of pure epoxy resins

The above comparison shows that microwave heating is superior to thermal
heating in processing of thermoset and thermoset based composites. A technique
based on microwave heating is highly desirable for composite processing in order to

5
obtain composites of superior properties, to process thick-section composites, and to

shorten processing cycles.

1.2 Research Topics

The main objective of this research is to develop a low-cost, high-speed
composite processing technique using microwave energy. In order to understand the
full potential of microwave curing, the interaction between microwave radiation and
material is studied and different processing techniques are developed. Speciﬁcally,
the following sub-objectives are used to support the core objective:

1. To investigate the microwave radiation effects on epoxy curing rates and the glass

transition temperatures of the cured epoxy samples.

2. To demonstrate the possibility of using microwave energy to batch process graphite
ﬁber/epoxy composites using 2.45GHz microwave radiation.

3. To study the scalability of the batch microwave process technique by comparing
the heating in a 17.78 cm cavity and a 45.72 cm cavity.

'4. To demonstrate the possibility of continuous processing of continuous graphite
ﬁber reinforced epoxy composites using microwave energy.

5. To develop a microwave power absorption model for prediction and controlling of

the energy coupling during microwave processing.

6. To develop a microwave and thermal process model for precess simulation and
control.

CHAPTER 2
BACKGROUND FOR MICROWAVE PROCESSING

2.1 Introduction

The name "Microwave" generally refers to electromagnetic waves in the
frequency range of 300MHz to 3OOGHz or the characteristic wavelength range of 1.0
m to 1 mm. Although the microwave generator was ﬁrst invented by Kassner in
1937”, substantial industrial use did not start until after World War 11“. Heating
is one of the major non-communication applications of microwave power. However,
heating is conﬁned to ofﬁcially assigned ISM frequencies, such as 915MHz and
2.45GHz. To fully illustrate the microwave heating mechanism and its advantages,

the basic principles of microwave heating are reviewed in this chapter.

2.2 Interactions Between Electromagnetic Radiation and Materials

Most polymers and composites are non-magnetic. For non-magnetic materials,
the electromagnetic (EM) energy loss in a given material is only dependent upon the
electric ﬁeld. All non-magnetic materials are cataloged as dielectrics, conductors, and
composite materials consisting of dielectrics and conductors. For conductors, the
conductivity determines the interaction between EM radiation and the material. For a
perfect conductor, the conductivity is inﬁnite and all incident EM radiation will be
perfectly reﬂected. For a conductor of ﬁnite conductivity, the incident EM radiation
can penetrate into the conductor as deep as its skin depth, 2,. The skin depth is
deﬁned as the distance from the surface at which the ﬁeld strength falls to Us of its
strength at the surface. The skin depth in a conductor can be calculated as

z = 2 (2-1)
’ (0900

where o) is the frequency of the EM waves in rad/sec, u, is the permeability of the
free space, 41x10" H/m, and a is the conductivity of the conductor in mhos/meter.
The ﬁeld strength beyond the skin depth is usually negligible. For graphite ﬁber,
a=7x10‘ mhos/meter, and Z, is 38.4 um at 2.4SGHz. The skin depth for AS4
graphite ﬁber at 2.45GHz is therefore about four times the ﬁber diameter. The total
EM power loss in a unit conductor element of inﬁnite length can be calculated by:

 

p = l w "'0 H (2'2)
2 2 0 °
where H, is the magnetic ﬁeld strength at the surface in Amperes/ meter.

 

For dielectrics, the interaction between EM radiation and the dielectrics are
determined by the dielectric properties of the materials, e'=e'-je", at corresponding
EM frequency. 5’ is the dielectric constant representing electrical polarizability which
reflects the ability of a material to store electric energy, and e" is the dielectric loss
factor, representing the molecular relaxation phenomenon which reﬂects the ability of
a material to dissipate electrical energy. The skin depth for dielectric materials is

20 = 2
II I"a (2-3)
7

For fully cured DGEBA/DDS epoxy, e°=eo (3.5-j 0.1), the skin depth 20 =0.729m at
2.45GHz. The microwave power loss in a dielectric can be calculated using

(DC

Equation (1-1). For homogeneous materials in general, the skin depth is calculated
by‘1
1 2 ‘ e” 2 ‘ 'i
2.=—1——1’1(1+(—) )7-11 (24)
2w pou’eos’ 8’

The skin depth in an anisotropic composite can be calculated in its principal directions
using Equation (24). For AS4/3501-6, the effective complex permittivity along the

8
ﬁber direction is (=5, (1-j2500), and perpendicular to ﬁber direction is e'=eo (14.5-
j75.8) '. Assuming the ﬁber direction is the principal direction, the skin depth for
this type of composite is 9.8mm and 3.2m for electric ﬁelds along and perpendicular
to the ﬁber direction, respectively. The power absorption rate inside the composite
can be calculated by Poynting’s theorem‘2

(2-5)

 

where a; and 53’ are relative effective dyadic loss factor and relative dyadic loss

factor
due to dipolar distribution,

a is the dyadic conductivity, S/m, and

E‘ is the conjugate electric ﬁeld vector, V/ m.

When a conductor is placed in an alternating EM ﬁeld, the free electrons will
move against the applied ﬁeld at the alternating frequency. When a dielectric is
placed in an alternating EM ﬁeld, the dielectric material will undergo four different
kinds of polarization.

The four polarizations are electron or optical, atomic, orientational or dipolar, and

interfacial or space-charge orientations.
2.2.1 Electron or Optical Polarization
This type of polarization is due to the shift of the nuclei center of electron

orbits caused by the applied electric ﬁeld, as shown in Figure 2.1 (a). When no
external ﬁeld is present, the center of positive charges, which are concentrated in the

9

nucleus, coincides with the center of negative charges, which are circulating around

the nucleus. When an external ﬁeld is applied, the electrons are pushed away from
their original path and a dipole moment, a , is induced opposite to the applied electric
ﬁeld. The induced dipole moment is proportional to the ﬁeld strength, E, but

independent of the frequency. There is no phase displacement with regard to the
electric ﬁeld and therefore no dielectric losses will occur. The typical relaxation time
for electron polarization rs 10‘"sec .This type of polarization occurs in all
materials and is responsible for refraction of visible light. The breakdown of this
polarization occurs in the ultra-violet and visible range of EM spectrum.

No field Field _ooohed
Electronic mutation

e 0

Atom: polarization

9” M We
Orientation admiration
(c) o— \ o— .-
‘t ‘1
_ Space atone contraction
@6999 @9696
cases @6999
(d) @9999 ©9666
@9969 GOOSE-D

Figure 2.1 Four Types of Polarization Mechanisms

10
2.2.2 Atomic Polarization

This type of polarization is observed when molecules consisting of two
different kinds of atoms are placed in the electric ﬁeld, as shown in Figure 2.1 (b).
When there is no external electric ﬁeld, the molecule is neutral as a whole either
because the center of the positive and negative charges coincides, as in a non-ionic
molecule, or because individuals ions are not freely mobile due to a strong internal
electric ﬁeld (or dipole moment), as in an ionic molecule. When an external electric
ﬁeld is applied, the positive charges move in the direction of the ﬁeld while the
negative ones move in the opposite direction. The displacement from their rigid
position of equilibrium is proportional to the external electric ﬁeld strength. This
type of polarization mainly causes the bending and twisting motion of the molecules.
For example, the elastically joined molecule, such as Na+ Cl', can be excited into
compulsory oscillation which behaves as resonators with very small attenuation under
an alternating electric ﬁeld. The magnitudes of atomic polarization of non-ionic, or
non-partially ionic polymers are much less than those of ionic or partially ionic
polymers. The typical relaxation time of this type of polarization is 1013 sec. The
breakdown of atomic polarization occurs in the infra-red region of the EM spectrum.

2.2.3 Orientation or Dipole Polarization

This type of polarization only occurs when dipolar or polar molecules are
placed in the electric ﬁeld, as shown in Figure 2.1 (c). In dipolar molecules, the
centers of gravity of the positive and negative charges do not coincide but are
separated by a small distance. The strength of this permanent polarity is measured as

a dipole moment, a . When there is no external electric ﬁeld applied, the material as

whole is neutral due to the random motion of molecules. When an external electric
ﬁeld is applied, the dipolar molecules will rotate until they are aligned in the direction

11

of the ﬁeld. Orientation polarization is fundamentally different from electronic and
atomic polarization. The latter is due to the displacing force exercised by the external
ﬁeld upon the electrons and atoms via the induced dipole moments, while the
orientation polarization is due to the torque action of the ﬁeld on the preexisting
permanent dipole moments of the molecules. The dielectric loss of orientation
polarization is mainly due to the frictional resistance of the medium during dipole
rotation. The typical relaxation time of this type of polarization is 10” sec. The
breakdown of this polarization occurs in the microwave range of the EM spectrum.
This type of polarization has most interesting to us because it occurs in the
microwave range. The calculation of dielectric properties for polar materials will be
presented in further detail in section 2.3.

2.2.4 lnterfacial or Space-charge Polarization

The previously discussed polarizations; electronic, atomic, and orientation are
caused by the displacement or orientation of bound charge carriers. The atoms and
molecules are affected by the local ﬁeld and modulated by the polarization of the
surroundings. lnterfacial polarization is caused by the migration of charges inside
and at the interface of dielectrics under a large scale ﬁeld. It occurs at radio

frequency range.

2 . 3 Dielectric Properties of Polar Materials

The dielectric properties of a material are characterized by a dielectric
constant, e', and a dielectric loss factor, 6". e' is deﬁned as the ratio between the
capacitance of a condenser ﬁlled with dielectrics and the capacitance of the same
condenser when empty (Faraday called 6' speciﬁc inductive capacity). The higher
the polarizability of a molecule, the higher its dielectric constant.

12

Figure 2.2 shows the frequency dependence of e’, and 5". At low frequencies,
alltypesofpolarizationcanreach the steadystatevaluesasthey wouldinastatic
ﬁeld. Therefore polarizability or e' is high. As frequency increases, some types of
polarization may no longer have time to reach their static ﬁeld values and the
polarizability, that is e', is decreased. The polarization fails at its natural frequency
to, which is the reciprocal of the polarimtion relaxation time. This is related to the
occurrence of electric energy dissipation due to dielectric relaxation at the frequency,
to... At the breakdown of a polarization, 5', e" is usually increased dramatically.

 

   

 

 

 

 

 

 

i
‘ tonic
LEE Orietaationet
I: """"
°'.'-. Molecnhr
‘: --------------
355 Electronic
8: ---- - --- ----. ......
L5
O
a . 2 . J . . . . .
t 102 104 106 tel row roll to“ 1016 10"
m mm W -—"'¢sm7 (‘1‘)
"“° ma. «11mm
_1_ 1 r r 1 1 L l 1
10' to6 10‘ 101 1 111°2 10" 10“ 10" movement-l
.3".
98.
‘5
:2
2:
:5 z i i ' ‘ 'r ' " 4
1 to 10‘ 10‘ 1:1‘1 10‘0 10‘2 to“ to" 10W

—-treeusae7 (ell)

Figure 2.2 Frequency Dependance of e' and e”

13
The dielectric properties of the material is very important in understanding
how the microwave energy is coupled into the dielectric materials. For polar
molecules in a liquid or solid, Debye derived formulae for the dielectric constant e'
and the dielectric loss factor 5"“.

 

 

 

e’=e “+£1wa
0 2 2
1+0) 1, (2-6)
ll (Ki-watc
c =e
0 2 2
1+1.) 1:,
with
K.’+2
r‘=r
K92
1,=4rtr311
KT

where u=1.380x10'23 J/°K, Boltzmann’s constant,
K,’ and K.’ are the relative static and optical permittivities,
1 is the dipole relaxation time, in sec.,
r is the radius of the assumed spherical molecules, in m,
11 is the viscosity of the medium, in poise OM”), and

T is the temperature, in °K.

The dielectric conductivity 0,, and dielectric loss tané of the material are

 

agree”
8” (Ki-KIA) ‘9 to (2-7)
18110 =-; =

I 2
e K1+K,o2z,

Other equations for the determination of relaxation time of polymers are the
Arrhenius relation and the WLF relation. The Arrhenius relation is for relaxation
dominated by local molecular transitions while the WLF relation is for relaxation

14
dominated by structural transitions. Their expressions are

Arrhenius relation: 1 =10, 1"

-4051?
WLF relation: t “.9 ”-7.

where E is the activation energy in call mole, Tg is the glass transition temperature in
°K, T is polymer temperature in °K, R is the gas constant, R=1.987 callmole/°K,
and A,B are constants.

2.4 Microwave Heating Applications

The usage of microwave energy can be classiﬁed into communication and non-
communication applications. The major non-communication applications of
microwave energy are in medicine and heating. Microwave heating has been used in

food processing, drying, material processing, waste treatment, and organic synthesis.

In the forest industry, microwave energy has been used to process large
bundles of wood and plywood veneer". In the food industry, microwave energy has
been used to dry potato chips since the 1960’s“. Other materials studied in
microwave drying included acid hydrolysis of corn stover”, 13X zeolites“, and
polymers". All of these studies show that microwave drying is signiﬁcantly faster
than conventional thermal drying. However, extra care is required during microwave
drying. For example, the bound water inside nylons is very difﬁcult to remove and
may cause thermal runaway in the material if the microwave drying is not well
monitored".

The most common applications of microwave processing are food processing,

polymer and composite processing, coal processing, and ceramics processing. In food

15

processing, bacon cooking and meat tempering accounted for 79% of installed
microwave heating capacity as of 1984”. Other applications in food processing are
seed treatment for insect control“ and soybean treatment for quality

improvement”. In the processing of polymers and composites, microwave energy
enhanced the polymerization rate of epoxy‘°""“"°-”'”’3, increased glass transition
temperature (1;) of cured epoxy‘°-‘°, improved the interfacial bonding between
graphite ﬁber and matrix“, and increased the mechanical properties of the
composites”. However, no increase in reaction rates in microwave curing of epoxy
were also reported as compared to those of thermal curing""’. In the processing of
coal, microwave energy was able to produce a greater number of low molecular
weight hydrocarbons than is possible in thermal processing“. Microwave energy was
able to change non-magnetic pyritic sulfur into magnetic material. Microwave energy
was also used to process ceramics and a much faster heating rate and better properties
of sintered ceramics were observed”. Overviews of microwave heating in the

processing of polymers“ and ceramics” have recently been published.

Microwave heating was effectively applied to the processing of waste
materials“"7-”. A wide range of waste materials has been studied, including
infectious medical waste, solvent-laden waste, sludge, used rubber tires, contaminated
soils, petroleum products, coal, activated carbon, gases, plastics/elastomers, nuclear
waste, and sewage/waste water”. A faster processing speed and lower processing
cost were reported. Microwave heating was also used as an alternative to thermal
heating in organic synthesis°°'°"°2'°3. Again, a signiﬁcant reduction in the time
necessary to complete the reaction under microwave conditions was observed,

especially when lossy solvents were involved.

16

2.5 Microwave Heating Systems

Three kinds of microwave applicators, waveguides, commercial multimode
microwave ovens, and tunable resonant cavities are commonly used in the microwave
processing of materials. All the three ldnds of applicator have been studied in
microwave heating of polymers and composites. However, only the resonant cavity
can easily focus the microwave power into materials and was successfully used in
processing of crossply and thick-section graphite ﬁber reinforced composites’”.
Cylindrical tunable resonant cavities were used in the present study. The
characteristics of resonant cavities, microwave circuit systems, and electric ﬁeld

patterns inside a cavity are discussed in this section.

2.5.1 Cylindrical Tunable Resonant Cavity

The cylindrical cavity in this study is made out of a length of metal circular
waveguide with both ends shorted by the same metal, such as copper or aluminum, as
shown in Figure 2.3. The top short-plate is adjustable so that the cavity length L, can
be changed. The bottom plate is removable for sample loading. The microwave
energy is introduced into the cavity by a coaxial coupling probe. The coaxial
coupling probe is adjustable in the radial direction so that the coupling probe depth L,
can be changed for locating critical coupling conditions.

17

 

 

 

 

 

 

 

 

<-——>
/

\Top plote

' LP Coupling probe

 

 

  

 

a--—----—-——---——---—- r- --———---———-—>
0

 

Bottom plote

 

Figure 2.3 Diagram of Cylindrical Tunable Resonant Cavity

In order to conﬁne microwave energy inside the cavity, the ratio of cavity
length/cavity radius must be such that the electric ﬁeld patterns inside the cavity reach
a resonant condition. The resonant electric ﬁeld inside an empty cavity can be
theoretically calculated according to Maxwell’s equations and the electric and
magnetic boundary conditions at the cavity wall. The boundary conditions at the

cavity wall are that the electric ﬁeld, E , has only a normal component while the

magnetic ﬁeld, ['1' , has only a tangential component. There are two types of resonant

conditions or modes. Transverse electric (TE) and transverse magnetic (TM). For
TE modes, the electric ﬁeld is aligned perpendicular to the axial direction. For TM
modes, the electric ﬁeld is aligned along the axial direction. Three subscripts, n,p,q,

18

are used as mode nomenclature to represent the physical appearance of the
corresponding mode in an empty cavity, i.e. TE," and TM“. n denotes the number
of the periodicity in the circumferential direction, n=0,l,2, ..... p denotes the
number of ﬁeld zeroes in the radial direction, p=l,2,3,.... q denotes the number of
half wavelengths of the equivalent circular waveguide, q=0, 1,2,... for TM modes and
q=1,2,3,... for TB modes. Figure 2.4 shows the physical appearance of the EM
ﬁeld of the TM,” mode. The dashed line represents the magnetic ﬁeld while the
solid line represents the electric ﬁeld. In the diagram, the higher the density of the
lines, the stronger the local ﬁeld is. This mode is very efﬁcient for heating rod
shaped materials when the rod is placed at the center of the cavity.

The relationship between the cavity diameter/length ratio and the product of
cavity diameter and frequency for a given resonant mode can be calculated
theoretically and presented as a mode chart. The theoretical equations for TM and

TE modes in an empty circular cylindrical cavity are‘2

 

 

x 2 2 I 2
.__ v ‘1 11 (2-8)
”01“.. “zeopo + “op“: Le)
x, 2 2 d 2
«(1)21: = "P + q (_) (2-9)

“ “Zappa 480 “0 L:

where f is the frequency, Lc is the cavity length for the resonant mode, d is the
diameter of the cavity, x", and x,’ are the ordered zeros of the Bessel function J,(x)
and its derivative J,’(x’) with the ﬁrst subscript referring to the order of the Bessel
function and the second to the order of the zero, and u, is the permeability of free
space. Figure 2.5 shows the mode chart. For a given cavity, the cavity length of a
certain mode is strongly dependent on the frequency, except for those TM modes with
q=0.

   
  

cavity

loaded sample

 

 

 

 

Figure 2.4 Theoretical EM Field of TM,” in a Cylindrical Empty Cavity

In this study, three sizes of cylindrical tunable resonant cavities are used.
Their inner diameters are 15.24, 17.78, 45.72 cm. Both 15.24 and 17.78 cavities
are made from brass and the 45.72 cavity is made from aluminum. Figures 2.6, 2.7,
and 2.8 show the cavity length versus frequency mode chart for the empty 15 .24,
17.78, 45 .72 cm cavities, respectively. The 15.24 cm cavity was used for polymer
kinetics study and the 17.78 cm and 45.72 cm cavities were used for the processing
study. A thin ﬁlm sample was used in the 15.24 cm cavity during the kinetics study

20

and the electromagnetic ﬁeld inside the cavity is very close to that of the empty
cavity. During the processing studies in the 17.78 cm and 45.72 cm cavities, the
sample size was relatively large and the electromagnetic ﬁelds were greatly perturbed,
especially when the loaded material is graphite ﬁber composites. In these cases, the
location of the coupling probe is also important. For 17 .78 cm cavity, the coupling
probe was located 5.461 cm above the bottom plate. To measure the radial electric
ﬁeld strength along the axial and circumferential direction, holes 0.216 cm in
diameter were drilled in the cavity wall. To prevent microwave leakage from the
holes, a 1.27 cm thick copper strip was used to surround the holes. In the axial
direction, the holes were drilled in the cavity wall with 90° respect to the coupling
probe. The ﬁrst hole was drilled 0.605 cm above the bottom plate and the rest of
the holes were drilled 0.922 cm apart from each other. In the circumferential
direction, three rows of holes were drilled at the heights of the third, fourth, and ﬁfth
holes of the axial hole line because the composites were usually loaded at this height,
2.5 to 4.3 cm above the bottom plate. Using the axial hole line as reference, the
holes were drilled every 10° in the circumferential direction. For the 45 .72 cm
cavity, the coupling probe was located 10 cm above the bottom plate. Only an axial
hole line was drilled with ﬁrst hole at 3.13 cm and the rest of the holes were 0.922
cm apart from each other.

f, 6111 (resonant frequency)

12

21

11221112

 

it, «11’

 

 

 

 

(d/L) 2
Figure 2.5 Mode Chart for Circular Cylindrical Cavities

 

28-‘

1

L4

 

2.4-4

2.0d

L6

 

..,I...I,,,I"'
“ﬂ,
‘3',
TMnr
u:
‘I'

\

 

 

 

I 7 r v I , v ' I v
12 16 20

L, cm (cavity length)

m—

Figure 2.6 Mode Chart for 15.24 cm Cylindrical Cavity

24

f, 6112 (resonant frequency)

3.2

22

 

 

 

 

 

 

0d
d
N
d
at
N
O

24
Lc cm (cavity length)
Figure 2.7 Mode Chart for 17.78 cm Cylindrical Cavity

fo MHz (resonant frequency)

23

 

 

 

 

/F
1000 1 a a
.. u

no " T

960 1

no

920 i

900 4

no

no .

no l

are

soo .

no

no , , f . A]

to 20 30 so so so 70 so

L, cm (cavity length)

Figure 2.8 Mode Chart for 45 .72 cm Cylindrical Cavity

2.5 .2 Q Factor of the Resonant Cavity

The Q factor of the resonant cavity is deﬁned as ratio of the energy stored

inside the cavity to the energy lost at the cavity wall per unit u'me.

24

(2-10)

 

2
energy stored _ CLIEI dt
Q ‘Zﬂf, - - rt ,—
energy lost Rf ”1'24,
8

where R is the intrinsic wave resistance of the metal walls.

The Q factor of the resonant cavity in the microwave frequency range is
usually very high, 10‘ or more“, and is a function of the resonant modes. The Q
factor of a single mode cylindrical cavity of ﬁnite conducting walls can be derived
theoretically using the cavity-wall perturbation method".

 

 

 

 

W ._. _32__ 2-11
(0.)... 212mm,) ( >
2 qna 2
a. "\ TIT] ‘2'”)
(Q)m= 2R ‘
(1+ach)
2 3’2
na
,, 1m 1)
(0,1,, = ‘ (2-13)

 

2 2
is?) m—(—] W}

e c e
where n is the intrinsic impedance of the dielectric, R is the intrinsic wave resistance
of the metal wall, a is the cavity radius. However, the theoretical Q values are
higher than the Q factors of the actual cavity because the actual cavity has the coaxial
coupling probe at the side wall and the ﬁnger stock at the top and bottom plates. The
Q factor of the actual cavity is usually measured experimentally from the power
absorption curve, as shown in Figure 2.9. The procedure is ﬁrst to locate the desired
resonant mode at frequency f, by adjusting the coupling probe depth, L,, and the
cavity length, L,, using a swept frequency microwave source, then measuring the
bandwidth at the half power points (if. The Q factor for a cavity in a single resonant
mode can be determined as

25

Q = 2Q (2-14)
5f
Equation (2-14) can not be used to determine the Q factor for a cavity resonated in

multimode conditions.

Power

 

 

 

. Frequency
Figure 2.9 Q-factor Determination From Power Absorption Curve

2.5.3 Electromagnetic Field Inside the Resonated Cavity
In a homogeneous, source-free cylindrical cavity with perfectly conducting

walls, the EM ﬁeld inside the cavity can be derived from Maxwell’s equations and the
boundary conditions". For TM modes:

2

E =16 I H “1&1

' 9 61:3: " p at»

2
£3.11" "' ”Fine (2-15)
99 6462 69

E =l(_a_2. +E)¢ H :0

’ 9 a:2 ’" ‘

where k2=w2uo£ and ll," is the wave potential for TM," modes.

x p sinnd
= _'L 91 (2-16)
t... J..( a ll Iboetdz)
ForTEmodes:
2
H .1119: 5 ”lg-z:
’ 2 6sz ' 9 do
H Jim 5 =11 (2-17)
’ to are: t 69
514$ +13) E .0
l f 622 '1'] l
where it... is the wave potential for TE." modes.
x’ p sinndl
= ._‘L ° .41 (2-18)
11,, J.( a )1 llsm( a"

For the loaded cavity, Equation (2-15) to (2-18) are no longer applicable.
Cavity perturbation theory is usually used to calculate the EM ﬁeld in a cavity loaded
with a small object which only perturbs the resonant frequency by a few percent. The
object can be dielectric, magnetizable, or conducting. Using the material-cavity
perturbation method, a cylindrical cavity resonated in the TM012 mode has been used
for measurement of material dielectric properties". For a cavity coaxially loaded with
homogeneous lossy materials, a mode-matching method was used to calculate the EM
ﬁeld inside the cavity“. However, for a cavity loaded with an anisotropic composite
plate, especially graphite ﬁber reinforced composite, the EM ﬁeld inside the cavity is
very complicated. A ﬁve parameter formula for calculating the electric ﬁeld inside
the loaded composite is presented in Chapter 8.

2.5 .4 Microwave Processing System

Two microwave systems, one operating at 2.45GHz with 15.24cm and
17.78cm inner diameter cavities and one operating at 915MHz with a 45.72cm inner

27

diameter cavity, were used in this study. Figure 2.9 shows the microwave circuit
setup for each of these systems. Both systems have four basic units; a microwave
source Circuit, a tunable circular cylindrical cavity, a data acquisition unit, and a
temperature and electric ﬁeld (E-ﬁeld) measurement unit. Either a Luxtron 755
multichannel ﬂuoroptic thermometer or an infrared thermometer was used to measure
the temperatures and an antenna probe was used to measure the E-ﬁeld strength. The
radial E-ﬁeld strength in the axial direction and the axial E-ﬁeld at the top of the
cavity can be measured. The input power, P,, reﬂected power, P,, and the detected
power at cavity wall, P,, are measured by power meters (HP4356). The data
acquisition unit is a data collection board in a personal computer driven by software
described elsewhere7. Each microwave source circuit consists of two parts: a swept
frequency, low power microwave source circuit and single frequency, high power

microwave source circuit.

In the 2.45GHz system, the swept frequency microwave source is a HP 83508
Sweep Oscillator with the HP 86235A RF Plug-In (frequency range 1.7 to 4.3 GHz).
A power absorption curve is observed on an oscilloscope (Tektronix 2213) over a
swept range of microwave frequencies from the power reflected back into the
microwave circuit from the cavity. Resonant modes exist at those locations on the
oscilloscope trace where there is zero reﬂected power. The dimensions of the cavity
are adjusted so that zero reﬂected power is observed at 2.45GHz. Once the resonance
condition at 2.4SGHz is found, single frequency microwave power from an Opthos
MPG-4M is introduced into the cavity for processing.

In the 915MHz system, the swept frequency microwaves are generated by a
HP 83508 Sweep Oscillator with HP 86220A RF Plug-In (frequency range 10 to
1300MHz). The power absorption curve is displayed on a Tektronix 465
Oscilloscope. High power is generated by supplying a continuous wave of 915MHz
from the sweep oscillator to a Logimetrics Model A610/P (frequency range .5 to 1.0

GHz) microwave power ampliﬁer.

 

 

 

 

 

 

  
    
 
  
  

 

 

  

28
915 MHz system

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D/A o.c.
Dover
m" Resonant
Cavlty
Microwave 000.91 rm
Prob.
Mill-tor
Circulator Clmiltﬂ' j
Mlcromv. Sam I.
Mpl lflor m“
i Olroetlor-l J' DIP-ﬂiﬂil
I metlleooope } Captor Pl
bunny Fb
Lood Load
A/ D
hormonal
Corrputor
Flwoptlc
135-2320 Thermometer
2.45 GHz system
0. C.
”we" supp ' y Comoostto
D/ A Pol
“ 01' W
L
Dlrectlonal
Clrculator Coupler Coaxial
SwitCh
._ -a-
l
Sweep
Microwave Oocl l later
5"" 1:3
_ l - _ _ _ - _ .. 'i
Dummy PI 1
Load Cool I loscope '
PC I IE
A/ D
FIS- 2320

 

 

 

Fl uoroot lc
Thermomter'

Figure 2.10 Microwave Processing Systems

29
A coaxial switch (micronetics RSN-2D-I/12V) is installed in the 2.45GHz

microwave source circuit to turn the microwave power on and off for maintaining
isothermal conditions after the sample. has reached the control temperature. The
switch is regulated by a dc power supply (Electronic Measurements HCR-30-8-11 l)
which is controlled by an I/O interface board (Omega DAS-l6) in a personal
computer. Actuation of the on/off switch is in response to temperature feedback
from a ﬂuoroptic temperature sensing system (Luxtron 755).

For an empty cavity, the power absorption curves are usually single peak
curves and the location of zero reﬂected power can be calculated theoretically. When
either a polymer or composite plate was loaded into the cavity, however, the power
absorption curve on the oscilloscope was not always a single peak curve. Especially
when the resonant standing electromagnetic ﬁelds interact with an anisotropic
composite, the electric ﬁeld strength distribution inside and outside the composite is
complicated. The resonant heating modes in a composite loaded cavity are not
identical to any theoretical TM or TE modes. A power absorption curve with
several resonant peaks (at different frequencies) can be generated by adjusting the
cavity length and the coupling probe depth. The position of resonant peaks changes
and the number of resonant peaks is increased or decreased as the cavity length
or/and coupling probe depth are changed. The cavity length and coupling probe
depth, however, play different roles in the presence of resonance condition. The
cavity length determines the possibility of resonance at a given frequency and the
coupling probe depth determines the realizability of the resonance. With improper
cavity length, the resonance condition can not be achieved no matter what the
coupling probe depth is. With proper cavity length, the resonance condition can be
obtained only at a certain coupling probe depth. For a given loaded cavity, the
presence of resonant peaks was a function of frequency, cavity length and the probe
depth. Each peak corresponded to a resonant heating mode of a particular electric
ﬁeld strength and pattern as well as the cavity length, probe length and the resonant
operating frequency. At the ﬁxed frequency, several resonant heating modes were

30

generated depending on the combination of cavity length and probe depth. Based on
the appearances and behaviors of the resonant peaks, three types of resonant heating
mode, pseudo-single (PS), controlled-hybrid (CH), and uncontrolled-hybrid (UH)
resonant modes, were observed on the oscilloscope as shown in Figure 2.11. The
pseudo-single mode looks and behaves as an empty cavity mode. The hybrid mode
curves look like curves of two or more peaks coalesced and each peak represents a
different resonant mode. If all peaks can be located at 2.45 GHz for processing, the
hybrid mode is a controlled-hybrid mode. If only one peak can be located at 2.45
GHz for processing, the hybrid mode is a uncontrolled-hybrid mode. When a hybrid
mode was used to process the composite, the resonance of the loaded cavity could be
maintained at a chosen peak (mode) by properly adjusting the cavity length and
coupling probe depth constantly during the processing.

 

 

 

 

 

 

 

 

 

100 100

g s

E 5

.. r:

s e

E 50 — 9, 50

< ‘3:

he I-

i’ i’

e — o

O- D-

0 l o l
2.45 2.45

Frequency (GI-12) Frequency (6112)
(a) hybrid mode (b) pseudo-single mode

Figure 2.11 Deﬁnition of Pseudo-single, Controlled-hybrid, and
Uncontrolled-hybrid Modes

CHAPTER 3
MICROWAVE CURE OF EPOXY

3.1 Introduction

The general advantages of microwave heating over thermal heating were
discussed in Chapter 1. Chapter 2 summarized the interactions between microwave
radiation and materials, microwave processing systems, and other fundamental
principles relating to microwave heating. This chapter discuses how microwave

radiation affects the curing of one of the most widely used thermoset, epoxy resin.

Microwave processing of polymers and polymer composites has been studied
as an alternative to conventional thermal processing. In general, the unique
microwave heating mechanism3'7 allows selective, global, faster and more
controllable heating than is possible with thermal heating. To determine whether
microwave energy is superior to thermal energy in the processing of thermoset and
composites, some fundamental questions must be studied. They include: 1) whether
microwave radiation enhances the polymerization reaction rates 2) does the
microwave radiation induced reaction follow the same reaction path 3) whether these
two types of processing will result in the same polymer network structure 4) do
microwave processed polymers have different physical properties (such as density),
mechanical properties (such as strength and modulus), and thermal properties (such as
thermal expansion coefﬁcient and glass transition temperature) compared to those of
thermally processed samples.

In order to understand the effect of microwave radiation on polymer reaction
kinetics, epoxy resins have been microwave cured using single-mode
cavities‘°"°'2°'22'25, commercial multimode ovensm‘, and waveguides“”"”'”.

However, literature reports disagree on whether microwave heating accelerates or

31

32

decelerates the reaction rate compared to thermal heating. This discrepancy may be
due to effects of the temperature gradient within the material during microwave cure,
caused by the low thermal conductivity, heat transfer area of curing epoxy resin, and

non uniformity of power absorption inside the sample.

To elucidate the microwave radiation effects on the reaction kinetics, efforts
have been made to develop a novel thin ﬁlm technique to ensure spatial uniformity of
the temperature and the extent of cure“25 inside the samples. A parallel isothermal
microwave and thermal cure of DGEBA/mPDA epoxy resin has been done using this
novel technique, and only a slight increase in reaction rate was observed in the case
of microwave cure”. This increase may be due to a change in the microwave
radiation excited reaction kinetics and/or localized 'hot spots” on a molecular level
by rotational excitation resulting from the dipolar response to the electromagnetic
ﬁelds”. A similar study was carried out using DGEBA/DDS epoxy resin“, and
signiﬁcant increase in reaction rate was observed in microwave cure. An increased
glass transition temperature (Tg) was found in the microwave cured samples as
compared to the thermally cured samples at a high extent of cure. Similar Tg values
were obtained for microwave and thermal cured samples at a low extent of cure“.
However, Mijovic et. all“:19 used DSC to determine both extent of cure and Tg, and
decreased reaction rates and Tg in the microwave cure of DGEBA/DDS were
observed as compared to those of thermal cure.

This chapter reports the results of a study of the reaction kinetics and the glass
transition temperature of microwave and thermally cured epoxy resins. Isothermal
cure of DGEBA/DDS and DGEBA/mPDA epoxy resins was done at three different
temperatures in a single mode resonant cavity and a thermal oven. A cavity
resonated in the TEm mode was used in microwave cure. Thin ﬁlm samples were
used in this study and FTIR was used to measure the extent of cure. TMA was used
to determine the glass transition temperatures of the cured thermoset.

33
3.2 Thermal Cure of Epoxy Resins

In modelling Tg and the reaction kinetics of microwave cured epoxy resins,
models based on the thermal cure kinetics can be used and parallel parameters
calculated and compared. To elucidate the critical parameters in the analysis of the
microwave cure of epoxy resins, a general literature review of the Tg and reaction
kinetics of the thermal cure of epoxy resins is necessary. A complete review should
address cure kinetics, reaction mechanism, reaction by-products, crosslinked network
structure, glass transition temperature, mechanical properties and other physical
properties of the cured epoxy resin, effects of the stoichiometry, diluents, ﬁllers, and
cure conditions on the reaction kinetics and the thermal properties of the curing epoxy
resins. While general reviews are available elsewhere°7'°"°°, a review of the
reaction kinetics model and glass transition temperature of cured epoxy is provided

here.

3.2.1. Cure Kinetics

Two different types of models for thermal cure kinetics have been proposed, an
11" order reaction kinetics model”71 and an autocatalytic reaction

model‘n.73,74.75

3.2.1.1 The 11'“ order kinetics model

For the n‘ll order assumption, the reaction kinetics can be expressed as

dog -
C—E k(T)f(a) (3 1)

where a is the extent of cure, t is the time, the function f(a) is expressed as (l-a)“,

34

and k(T) is the overall reaction rate constant and obeys an Arrhenius relation.
k < n =2: exp < - E/T—‘h

where A is the pre-exponential constant (min'l), E is the overall activation energy
(cal/mole), R is the gas constant (1.987 cal/mole/°K), and T is cure temperature
(°K). Substituting the Arrhenius relation into Equation (3-1) and rearranging
produces

ln(j;I f_(—1a —da)=1nk(T) +1n(t:) (3'2)

The left hand side is only a function of a, or equivalently, only a function of the glass
transition temperature, Tg, as there is a unique relationship between a and Tg.
Equation (3-2) can be rewritten as i

E/ R

f(Tg) =-—— +1n(A) +1n(t) (3-3)

The left hand side of Equation (3-3) has the same value for different cure
temperatures at a given a or Tg. Therefore, the activation energy can be obtained by

relating the cure time to cure temperature at the same a or Tg.

1n(

 

 

t -g -
)-R( (34)

ref

(3

ER is the slope of the ln(t/t,,,) versus 1/T line. Equation (3—4) also suggests that the
a or Tg versus ln(t) curve at various cure temperatures can be shifted to a single
curve at a reference temperature T", using the shifting factor SF=(E/R)(l/T-1/T,,,).
This single curve is generally referred to as the master curve". Once E is
determined, the value of A and n can be obtained from Equation (3-1) using
experimental data at the reference temperature.

A and n can be determined from the intercept and slope of the ln(da/dt)+(ElR)/T

versus ln(l-a) line.

35

do 5712' _ _
ln(Ti—E)+ T ln(A)+nln(1 a) (3 5)

 

3.2.1.2 Autocatalytic Kinetics Model

The 11" order reaction kinetics has the advantage of simplicity. However, the
maximum reaction rate should occur at the beginning of the reaction according to the
n"I order assumption. In reality, the maximum rate usually happens at an extent of
cure, a = 0.3 ~ 0.4. This phenomenon has been explained by an autocatalyzed
reaction mechanism and the phenomenological cure kinetics expression for a
stoichiometric epoxy-resin has been given asm‘”

$3. = (k,+k,a") (1-ct)n (3-6)
where
k2 is the autocatalytic polymerization reaction rate constant,
k2 = A2 exp(-E2 / RT)

k1 is the non-catalytic polymerization reaction rate constant,
k, = A, exp(-EI / RT )

m is the autocatalyzed polymerization reaction order, and

n is the non-catalyzed polymerization reaction order,

Based on the reaction mechanism for epoxy resin, a general cure kinetics

expression was derived as follows”

% = (k,+k,a) (1-a) (B—a) (3-7)
where B is the ratio of initial hardener equivalents to epoxide equivalents. For a

stoichiometric mixture, B = 1. The proposed reaction kinetic mechanism is"8

36

K1 (3’8)

al+e --0 a2+0H

e

a2+e -- «Bat-OH (3-9)

0H+e -~ et+0H (3-10)

where a,, a2, a,, e, and et are primary amine, secondary amine, tertiary arrrine,
epoxide, and ether group, respectively. Ki and K,’, i=1,2,3, are speciﬁc reaction
rate constants for the catalytic and noncatalytic reactions, respectively. The
experimental evidence shows that for a stoichiometric mixture of epoxide and amine,
the etheriﬁcation reaction can be ignored7"7’"°""”'”'“ at low cure
temperatures. Equation (3-7) for a stoichiometric mixture can be easily derived from
the proposed mechanism and the following assumptions.

1) There are no substitution effects, i.e. the reaction rate constant is the same

for primary and secondary amine.

2) No OH impurity, ie. initial OH concentration equals zero.

3) Extent of cure is deﬁned as a=(e,-e)/e,, where e, is the initial epoxide

concentration.

Most of the derivation is straightforward except the kinetics equation for a,.

da,
dt

 

=2a,e[K{+K,(0H)] (3-11)

The factor 2 is due to the fact that the neighboring amine hydrogen becomes a
secondary amine hydrogen as the primary amine hydrogen is reacted.

37

If k,/k2 in Equation (3-7) is independent of temperature, the approach to
generate the master curve used for rl‘h order reaction kinetics can be applied here.
The Tg data can be represented by a single master curve at a reference temperature
using the Tg versus ln(t) curve. The Tg versus ln(t) curves of all other temperatures
can be obtained simply by shifting the master curve horizontally using a shifting
factor SF= (l/T - l/’T,,,) EIR. As a has a single relationship with Tg, the (1 versus
ln(t) curve at the reference temperature can also be used as the master curve for
reaction rate. The reaction kinetics at all other temperatures can be obtained simply
by shifting the curve using the same SF used for shifting Tg master curves.

Equation (3-7) is no longer applicable if there is an etheriﬁcation reaction
and/or OH impurity, or the reaction rate constants for primary and secondary amine
are not the same. For epoxy/amine systems, the reaction rate constant ratio between
the secondary amine-epoxy and primary amine-epoxy, deﬁned as n=K2/K,=K2'/K,',
is not always equal to unity and is system dependent. For example, n for thermal
cure of diglycidyl ether of bisphenol A- (DGEBA)/ ethylenediarnine (EDA) is equal to
1 while it for thermal cure of diglycidyl ether of bisphenol A (DGEBA)/
diaminodiphenyl sulfone (DDS) is equal to 0.4”. Further more, for an
epoxy/aromatic system curing at high temperatures or having excess epoxy, the
etheriﬁcation reaction can no longer be neglected. The reaction rate constant ratio
between etheriﬁcation reaction and primary epoxy-amine reaction is deﬁned as
L=K,/K, =K3'IK1'. For a stoichiometric epoxy/amine system having an OH
impurity, a non-unity reaction rate constant ratio between secondary amine-epoxy and
primary amine—epoxy, and an etheriﬁcation reaction, the cure ln'netics can be derived
from the fundamental reaction steps, Equations (3-8) to (3-10).

38

 

get . ( 2 ”-anth +LF<¢>1 (1-«1 [k1+kaF(¢)l (3-12)

Full =1+10H1,/e,- [<1-n)¢+¢"’=1/(2-n) (3-13)

=___].’__ - - - DI: -
2(2_n) ((1 o) (1 n) (2 L) +2(1-¢ )(1 L/n) (344)

—(2-n)L(1+[0H],/ea)1n¢)

where [OH], is initial OH concentration, ¢ is a,/e,, k,=eoK,’, and k2=eo’K,.
Equation (3-7) for a stoichiometric system can be easily obtained from Equations (3-
12), (3-13), and (3-14) with [OH],=0, L=O, and n=1.

Equation (3-7) has been used to describe the reaction kinetics of Hercules
3501-6 epoxy resin. It holds very well for the early stages of cure, but it is not valid
after the gelation point. To model the whole curing reaction, an equivalent ﬁrst
order reaction has been suggested to model the reaction kinetics after the gelation
point“"7. The reaction kinetics are given by
$4 (k1+k2a) (1-a) (B-a) an,“ (3-15)
dt k3(1-a) out”,
where k, is the ﬁrst order reaction rate constant and a“, is the extent of cure at the
gelation point. Two expressions have been proposed for k,, they are

k, = A3 exp (-Fq / RT ) in one reference”, and

k3 = A, exp(-1.9a) exp (-E, / RT ) in the second reference“.

Clearly, the above model has a lack of continuity at the gelation point. To couple the
diffusion effect into the kinetics of the reaction, Havlicek and Dusek“ suggested the
Rabinowitch model"9 to calculate the overall reaction rate constant

39

1 1 1

_—k,(a,T) ”Tm" +—kd(a.T) (3'16)

where k,(a,T) is the overall reaction rate constant, k1-(T') is the ln'netics rate constant,
and k,(a,T) is the diffusion rate constant. k,(a,T) refers to the overall non-catalytic
polymerization reaction rate constant. There are two extremes during the reaction.
On one hand, k, > > kT for the reaction prior to vitriﬁcation, especially at a low
extent of cure. On the other hand, k7) >k, for the reaction near and after
vitriﬁcation, where the diffusion effect is signiﬁcant. The diffusion rate constant, k,,
is controlled by the diffusion of chain segments and is expected to be proportional to
the diffusion constant of the reactants. Equivalently, k, is inversely proportional to
the relaxation time of the polymer segments”. Based on the above assumption and
the modiﬁed Williams-landel-Ferry (WLF) equation for molecular relaxation
time"'”, the temperature dependence of k, is suggested to be"3

lnlka(T)]=1n(ka)+5—1% (3-17)
where k.» is the diffusion rate constant at T, and is assumed to be a constant and B is
a constant. Both km and B need to be determined from the experimental data. ‘
Equation (3-17) is. identical to the modiﬁed WLF equation with the reference
temperature taken to be Tg. The Arrhenius relation can be also used to relate the
relaxation time of the polymer segments to the temperature.

From Equation (3-7), the overall reaction rate constant k, can be calculated
from a and da/dt by assuming kZ/kl remains the same.

da/dt

k°='(T-T)T§-a) (Io/knot) (3'18)

 

As the kinetics rate constant remains the same, k,(a,T) can be obtained by
rearranging equation (3-16).

4o

kok,

k‘g‘kT-k’ (3-19)

A brief review of the general nature of curing will be helpful to fully describe
I the curing characteristics. Several distinct steps are observed during the complex
curing process. The polymerization starts off with formation and linear growth of the
polymer chains. At this stage, there is no hindrance for the reacting molecules and
the effective reactant concentration is equal to the existing reactant concentration.

The chains begin to branch and then to crosslink as the reaction proceeds. A sudden
and irreversible transformation from a viscous liquid to an elastic gel will occur when
the crosslinks develop into a network of inﬁnite molecular weight. This phenomenon
is called gelation. The extent of cure at gelation, a“, is only a function of the epoxy
system. For a system containing a mixture of A1, A2, Ai moles of monomer with
a functionality of f1, f2, ﬂ, and Bl, B2, Bj with a functionality of g1, g2,
gj , Macosko and Miller have proposed an equation to calculate a“, by assuming A’s
can only react with B’s"

= 1
JIf,-1) (9,-1)

 

a gel

 

2A '
where f. = .2. 12A: (3-20)

2 g 3 E
g =
a :1 91 Bi
For a difunctional monomer reacting with a tetrafunctional monomer (e.g.

DGEBA/DDS), Flory proposed a simple equation for a". prediction"

a =

1 is (3-21)
99 3

Again, B is the ratio of initial hardener equivalents to epoxide equivalents.

41

There are two types of material in the post-gel reactants, soluble material and
gelled material”. Usually these two materials are mixed together in a macro-uniform
fashion. Because of this, the number of reactants available for reaction will be less
than the number of reactants that exist in the mixture. Therefore, an effective
reactant concentration should be used instead of the existing reactant concentration in
order to properly model the reaction kinetics. For a difunctional epoxide and
tetrafunctional amine system, the weight fraction of solubles, w,, has been calculated
theoretically by assuming a single rate constant”.

 

ttr‘,=w,,p‘+ttr,,(titixppi+1-a,,,,,,)2 (3'22)
_ 1 _3 1/2_1
p"(a2 '4') -2-
exp

where w A and WE are the weight ﬁactions of amine and epoxy molecules,

respectively. w, is 0.135 and 0.264 for the DGEBA/mPDA and DGEBA/DDS
systems, respectively. P is the probability of ﬁnding a ﬁnite chain when looking out
from a randomly chosen amine molecule. a“, is the experimentally determined extent

of cure.

Another distinct step during cure is the onset of vitriﬁcation. Vitriﬁcation
occurs when the Tg of the thermoset reaches the cure temperature which transforms
the elastic gel into a glassy solid. The curing may enter vitriﬁcation without passing
through gelation if the cure temperature is less than NT:- ‘dT, is the temperature at
which gelation and vitriﬁcation occur simultaneously. The onset of vitriﬁcation
switches the reaction from chemical control to diffusion control which brings an
abrupt halt to curing. Vitriﬁcation is a reversible transformation and the cure can be
resumed by heating to devitrify the partially cured thermoset. The effects of
vitriﬁcation on the kinetics can be modelled by using an overall rate constant instead

of a reaction rate constant as described in Equation (3-16).

42
3.2.2 Glass Transition Temperature

The glass transition temperature, Tg, is the temperature of transition between a
glassy state and a liquid or rubbery state. Because of the dramatic change in the
physical properties of the polymer at this transition, Tg is an important material
parameter in polymer applications. Physical properties useful in determining Tg
include; polymer density, speciﬁc heat, mechanical strength and modulus, stiffness,
dielectric properties, and rate of gas or liquid diffusion through the polymer. Glass
transition is a complicated phenomenon and a signiﬁcant amount of uncertainty exists
in Tg measurement. Many factors affect Tg, including the structure, crystallinity
and the molecular weight of the polymer, diluents in the polymer, thermal history and
the method of measurement. Many physical and chemical properties will change at
the glass transition. However, only the sensitive ones will change signiﬁcantly and
can be used for Tg measurement and process monitoring. The sensitive properties at
the glass transition also vary with the polymer system. Therefore, the measurement
conditions and methods should be reported with Tg data. To obtain a reliable Tg
value, the choice of the right measurement method is very important” . TMA
(thermal mechanical analysis, which traces changes in polymer dimensions) and DSC
(differential scanning calorimetry, which traces changes in polymer enthalpy) are two
classical methods for Tg measurement. Figures 3.1 and 3.2 show the DSC and TMA
thermographies for both microwave and thermally cured DGEBA/DDS samples”.
The interpretation of Tg in the DSC thermograph is rather arbitrary, especially in the
case of the microwave cured sample. On the other hand, the TMA thermographies
reveal well-deﬁned transitions at Tg. For this reason, TMA was chosen to measure

the Tg of both microwave and thermal cured samples.

The glass transition temperature of epoxy resins has been intensively studied
due to its importance in industrial applications”. These studies include the effects of
the extent of cure and crosslinking density, polymer structure, impurity, and
annealing on the Tg. Different experimental methods have been investigated to

43
determine Tg according to the magnitude of change of different physical properties of
cured epoxy matrix in the glass transition region. Glass transition temperature effects
on physical properties have also been studied. Tg in epoxy matrix-based composites
were measured and found to be different from pure epoxy. Generally speaking, Tg
increases with extent of cure until reaching an ultimate value corresponding to the
ultimate conversion at a given temperature. This is the so called vitriﬁcation point.
In general, vitriﬁcation is deﬁned as the point where Tg equals the cure temperature,
Tc. Experimental results show that Tg at the ultimate extent of cure actually equals
Tc+C, where C is a constant. For the DGEBA/mPDA system, C is 10-13°C'”.
Impurities present in the epoxy matrix usually lower Tg. For partially cured epoxy
resin, Tg may also depend on the thermal history. Because Tg is very sensitive to
small changes in conversion when the diffusion effect is important .and when the
reactant concentration is low, Tg has been used to monitor the cure process”. Two
stages of Tg behavior have been proposed with cure temperature as the dividing point.
Tg has been found to be a function of the extent of cure, cure temperature and cure

stage”.

 

 

 

 

 

 

 

 

 

20
27% thermally cured sample
A 15 2
a i
3
l:
.3 10
m J
l:
01
e
«-i
a 5 4
T9
0 , , . '
o 50 100 150 200 250
Temperature(°C)
TMA Thermograph
0.6
o 4 J 25% thermally cured sample
2‘ 0.2 .
E 1
S o o.
H
it.
1; -o.2 i
g 1
T9
-006 r l ' I V I T I'
30 90 150 210 270 330

Temperature(°C)

DSC Thermograph

Figure 3.1 Comparison of Tg Determination for Thermally Cured DGEBA/DDS

45

 

 

 

 

20
44% microwave cured sample

’E‘ 15 -t
3
c
e
-—l
m 104
c
o
a
.8 T

5 - 9

0 . I ' r . l . 1 .

0 50 100 150 200 250

Temperature(°C)

TMA T'herrno graph

 

 

 

 

0- 0 - 43% microwave cured sample
’5
E -002 "‘
3
T
1-1
In -o.4 . g
4.1
to
Q)
=
-o.6
i
-008 I I I T I r v I y I
30 90 150 210 270 330

Temperature(°C)

DSC Thenno graph
Figure 3.2 Comparison of Tg Determination for Microwave Cured DGEBA/DDS

46

In the study of physical aging effects on the thermal and mechanical properties
of cured epoxy polymers, Jo and Ko found that both properties are strong functions of
the extent of cure and the thermal history“. In aged samples, where the free
volume is low, the major factor which determines the properties of the cured epoxy
polymer is the packing density rather than the extent of cure. However, in the
quenched samples where sufﬁcient free volume exists, the major factor which
determines the properties of the cured samples is the extent of cure. The thermal
history effects on Tg have not been fully explored yet. Several predictive models
for Tg have been proposed‘m'm'm. The most widely accepted one is the
DiBenedetto model, derived from the lattice energy of the polymer:

 

E F
T-T ‘35-?!”
_LT._22= I F- (3—24)
7° 1-(1-J)a
F-

where
T,, is the glass transition temperature of the unreacted sample,
E, is the unreacted epoxy-monomer lattice energy,
E, is the full-cure epoxy-monomer lattice energy,
E, is the unreacted epoxy-monomer segmental mobility,
F, is the full—cure epoxy-monomer segmental mobility, and

a is the extent of cure.

3.3 Experiments

Stoichiometric epoxy/amine mixtures were isotherrnally cured in a 15.24 cm inner
diameter single-mode resonant cylindrical microwave cavity and in a conventional
thermal oven. The epoxide used was DGEBA (diglycidyl ether of bisphenol A, Dow
Chemical DER 332) with an epoxy equivalent weight of 173. The crosslinking

amines used were DDS (diaminodiphenyl sulfone, Aldrich Chemical) with an amine

47

equivalent weight of 62 and mPDA (meta phenylene diamine, Aldrich Chemical) with
an amine equivalent weight of 27. Figure 3.3 shows the chemical structures of DER
332, DDS, and mPDA. To ensure spatial uniformity of the temperature and the
extent of cure in the sample during the cure, thin ﬁlms were used. The
stoichiometric mixtures were dissolved in a solvent, acetone for DGEBAIDDS system
and diethyl ether for DGEBA/mPDA system. Ether was used as the solvent for
DGEBA/mPDA because of its higher volatility, which is required for mPDA.
Samples were prepared by casting several drops of solution onto a 13 mm diameter,
1mm thick potassium bromide (KBr) disk. The recipe used for the solution in this
study was 5g of mixture in 80cc of solvent and casting six drops of this solution onto
a KBr disk to form a thin ﬁlm sample. An approximately 10 micron thick ﬁlm
remained on the KBr disk after the evaporation of solvent at room temperature. Time
periods of 1 hour and 15 minutes were allowed for the evaporation of acetone and
ether, respectively. KBr was chosen because of its transparency to infrared and
microwave radiation (dielectric constant measured in our lab is e' = 6° (2.71 - j
0.002) at 2.45 GHz). The detailed procedure of thin ﬁlm sample preparation is
available elsewhere”. Another KBr disk was placed on the top to protect the sample
after the solvent evaporated. Fresh samples were scanned using a Fourier Transform
Infrared(FI‘IR) spectrometer (Perkin-Elmer 1850) for later reference in the extent of

cure determination.

The 2.4SGHz system was used with a 15.24cm inner diameter cylindrical
cavity for microwave curing. The detailed description of the system was provided in
Chapter 2. The TE", resonant heating mode was used to isotherrnally cure the thin
ﬁlm samples at three different temperatures. Figure 3.4 shows the electromagnetic
ﬁeld pattern of the TB," mode in the cylindrical resonant cavity. The line density
represents the magnitude of E-ﬁeld strength. The thin ﬁlm samples were placed in
the center of the cavity where electric ﬁeld is strongest. As the sample size is very
small relative to the cavity size, the electric ﬁeld pattern inside the thin ﬁlm loaded

cavity is almost the same as that of the empty cavity. The electric ﬁeld is fairly

48

uniform across the sample. Therefore, the temperature is fairly uniform across the
sample. The center temperature was measured by placing the Luxtron ﬂuoroptic
probe directly on the top of the thin epoxy ﬁlm through a hole in the top BKr disk.
An initial heating rate was obtained with a typical value of 2°C per second. The
reaction was quenched after each sample had been heated for a given time.
Quenching the cured samples was necessary to ensure that the T, is determined
mainly by the extent of cure'°°. Parallel thermal cures with the same sample
conﬁgurations were run in a conventional thermal oven. The experimental
temperatures used are listed in Table 3.1.

vie-wwiwrﬁel
cherries.
“at" “'me

mPDA DDS

Figure 3.3. Chemical Structure of DGEBA, DDS, and mPDA

49
Each cured sample was scanned in the FTIR and the transmission infrared spectra
of the cured sample was used to determine the extent of cure. As the concentration
of the absorbing species is linearly related to the area under absorbance peak, the
extent of cure (conversion of epoxy) can be calculated as

 

Jarred
A2962 (3_25)

where A9,, and Am are the areas under the analyte absorbance peak and the
reference absorbance peak, corresponding to the infrared absorption of an epoxy ring
vibration at wavenumber 915 cm" (891-927 cm") and non-reactive aromatic vibration
at wavelength 2962 cm'1 (277 8-2992 cm“), respectively. Normalized analyte
absorbance peak areas were used to compensate for the possible difference in the ﬁlm
thickness from experiment to experiment.

Table 3.1. Temperatures Used for Microwave and Thermal Cure

 

 

 

 

 

 

 

Cure Temp. (°C) I
DGEBAIDDS microwave ~ 125 135 145
thermal 135 145 165
DGEBA/mPDA microwave 80 100 120
thermal 80 100 — 120

 

 

 

 

FTIR has been used to determine the reaction extent of epoxy/amine systems
previously‘°'2‘"°‘"°5"°°. Although the analyte peak used was always 915 cm",
several reference peaks have been used in the literature. In the analysis of the
DGEBA/mPDA system, the 1034 cm" peak that was declared to be due to an out-of-
plane CH deformation of the aromatic ring was used as the reference peak'“. The
1184 cm'1 peak”5 and the 1508 cm" peak” were also used as reference peaks to

50

analyze the same system. In the analysis of the DGEBAIDDS system, the 2962 cm"
peak was used as the reference peak". However, according to the SPECTRA-
STRUCTURE CORRELATIONS released by Stamford Research labs of the
American Cyanarnid Co. , which describes the possible positions of characteristic
infrared absorption bands, the 2962 cm" peak seems to be the most reasonable
reference peak. The NH band, which decreases as the reaction proceeds, may also
appear at 1508 cm". The C-N stretching, which increases with reaction, may appear
at 1034 cm" and 1184 cm". For this reason, the 2962 cm" peak, which is due to an
aromatic vibration, was chosen as the reference peak in this study.

The thin ﬁlm disk was cut into ﬁve pieces after being scanned in the FTIR. The
small pieces of the sarnple were used to measure the dimensional change in the
thickness when it was heated from room temperature to 350°C in a TMA furnace.

A small tin ball was loaded on the top of the probe to ensure the proper contact
between the probe tip and the sample surface. A temperature ramping rate of 10 °C
per minute was used in the experiments. The glass transition temperature was
determined from the TMA thermograph.

613

 

Figure 3.4. Electromagnetic Field Patterns for TE," Mode in the Cylindrical Cavity

51

3.4 Results and Discussion

3.4.1 Reaction Kinetics

Infrared spectra of microwave and thermal cured DGEBAIDDS were compared to
those of fresh samples, as shown in Figure 3.5. Figure 3.6 shows the comparison for
the DGEBA/mPDA system. The transmission spectra of microwave and thermal
cured samples showed no signiﬁcant difference for both systems.

The extent of cure of both microwave and thermally cured samples at various
times for the DGEBAIDDS and DGEBA/mPDA systems are shown in Figure 3.7 and
Figure 3.8, respectively. The symbols th and mw mean thermal and microwave,
respectively. The numbers after th and mw are the curing temperatures. The solid
lines are the least-squares ﬁt of the experimental data using the equation a=bl ‘(b2-b3
* exp(-b4 * t)), where b1, b2, b3, b4 are the ﬁtted parameters and t is the curing
time. Table 3.2 lists the ﬁtted values of bl, b2, b3, and b4 and the standard
deviation of the ﬁt, e, for each curing experiment. Low standard deviation of ﬁt
indicates a good ﬁt between the experimental data and ﬁtted lines. With thin ﬁlm
FI'IR technique, DGEBA/mPDA system has been previously cured using both
microwave and thermal energy”. As listed in Table 3.2, the standard deviation of
microwave and thermal cure experiment at 120°C for the DGEBA/mPDA system are
0.032 and 0.047 respectively in this research. With the ﬁtted expression obtained in
this research to analyze the experimental data reported in the literature 25, the
calculated standard deviation of microwave and thermal cure experiment at 120°C are
0.038 and 0.053. The comparison between the curing results in this research and the
results reported in the literature 25 suggests that the curing using thin ﬁlm FTIR
technique is experimentally reproducible.

52

Table 3.2. The values of bl, b2, b3, b4 and e from the least-squares ﬁt

 

 

 

 

 

 

 

 

 

 

 

 

 

I- -——-r————1
b1 b2 b3 b4 6'

DDS,mw125 0.6161 0.9476 0.9491 0.0582 0

DDS,mw135 0.4064 1.5292 1.5249 0.1211 0.063
DDS,mwl45 0.4713 1.9592 2.0298 0.1302 0.079
DDS,thl35 0.5731 1.0867 1.0916 0.0369 0

DDS,thl45 0.5478 1.1798 1.1862 0.0673 0.016
DDS,thl65 4.7102 0.2106 0.2140 0.0931 0.027
mPDA,mw80 1.4783 0.6574 0.6641 0.0346 0.024
mPDA,mw100 0.0761 12.522 12.596 0.1300 0.028
mPDA,mw120 0. 8430 1.1292 1.1218 0.4285 0.047
mPDA,th80” 0.029
mPDA,th100 2.7330 0.3382 0.3410 0.1001 0.043
mPDA,th120 9.2797 0.1011 0.1021 0.1862 0.032

 

h

 

 

 

 

 

 

 

 

‘

‘ e=\J 37%: (“.m'auclz , where N is the number of the data points.

” A polynomial ﬁt was used in this case. The ﬁtted equation is oz=2.302xl(r3 +
1.158 x102 t - 2.250x10‘ t2 + 1.324x10’ t3 - 1.6981(107 t‘. The standard deviation

. . - 1
forthrsﬁtrs e- ﬁ£(aem-¢m)2.

 

53

 

 

 

 

 

99—

 

5.25:“ 0
ar.--- ~-~—.Z2:12:If:'.‘’:."'..':.'-:.m»..-.. — 8
O
,_O
O
O
,_0
Ln

L
O
_O
O
N
0
PO
L0
J: N

m b

“J

E C’ ‘1. O
E .— m _o
O E o O
‘0 O C) n

L0. N,\ b

'd' In at)
3 s— (.9 ho
E :5 e um)
u : O
O
l I l I r 1 ﬂ I O
03 03 05 m d-
1\ L0 r0 .—

Figure 3.5. Comparison of FI'IR Spectra of Microwave Cured,

(Z)

uogssluusumi

-1

Thermal Cured, and Fresh DGEBAIDDS samples

Wovenumber

54

000
P

oo_o F 00.0 F

$3E3cm>o>>
OO_ON oo_mN COMM

b P

000..N

 

-5“
_ _----’..
38-

     

- -'---'

 

        

     

 

 

 

58:.<an\<88
.EEoﬁong
$826235 .|

PI

.
u .
. u r
s
s ...
... . .. n
.. “ ~é . a
.. . . . . 12
.. . . .. s.
.. . _ . 1....
.. . . a . . .:
. o - 0? ....
. . . .. a . .. . s
. .. . ~ . I o l
. c a . 2 u
.
. . e . a . .
. ~ ~ ” .
. .
. . . . . Ion
. - 0 § - c
n u . . x
_
x . . u .t. _ .
. . . . . . .
. . . . . 1 .
- . c u \ o a I.
a s o. c s ' c
s . . . w
. o. . I
I . ..l
. mm
. .
.3. s \ . I; j
o 4 so 3 . v
. . . : . 3
rs, . s. a . D N... v
’ \- ' ~ I s I I
r f\.t

1mm

 

[mm

 

) uolssguusuoli

%

(

Figure 3.6. Comparison of FTIR Spectra of Microwave Cured, Thermal

Cured, and Fresh DGEBA/mPDA samples

55

As seen in Figure 3.7 and Figure 3.8, instead of a decrease in the reaction rate
in microwave cure as reported in some earlier work""’, microwave cure has a higher
reacﬁonratewharcomparedwthatofdtemtalcureatthesamecuretempemmre.
This result agrees with results in most of the literatures‘°""‘°m'”.

Figure 3.7 shows the extent of cure versus time curves for the DGEBAIDDS
system. Signiﬁeant increases in the reaction rates for microwave curing were
observed as compared to those for thermal curing at the same temperature. Figure
3.8 shows the extent of cure versus time curves for the DGEBA/mPDA system. Only
slight increases in the reaction rates for the microwave curing were observed as
compared to those for thermal curing at the same temperature. This may be due to
the chemical structure difference in DDS and mPDA as shown in Figure 3.3.
Apparently, the effects of microwave radiation on the cure of a thermosetting polymer
depend on the particular curing agent used. The enhancement of the polymerization
reaction rate of epoxy resins by microwave radiation is expected. Comparing the
microwave and thermal heating mechanisms, microwave cure should have a faster
reaction rate than thermal cure providing that the reaction pathway is the same for
both microwave and thermal cure. For the same bulk cure temperature, the local
molecular temperature will be higher in microwave heating than in the thermal
heating due to the difference in microwave and thermal heating characteristics. In
microwave cure, microwave radiation heats up polymer molecules directly due to the
relaxation of the polarization of the polymer dipoles along the electric ﬁeld. The
measured bulk temperature therefore is less than the local molecular temperature
which is the effective temperature for the polymerization reaction. On the other
hand, the local molecular temperature is at most equal to the bulk temperature during

the thermal cure.

56

 

 

 

 

 

 

1.0
1 11 9
0.8- g
y 0.6-
C a
0 .4
4.:
3.3 / -
0.44
q / , a th 145
‘ a mu 145
0.2- I th 135
D mw 135
0 th 165
a MN 125
000 I I 1 I 1 I U I
I 20 40 60 80

Time(min.)

Figure 3.7. Reaction Rates of Microwave and Thermally Cured DGEBAIDDS

 

 

 

 

1.0
. . ~
0.8- ' _ o
A
o
‘ ‘ . a o
H 0.6- o
C
8 .
83
0.4- .
A th 120
A mw 120
0.2 e I th 100
D mw 100
‘X' 0 th 80
O O o mw 80
o T I I I
0 2'0 40 so

Time(min.)

Figure 3. 8. Reaction Rates of Microwave and Thermally Cured DGEBA/mPDA

57

The opposite results of Mijovic’s work""’ may due to the following reasons:
a) Higher curing temperature in the thermal cure. Although the oven
temperature is set equal to the microwave cure temperature, the sample
temperature is higher than the oven temperature due to the heat released during
exothermic reaction for the thermal cure, while the sample temperature ean be
controlled exactly at the desired cure temperature for microwave cure.
b) Due to a different reaction pathway for microwave and thermal cure. In the
current work, the extent of cure is determined using FTIR and deﬁned as one
minus the ratio of residual amount of epoxide group to the initial amount of
epoxide group. For Mijovic’s work, the extent of cure is determined using
DSC and deﬁned as one minus the ratio of the residual exothermic hat to total
exothermic heat of unreacted sample. One assumption in the DSC deﬁnition
of the extent of cure is. that there is no exothermic/endothermic reaction that
does not involve epoxide. This may or may not be the ease in both microwave
and thermal cures. Clearly, the DSC measurements take into account of all
reactions while the FTIR measurements only monitor the disappearance of
epoxide group. If the Mijovic’s DSC samples were taken from the area close
to the temperature probe in microwave cured samples and from the surface in
thermal cured samples, the opposite result of the extent of cure from FTIR and
DSC suggests a different reaction pathway for microwave cure compared to
thermal cure.

The ﬁtted parameters in Table 3.2 are used to analyze the kinetics. Assuming
that there is no steric hindrance and no etheriﬁcation has occurred, that is n=1 and
L=0, the reaction rate constants can be calculated from Equation (3-7) by plotting
dcr/dt/(l-or)2 vs. 0: curves. The intercept is kl and the slope is k2. Only the early
portion of the da/dI/(l-a)2 vs. a curve is used because Equation (3-7) is not valid for
systems having an etheriﬁcation reaction, which may be signiﬁcant when reacting

systems reach high extent of cure, and because the diffusion effect becomes more

58

signiﬁeant as the reaction proceeds. The activation energy E and pre-exponential
constant A of k, and k, are obtained through a linear ﬁt of ln(k) vs. l/T.

Table 3.3 lists the calculated kl and k; for the DGEBA/DDS and the
DGEBA/mPDA systems. The absolute values of k1 and k, in the microwave cure
weregreaterthanthoseoftherrnalcureatthesamecuretemperatureforboththe
DGEBA/DDS and the DGEBA/mPDA systems. Table 3.4 lists the activation
energies and the pre-exponential constants for the DGEBAIDDS and DGEBA/mPDA
systems calculated from Table 3.3. The 0: versus t curves were regenerated using
parameters in Table 3.4. Figure 3.9 and 3.10 show the comparison between
regenerated curves and the experimental data for the DGEBA/mPDA and the
DGEBA/DDS systems respectively.

Table 3.3 Reaction Rate Constants for Both Thermal and Microwave Cure
of DGEBA Reacting with DDS and mPDA With Assumption of n=1, L=0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

k2(min") kr/kz
DDS, 01135 0.0230 0.0090 2.56
DDS, 01145 0.0435 0.0195 2.23
DDS, 01165 0.0923 0.0972 0.95
DDS, mw125 0.0340 0.0087 3.91
DDS, mw135 0.0753 0.0258 2.92
DDS, mw145 0.120 0.116 1.04
mPDA, 0180 0.00930 0.0941 0.0991
l mPDA, 01100 0.0341 0.383 0.0889
I mPDA, 01120 0.0640 0.721 0.0888
mPDA, mw80 0.0206 0.116 0.178
mPDA, mw100 0.0694 0.395 0.176
I mPDA, mw120 0.133 1.97 0.0674

 

 

 

 

59

Table 3.4 Activation Energies and Pre-exponential constants

for DGEBA Reacting with DDS and mPDA With Assumption of n=1, L=0

., I...» * “ k. m,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. E(keallmol) A E(kcal/mol) A
DDS, mw 21.0 9.745E+9 42.8 1.907E+2l
DDS, th 16.1 1.085E+7 28.3 1.069E+ 13
mPDA,mw 12.9 2.185E+6 19.5 1.203E+11
mPDA,th 13.3 1.882E+6 14.1 5.768E+7
1 .0 f f 1 f I T I I0 I I I I I I I I I I
0 a 4
0.9 - ° .
.- * - o “
0.8 '- 0 ' g .-
O.7 ' ‘ + o o .,
.. 0.6 - ' ' o o .
1: ~ 4
3 0.5 - , ° mw 80 -
:5 . - ° 9 mw100 .
0.4 — ' . <> mw120 -
» - ° . th 80 .
0.3 " o ' “1100 "
* + th120 ‘
0.2 .
0.1 g -
/
0.0 4 L 1 l 1 l 1 l 1 l 1 l 1 l 1 l 1 l

O 6 1 2 1 8 24 30 36 42 48 54 60

time(min.)

Figure 3.9 Regeneration of Reaction Rates for Microwave and Thermally
Cure of DGEBA/mPDA From the Model Using n=1 and L=0

60

For the DGEBA/mPDA system, the reaction kinetics can be described well by
Equation (3—7) for both microwave and thermal cure. This implies that for the
stoichiometric DGEBA/mPDA system, the reaction rate constants of primary amine-
epoxy are equal to secondary amine-epoxy and the etheriﬁcation reaction can be
neglected for both microwave and thermal cure in the temperature range studied. As
listed in Table 3.3, the kllkz ratio of microwave cure is higher than that of thermal
cure at low temperatures but lower than that of thermal cure at high temperatures. As
listed in Table 3.4, the activation energy of k, is about the same for microwave and
thermal cure while the activation energy of k, is larger for microwave cure than
thermal cure for DGEBA/mPDA system. The reaction kinetics of thermal cure
DGEBA/mPDA have been studied using DSC. The reported reaction rate constants
m7]

K, = 5.53 x 10‘ x exp(-9763.5/T)

K, = 8.06 x 10’ x exp(-5737.3/T)

Comparing the above values of E to the values of E in Table 3.4, the activation
energy of kl is lower in this study while E of k2 is higher in this study. The
difference may be due to the different deﬁnition of the extent of cure for DSC and
FI'IR.

Rewriting Equation (3-7) as

dd g 1‘1 _ _ _

For the DGEBA/mPDA system, the values of 111/k2 are small as compared one.
Especially in the thermal cure of DGEBA/mPDA, k,/k, is relatively independent of
cure temperature and the value is about 0.09 as in Table 3.3. This means the reaction
rate expression can be approximated by the product of two terms, one is a function of
temperature and the other is a function of the extent of cure, a. The reaction can be
described phenomologically by an n“I order kinetics model. Following the procedure

described in section 3.2.1.1, the order of reaction and the parameters for the reaction

61

rate constants are calculated. Table 3.5 lists the calculation conditions and results.
A master curve can be generated by combining n|l| order reaction kinetics with the
relationship between glass transition temperature and extent.

Table 3.5 Calculations based 11‘“ order assumption for DGEBA/mPDA

 

 

  

 

      
  
  
  
  
  
   
 

 

 

 

 

 

 

 

 

 

 

 

1—-
microwave cure thermal cure
T,“ (°C) 100.0 100.0
a“ 0.621 0.466
CPI)“ (°C) 76.4 69.0
E (kcal/mole) 17.3 13.7
Correlation Coefﬁcient for 13 0.9995 0.9914
A (min.") 2.42E+9 1.13E+7
Correlation Coefﬁcient for A 0.9395 0.9672
n 1.6 1.5

    

For the DGEBA/DDS system, the reaction rates are not be described well by
Equation (3-7) for both microwave and thermal cure as shown in Figure 3.10. At
lower cure temperatures, Equation (3-7) overestimates the reaction rates. However,
the overestimation was corrected at higher cure temperatures. This may be due to the
fact that the reaction rate of primary amine-epoxy is larger than that of secondary
amine-epoxy, that is n <1. At low cure temperatures, no etheriﬁcation occurred and
Equation (3-7) which assumes that n =1 overestimates the reaction rates. At higher
cure temperatures, etheriﬁcation reaction occurrs, i.e. L is not equal to 0. Since
n+L is closer to 1, the overestimation is partially corrected. To model the
DGEBA/DDS system, Equations (3-12) to (3-14) should be used. For our system,
[011], =0. Taking the derivative of Equation (3-14) with respect to time t, we obtain

62

£3
g” (l-n) (z-L) + (n-L)1 3 + (2-nlLQ'1 Q (3-27)
dt 2(2—n) dc

combining Equation (3-27) with (3-12) and (313), we have
d¢ldt=f(n.L.¢)(ki+kzF(¢)).

 

 

 

 

 

 

1.0 F I T T I I I I I T T —T
0.9 r + a
* 3
0.8 - 1 .
0
0.7 -
g 0.6 . , ' a " -
C 1' u .
.93 0.5 ' n o -
X ..
0 0.4 ~ . 01155 -
- a th145 .
0.3 - 0 th165 -
* 0 mw125 7
0.2 - - mw135 ‘
0.1 - + mw145 _
J
00 1 1 1 1 1 1 1 1 L 4 1 1 1 1 1 1 1

 

O 6 1 2 1 8 24 30 36 42 48 54 60

time(min.)

Figure 3.10 Regeneration of Reaction Rates for Microwave and Thermally
Cure of DGEBA/DDS From the Model Using n=1 and L=0

At low cure temperatures, L=0 and F(¢)=a. n and L can be obtained by a
least-squares ﬁt of experimental data to Equation (3-12), (3-13) and (3-14). The
procedure is

1) Assuming an n and L, an expression for d¢ldt as function of 41 can be

obtained from experimental data and Equation (3-27).

2) Obtain kl and k; values for all temperatures by plotting d4>ldt/f(n,L,¢)

63

versus F(¢) by varying 41 from 1.0 to 0.0. Calculate the activation energies
and pre-exponential constants of k, and k2.
3) Regenerate the 01 vs. t curve by using assumed n,L and the calculated
reaction constant parameters. Compare the regenerated values with
experimental results.
4) Repeat the above procedures with new 11 and L until the regenerated values
match the experimental data.
The reaction rate constants and the calculated parameters for DGEBA/DDS are listed
in Table 3.6 and Table 3.7 respectively.

As listed in Table 3.6, the reaction rate constants of microwave cure were
higher than those of thermal cure at the same cure temperature for the DGEBA/DDS
the system. However, the values of k,/k2 were lower in microwave cure than in
thermal cure. In other words, the microwave radiation enhances the catalytic reaction
of DGEBA/DDS system more than the non catalytic reaction, just as for the
DGEBA/mPDA system. The activation energies of microwave cure are higher than
those of thermal cure, for both kl and k2, as listed in Table 3.7. While microwave
radiation increases the reaction rate constant of primary amine-epoxy reaction as listed
in table 3.6, it decreases the reaction rate ratio of secondary amine-epoxy reaction to '
primary amine-epoxy reaction, 11, and reaction rate ratio of etheriﬁcation to primary
amine-epoxy reaction, L, as listed in Table 3.7. The absolute value of the reaction
rate constant for the secondary amine-epoxy reaction, nkl , nk2, and etheriﬁcation
reaction nL, are still larger in microwave cure than in thermal cure. Figure 3.11
shows the comparison of the calculated 01 vs. t curves based on the kinetics model
with n at 1, L560 and the experimental data for both microwave and thermal cure of
DGEBA/DDS systems. Good agreement is found between the two.

64

Table 3.6 Reaction rate constants for DGEBA/DDS system

ki/kz

 

 

 

 

 

 

 

 

 

k1 (mind) k2 (min")
mw125 0.0338 0.0448 0.754
mwl35 0.0749 0.1107 0.677
mwl45 0.1190 0.2701 0.440
111135 0.0229 0.0251 0.912
01145 0.0434 0.0510 0.851
01165 0.0917 0.1793 0.511

Table 3.7 Calculated reaction kinetics parameters for DGEBA/DDS system

 

 

 

 

 

 

 

 

 

 

microwave thermal
n 0.15 0.4
L‘ 0.6 0.8
kl E (kcal/mole) 20.9 16.1
A (min") 1.001E+ 10 9.946E+6
k2 E (kcal/mole) 29.7 23.2
A (min“) 9.156E+ 14 7.063E+10

1" etheriﬁcation reaction only occurred at high cure temperature, that is at 145 °C for
microwave and 165 °C for thermal cure.

All of the reaction kinetic parameters for the DGEBA/DDS and
DGEBA/mPDA systems are listed in Table 3. 8 for easy comparison. For
DGEBA/mPDA, both microwave and thermal cure have the same reaction rate for

primary and secondary amine and a negligible etheriﬁcation reaction. For this

system, the activation energy E for k, is about the same for both microwave and

thermal cure while the activation energy E for 112 is higher in microwave cure than

65

thermal cure. The same values of n and L imply a similar network structure for
microwave and thermally cured DGEBA/mPDA. Similar Tg versus a data is
expected from microwave and thermally cured DGEBA/mPDA samples. For the
DGEBA/DDS system, 11 and L of microwave cure are less than those of thermal cure.
For this system, the activation energy E for both k, and k, are higher in microwave
cure than in thermal cure. The different values of n and L may result in a different
crosslinked network structure in microwave and thermal cured DGEBA/DDS. The
lower values of n and L suggest that the primary amine-epoxy reaction is more
dominant in microwave cure than in thermal cure as compared to secondary amine-
epoxy and etheriﬁcation reactions. This implies that the epoxy molecules have a
better chance to form a long linear chain through reacting with primary amine at both
ends before crosslinking in the microwave environment. The network structure will
have higher T, if the two crosslinks are one at each end rather than both at one end“.
Also, the linear chains in the microwave cured epoxy are aligned with the applied
electric ﬁeld due to the effect of molecular polarization of both epoxide and amine
molecules. This alignment creates an oriented crosslinking network structure which
also increases the T, of the cured epoxy. Therefore, higher T, is expected for
microwave cured DGEBA/ DDS than thermally cured samples. Polarization of
molecules under microwave radiation may also lead to a different reaction pathway
from that of thermal cure.

66

Table 3. 8 Reaction kinetic parameters for both
DGEBA/DDS and DGEBA/mPDA systems

DGEBA

/ mPDA

 

DGEBA

 

/ DDS

 

 

microwave

   

thermal

 

 

microwave

thermal

      

 

 

n

1.0

 

 

1.0

 

0.15

 

0.4

         

 

 

  

L

0

 

0

0.6

 

0.8

             

 

        

 

 

 

E (kcallmole) 12.9 13.3 20.9 16.1
A (min") 2.185E+6 1.882E+6 1.001E+ 10 8.975E+6
k, E (kcallmole) 19.5 14.1 29.7 23.2

     

 

 

-1")

 

1.203E+11

 

5.768E-1-7

 

 

9.156E+l4

  

 

7.063E+ 10

 

. .0 .0 .0
\l on (O
V T V I I T

extent

 

 

 

45

+IOOOO

54 63

time(min.)

L L l A l

72

th135
th145
th165
mw125
mw135
mw145

l 4

81

 

90

Figure 3.11 Regeneration of Reaction Rates for Microwave and Thermally

Cure of DGEBA/DDS From the Model Using 11951 and L¢0

67

3.4.2 Glass transition temperature

The glass transition temperature, T,, is deﬁned as the temperature where the
step transition of the thermal expansion coefﬁcient occurs on the TMA thermograph,
as shown in Figure 3.1 and Figure 3.2. Figures 3.12 and Figure 3.13 show the Tg
versus extent of cure for the DGEBA/DDS and DGEBA/mPDA system, respectively.
While similar glass transition temperatures for microwave and thermally cured
samples. were obtained at low conversion, higher T,’s were observed in microwave
cured samples at an extent of cure higher than a,,., which occurs at a=0.58. The
increase is signiﬁcant in the DGEBA/DDS system but small in the DGEBA/mPDA
system. The increase of T, in the microwave cured epoxy at high extent may be due
to the different polymer crosslinked network structures in the microwave cured
samples as discussed earlier. A similar T, at low extent of cure implies that the
difference between microwave and thermal cured epoxy at this level is not signiﬁcant
enough to cause any difference in T,. Apparently, the microwave radiation effect is
stronger in DDS cured epoxy than in mPDA cured epoxy. The different effect of
microwave radiation on T, of DDS and mPDA cured epoxy may be due to the
different network structure resulting from the microwave environment for the two
systems as speculated in the kinetics analysis. Other techniques, such as a polarized
infrared technique, wide-angle X-ray diffraction, or solid state Nuclear Magnetic
Resonance (NMR) are required to detect these structure differences in the cured
polymer samples.

 

250. -

 

 

 

 

0.0 . 012 Y 014 036 . 0.8 t 1.0
extent

Figure 3.12. Comparison of Tg of Microwave and Thermally Cured DGEBA/DDS

200.

 

150.4

 

 

 

 

extent

Figure 3.13. Comparison of Tg of Microwave and Thermally Cured DGEBA/mPDA

69

Another interesting phenomenon is that microwave radiation affects Tg in the
same fashion as it does the reaction rates. The magnitude of increase in Tg by
microwave radiation is much higher in the DDS cured epoxy than in mPDA cured
epoxy, just like the microwave radiation effect on the reaction rate. DiBenedetto’s
model was used to ﬁt the TMA- determined Tg data. Table 3.9 lists the
experimentally determined Tg,, Tg., and the estimated values of Ex/Em, Fx/Fm,
along with the values reported in the literaure. As listed in Table 3.9, the values of
Ex/Em and Fx/Fm are lower in the microwave cured samples than in the thermally
cured ones. As Em and Fm are the same for both microwave and thermal cure, the
lattice energy and the segmental mobility of fully cured epoxy are lower in microwave
cure than in thermal cure for both systems. The magnitude of decrease is more
signiﬁcant in the DGEBA/DDS system than in the DGEBA/mPDA system. The
lower segmental mobility in the microwave cured sample may be due to a more rigid
network as a result of an oriented network structure. The ratio of segmental mobility
of the fully cured to unreacted epoxy resins, F,/F,_, has been related to the ratio of the
isobaric heat capacities of the fully cured to the initial systems, A”.

AC F
A = _Z a: __X (3-28)
AC,” 1;,

Where AC,,, and AC", are the isobaric heat capacities of the fully cured and the initial
systems, respectively. A is smaller for the microwave cured samples than for the
thermally cured samples. This is a possible reason why the T, transition in the DSC
thermograph is more difﬁcult to observe in the microwave cured epoxies than in the

thermal cured samples.

70
Table 3.9. The Parameters for DiBenedetto Model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lDGEBA/DDS,MW 22.2 258.3 TMA 0.32 0.18 this J
[DGEBA/DDS, Th 22.2 171.4 TMA 0.44 0.30 this

[ DGEBA/DDS, Th 5 DSC 0.751 0.419 3‘
DGEBA/DDS, Th 11 189 1.21 0.74 W
DGEBA/DDS, Th 0 190 DSC "
DGEBA/DDS, MW 0 190 DSC *4 i
LDGEBA/DDS, Th 210 TBA“ I“ I
DGEBA/DDS, Th 183 DSC 10°
DGEBA/DDS, Th 184 w "0 I
DGEBA/DDS, Th 0 DSC 0.3 0.18 "I I
DGEBA/mPDA, Th 158 DSC

DGEBA/mPDA, Th 150 TBA 5°
DGEBA/mPDA, Th 163 DSC 5* I
DGEBA/mPDA, 29 169.2 TMA 0.44 0.30 this
MW

DGEBA/mPDA, Th

 

 

 

 

 

 

 

 

* Torsional Braid Analysis
** Dynamic Mechanical Test.

71
3.4.3 Master Curve and TIT diagrams

The master curve and the time-temperature-transformation (TIT) diagram are
commonly used to interpret the curing process". The master curve describes the
relation between glass transition temperature (Tg) and cure time (t) at a reference
temperature. The Tg versus t curves at other temperatures can be obtained from the
master curve and the shifting factors which are calculated from the kinetics data. A
master cure is only possible when the kinetics can be expressed in the form of
Equation (3-1). Therefore, only master curves for microwave and thermal cure of
DGEBA/mPDA were generated. From the kinetics expression, Equation (3-2), and
the DiBenedetto model, the master curves at a reference temperature of 120°C were
constructed for DGEBA/mPDA system.

Figure 3.14 shows the master curves for both microwave and thermal cure of
DGEBA/mPDA at a reference temperature of 120°C. The data points are
experimental results obtained from 01 versus t and Tg versus a relations. The solid
lines are calculated from the modeling, Equations (3-2) and (3-24) and the kinetics
parameters listed in Table 3.5. The modeling results were compared to the
experimental data and a good agreement was found for both microwave and thermal
cure. Clearly, microwave cure is faster than thermal cure. Using the vitriﬁcation
point as a reference, for example, the time to reach vitriﬁcation is about 5 min. and
12 min. for microwave and thermal cure, respectively. Microwave cure is more than
twice as fast as thermal cure at 120°C.

 

microwave

     

'I'ITﬁU‘WrF‘r

thermal

thermal data
microwave data

 

 

1 J

1 0° ‘101 1O2
Time(min.)

 

Figure 3.14. Master Curves for Microwave and Thermal Cure of DGEBA/mPDA
at Reference Temperature of 120°C

Figure 3.15 shows the TIT diagrams for both microwave and thermal cure of
DGEBA/mPDA. Again, the point data are experimental values. The solid lines are
calculated based on Equation (3-7) and Equation (3-24). The vitriﬁcation time at a
given cure temperature was calculated based on the deﬁnition that vitriﬁcation occurs
when Tg equals the cure temperature. mw vitriﬁcation curve 1 and th vitriﬁcation
are vitriﬁcation lines based on this deﬁnition for microwave and thermal cure. The
calculated vitriﬁcation curves agree with the experiment data very well. Figure 3.15
shows that the time to gelation is shorter in microwave cure than in thermal cure,
especially at high cure temperatures. The vitriﬁcation times in microwave cure are
shorter than in thermal cure, and the time difference between the two increases with
cure temperature. The cure temperatures which display the minimum vitriﬁcation
times are about 152°C and 131°C for microwave and thermal cure, respectively. The
corresponding vitriﬁcation times are 2.4 min. and 12.6 min. for microwave and

thermal cure respectively. This implies that the processing time can be reduced by

73
approximately a factor of ﬁve when conventional thermal heating is replaced by

microwave heating.

The decrease in the vitriﬁcation time in the microwave cure is due to the
increase of reaction rate and Tg. The Tg of microwave cured samples are higher
than those of thermally cured samples at the same extent of cure. The extent of cure
at vitriﬁcation is, therefore, lower in the microwave cure than in the thermal cure. In
consideration of this effect, the vitriﬁcation for microwave cure was redeﬁned as the
point when the extent of cure of microwave cure is equal to the vitriﬁcation extent of
thermal cure at the same cure temperature. With this deﬁnition, the microwave
vitriﬁcation curve only take into account the effect of faster reaction rates caused by
microwaves. The curve, mw vitriﬁcation curve 2, was calculated based on the new
deﬁnition of vitriﬁcation for microwave cure. Comparing this curve to the thermal
vitriﬁcation curve, the minimum vitriﬁcation times for microwave and thermal cure
are observed at approximately the same temperature. The new cure temperature
which has minimum vitriﬁcation time for microwave cure is 137°C. The
corresponding vitriﬁcation time is 4.7 min. Based on this deﬁnition of the
vitriﬁcation for microwave cure, the processing time is reduced by approximately a
factor of two when conventional thermal heating is replaced by microwave heating.
Clearly, from curve th. vitriﬁcation to curve mw vitriﬁcation 2, the shortening of
processing time is due to the increase of reaction rate by microwave radiation. From
curve mw. vitriﬁcation 2 to curve mw vitriﬁcation l, the shortening of processing
time is due to the increase of Tg by microwave radiation. Also the increase of
reaction rate only shorten the processing time while the increase of Tg not only
shorten the processing time, but also increase the cure temperature which display the

minimum vitriﬁcation time.

Figure 3.16 shows the TI‘T diagram for microwave and thermal cure of the
DGEBA/DDS system. The notation in Figure 3.16 is the same as in Figure 3.15.
Only one experimental data point of vitriﬁcation can be obtained from Figure 3.7,

74

because the curing at the other temperature was not long enough to achieve
vitriﬁcation. Again, the data points are experimental results. The solid lines are
reconstructed from the kinetics expressions, Equations (3-12) to (3-14), the
DiBenedetto model, Equation (3-24), and ﬁtted parameters in Table 3. 8 and Table
3.9.

 

 

 

 

 

 

175 I mw vitrification curve1
160 mw vitrification curve 2
,1 145 E- th.vitrification
3 130 l
S i . , . . . th.vit
3 115 __ mw gelatlo \ 0 th. gel
$1100; 2 ,\ , mw.vit
E : ‘ . mw. gel
4) 85 f
f; 70 . mw 90119 \
L. _
3 y- \
. th.9°|T .
o 55 f g . \\ th.gelatlon
40
25'1111111 1 1 111.111 1 1 11111.1 1 1 1.1111
3 2 3
100 2 3 101 2 102 103
Time(min.)

Figure 3.15. Tl'l‘ Diagrams for Both Microwave and Thermal Cure
of DGEBA/mPDA System

 

530 ....... -........ ........, ......., -......., 4-..“... ......... 4..
505 ~
480 l
455 L
430 l
405 l
380 1
355 Z
530 l
305 C

. ........ . ........ ........1 2...... . ........ 1...“... . .......1
213133-22 10-12 1002 1012 1022 1032 1042 1052

time (min.)

Temperature (K)

 

 

 

 

Figure 3.16. TTI‘ Diagrams for Both Microwave and Thermal Cure
of DGEBA/DDS System

76

No analytical expression can be derived in the calculation of vitriﬁcation and
gelation curves for DGEBA/DDS system. Numerical methods have to be used in
these calculations. The FORTRAN code for this calculation is listed in Appendix I.
The calculating procedure is

1) Determine the extent of cure at vitriﬁcation for a given cure temperature

using Equation (3-24).

2) Calculate the extent vs. time curve using Equations (3-12) to (3-14). The

time required to reach gelation and vitriﬁcations at that temperature are

recorded.

3) Repeat for the next temperature.

In our experiments, the etheriﬁcation reaction is only signiﬁcant at high cure
temperatures, that is, above 145 °C for microwave cure and 165 °C for thermal cure.
In the generation of the TIT diagrams, the reaction rate constant ratio of
etheriﬁcation to primary amine-epoxy reaction, L, is considered to be zero for low
temperatures. The jump of the gelation and vitriﬁcation curves for the microwave
cure at 145°C is due to the introduction of a non zero L. In reality, L is a function
of temperature and both gelation and vitriﬁcation curves are smooth curves. To
observe this, more isothermal cures are needed above 135°C. A good agreement is
found between experimental data and calculated curves based on the models.

The temperature for the simultaneous occurrence of gelation and vitriﬁcation,
‘ ,, can be determined using Equation (3-24) with a=a,,,. They are 67°C and 62°C
for microwave and thermal cure of DGEBA/DDS respectively. Comparing the
microwave vitriﬁcation curve, vitl,mw, and thermal vitriﬁcation curve, vit,th, the
minimum vitriﬁcation time is much smaller in microwave cure than in thermal cure as
shown in Figure 3.16. Also the cure temperature at minimum vitriﬁcation is much

higher in microwave cure than in thermal cure. For thermal cure, the minimum

77

vitriﬁcation time is 758.9 min., occurring at the temperature 140°C and an extent of
0.951. For microwave cure, the minimum vitriﬁcation time is 0.067 min., occurring
at the temperature 249°C and an extent of cure of 0.998. Once again, the processing
time is reduced dramatically by using microwave energy. Microwave radiation
increases both the reaction rate and T, of cured epoxy. The vitl,rnw curve includes
both effects. The T, effects on the vitriﬁcation curve can be eliminated by redeﬁning
the microwave vitriﬁcation as discussed earlier. The vit2.mw curve only includes the
enhancement of reaction rate effect during microwave cure. Based on this deﬁnition,
the minimum vitriﬁcation time is 17 min. occurring at 157°C. The processing time
is reduced by 44.6 times only through the increase of the reaction rate by using

microwave energy.

Both Figure 3.15 and Figure 3.16 show that microwave radiation shortens the
minimum vitriﬁcation time in two ways. One is through the enhancement of the
reaction rate. The other is through increasing T, of the cured epoxy. Apparently, the
effect of microwave radiation is much more signiﬁcant in the DGEBA/DDS system
than in the DGEBA/mPDA system. This may be due to the fact that DDS has a much
higher dipole moment than mPDA. The dipole moment ratio of
DDS:mPDA:DGEBA is 19.8E-30:4.95E—30:8.5E-30 C-mm'm. Other
possibilities for stronger microwave effects in the DDS system are the activity of the
SO, group in DDS and the inhibiting effect of the proximity of amines in mPDA.
More research needs to be carried out to determine the key functional groups
responsible for microwave radiation effects.

3.5 Conclusions

Stoichiometric mixtures of DGEBA/DDS and DGEBA/mPDA epoxy resins
have been prepared using a thin ﬁlm technique. Parallel samples were cured in a

78

cylindrical resonant cavity resonating in a TE,“ mode and a conventional thermal
oven. FTIR was used for measurement of the extent of cure and TMA was used for
determination of T,. Faster reaction rates were observed in the microwave cure when
compared to those of thermal cure at the same cure temperature for both systems.
Effects of microwave radiation on the cure of therrnosetting polymers depends upon
the curing agent used. Similar values of Tg were obtained for microwave and
thermal cure at low extent of cure while higher Tg have been observed in microwave
cure at extent of cure greater than 01,... Microwave radiation has stronger effects in
the DGEBA/DDS system than the DGEBA/mPDA system. Tg data of microwave
cured samples were ﬁtted by the DiBenedetto model. The full-cure epoxy-monomer
lattice energy is lower in microwave cure than in thermal cure and the full-cure
epoxy-monomer segmental mobility is lower in microwave cure than in thermal cure
for both the DGEBA/DDS and DGEBA/mPDA systems.

It has been demonstrated that the cure kinetics of DGEBA/mPDA and
DGEBA/DDS systems can be described by an autocatalytic kinetic model up to
vitriﬁcation in both microwave cure and thermal cure. For the stoichiometric
DGEBA/mPDA system, the reaction rate constants of primary amine—epoxy are equal
to secondary amine-epoxy and the etheriﬁcation reaction is negligible for both '
microwave and thermal cure. For the stoichiometric DGEBA/DDS system, the
reaction rate constants of primary amine-epoxy are greater than those of secondary
amine-epoxy and the etheriﬁcation reaction is only negligible at low cure temperatures
for both microwave and thermal cure. Microwave radiation decreases the reaction
rate constant ratio of primary amine-epoxy to secondary amine-epoxy and the ratio of
primary amine-epoxy to the etheriﬁcation reaction. Microwave radiation increases the
activation energies of both k, and k2 in the DGEBA/DDS system while it only
increases the activation energy of k2 in DGEBA/mPDA system. The vitriﬁcation time
is shorter in microwave cure than in thermal cure, especially at higher isothermal cure

temperatures.

CHAPTER 4
MICROWAVE PROCESSING OF UNIDIRECTIONAL AND CROSSPLY
CONTINUOUS GRAPHITE FIBER/EPOXY COMPOSITES

4.1 INTRODUCTION

The advantages of microwave curing over thermal curing of epoxy resins have
been discussed in Chapter 3. This chapter discusses the feasibility of processing
continuous graphite ﬁber/epoxy composite using microwave energy in a 17.78 cm
resonant microwave cavity. The microwave radiation effects on the mechanical

properties of the microwave processed composite are also discussed.

Based on their investigation using a waveguide and continuous graphite
ﬁber/epoxy laminates, Lee and Springer‘ have reported that heating in a microwave
environment is effective only for unidirectional composites exposed to linearly
polarized TEM waves having a polarization angle of 90 degrees, and a laminate
thickness limitation of about 32 plies. Lee has also studied the microwave processing
of graphite/epoxy composites using a commercially available microwave oven‘ and
has reported that the microwave curing of multidirectional graphite epoxy composites
was not successful. Using a tunable resonant microwave cavity, Vogelz" has reported
that 7.62 x 7 .62 cm, 24 ply unidirectional graphite ﬁber/epoxy composites can be
successfully cured using microwaves with a relatively low input power. The heating
rate and the cure uniformity of the composite are strongly dependent upon the

electromagnetic processing mode.

For a given epoxy resin, the mechanical properties vary with the extent of
cure. Theoretically, mechanical strength should increase with increasing extent of

cure as the covalent bonds in a crosslinked network are much stronger than the Van

79

80

Der Waals forces in a non-crosslinked structure. In reality, however, the strengths of ‘
thermally cured resins increase with increasing extent of cure only to a certain point
and then begin to fall oft‘“'"""‘. This is attributed to submicroscopic cracks

induced by the internal thermal stresses resulting from resin shrinkage and non-
uniform temperature distribution during cure'“. A similar trend was also observed
for thermally processed graphite ﬁber/epoxy composites“. The strengths of
microwave cured epoxy resins, however, did not exhibit this trend, but rather showed
consistently increasing strength with increasing extent of cure'”.

A tunable microwave resonant cavity operating at 2.45 GHz has been developed
and successfully used to transfer microwave energy efﬁciently into loaded materials
and control the heating process by maintaining a selected resonant electromagnetic
ﬁeld during cure”"‘7'“'-“9. To further demonstrate the feasibility of microwave
processing of graphite ﬁber/epoxy composites in a tunable resonant cavity, this study
focuses on the microwave cure of 24-ply unidirectional and crossply graphite
ﬁber/epoxy composites and the effects of the resonant cavity mode on the mechanical
properties of microwave cured composites. Parallel thermal processing was
conducted for comparison of mechanical properties. The ability to cure relatively
thick, 72 ply crossply and unidirectional laminates is also demonstrated.

4.2 EXPERIMENTS

Microwave curing experiments were performed using a 17 .78 cm diameter
tunable resonant microwave cavity. The microwave processing and diagnostic
system used was the 2.45 GHz system described in Chapter 2. A coaxial switch
(micronetics RSN-2D-I/12V) was used to turn microwave power on and off to
maintain isothermal conditions. This switch was regulated by a dc power supply
(Electronic Measurements HCR-30-8—111) that was controlled by an I/O interface
board (Omega DAS-l6) in a personal computer. Actuation of the on/off switch was

81

in response to a temperature feedback from a fluoroptic temperature sensing system
(Luxtron 750).

Figure 4.1 shows the layup conﬁguration of the composite laminates before
microwave processing. Commercially available, continuous, graphite ﬁber/epoxy
prepreg (Hercules AS4/3501-6) was used for this study. The physical and chemical
properties of this material are available in the literature“. Unidirectional prepreg
tape was cut into 7.62 x 7.62 cm square sheets (about 1.5 grams/ply in weight and
0.21 mm/ply in thickness) and stacked with either uni- or crossply ﬁber orientation.
The composite laminate was then surrounded with an adhesive cork dam to prevent
composite deformation by not allowing excess epoxy resin ﬂow through the edges
during cure. A porous teﬂon release ﬁlm was placed on the top and bottom of the
laminate and several layers of polyester bleeder cloth (one layer of bleeder cloth for
every four plies of composite) were laid below the bottom porous ﬁlm. A nonporous
teﬂon release ﬁlm was then placed on each side.

 

Teflon plate

 

 

 

Nonporous Teflon release film

 

 

Porous Teflon release film

 

Graphite / epoxy composite I 1

 

 

 

Polyester resin bleeder l 7

 

 

 

 

 

Figure 4.1 Composite Layup for Microwave Process

82

The whole composite panel was sealed in a polyamide vacuum bag under a
vacuum pressure of 85 kPa (25.0 in Hg) and was packed between two teﬂon plates
(1.5” in thickness and 5.0" in diameter) to reduce the heat loss during microwave
processing. The bottom Teﬂon plate also lifted the composite laminate to a location
having a stronger electric ﬁeld strength since the tangential electric ﬁeld strength at
the bottom plate of the cavity is zero. The coupling probe was located slightly higher
than the composite laminate. The ﬂuoroptic temperature probes from the temperature
sensing system were placed in contact with the top surface of the composite panel by
insertion through the top teﬂon plate. Glass capillary tubes were used to protect the
probe tips from encasement in the composite. The laminate assembly was then
loaded into the bottom of the microwave cavity for processing. The conﬁguration for
comparative thermal processing was the same as in Figure 4.1 except that no Teflon
plates were used.

4.2.1 Resonant Heating Mode Selection and Maintenance

Low power swept-frequency microwave energy was coupled into the loaded cavity
and was used to locate the resonant heating mode. The swept-frequency power
reflected from the loaded cavity was shown as a curve of power absorption versus
frequency on an X-Y oscilloscope. The point of minimum reﬂected power was
indicative of a resonant mode and therefore called a resonant peak. For an empty
cavity, the point of zero power reﬂection was indicative of a critically coupled
resonant mode, as shown in Figure 4.2. Both over and under coupling resulted in a
non-resonant ﬁeld pattern inside the cavity and exhibited a certain amount of reﬂected
power. When either a unidirectional or crossply composite was loaded into the
cavity, a non-resonant power absorption curve was usually observed on the
oscilloscope as shown in Figure 4.3 at the critical cavity length and coupling probe
depth of an empty cavity mode. However, a resonant power absorption curve with
several resonant peaks could be generated by adjusting the cavity length and coupling

83
probe depth. Most of the resonant heating modes at 2.45 GHz found in the cavity
length range of 9 to 15 cm in the 17.78 cm cavity have been used to process 24 ply
unidirectional and crossply composites. Once the resonant heating mode, either PS,

or CH, or UH, was selected, the low power swept-frequency source was replaced by
a high power, 2.45 GHz single frequency source.

100

 

 

Power absorption (2)
S
l

 

 

2.25 2.40 2.“ 2.60 2.55

Frequency (GHz)

cavitywell

/ c/onpuu 11:! 5

:7 _—1

I. cvercenpled II. critically coupled In. udercenpled

Figure 4.2 Power Absorption Curve of Empty Cavity under
Various Coupling Situation

4.2.2 Microwave Processing and Mechanical Properties Test

The microwave processing and diagnostic system operating at 2.45 GHz was
connected to the loaded cavity after the resonant heating mode was selected. For

84
crossply composite processing, the composite was oriented with the top ply ﬁber
direction perpendicular to the coaxial coupling probe. For unidirectional composite
processing, the composite was oriented with ﬁber direction either perpendicular,
parallel, or 45 ° to the coupling probe. Four top surface temperatures, one at the
center and three evenly distributed at the edges, were measured during the processing.
The distance from edge to center probe, b, is 3 cm. The locations of the
temperature probes and the meaning of the ﬁber orientations are shown in Figure 4.4.

100

 

Power absorption (’8)

 

 

 

I I

2.40 _ 2.45 2.50
Frequency (6112)

Figure 4.3 Typical Non-resonant Power Absorption Curve in
the Composite Loaded Cavity

24-ply crossply composite laminates (about 30 grams each) were processed in
nine different resonant modes. Each composite was processed with an input power
of 60 W, corresponding to a power density of 2 W/g, for 90 minutes at a control
temperature of 160°C. No pressure was applied during the process. The 24-ply
unidirectional composites were processed in the same cavity under the same process

conditions except with a lower input power of 42 W, corresponding to a power
density of 1.5 W/g.

85

    

Compmlite Top Ply Fiber

Direction

Coupling Probe

Fiber Orientation

Figure 4.4 The Locations of Four Temperature Probes and
Meaning of Fiber Orientations

As the complex dielectric properties of the composites were functions of
temperature, extent of cure, and ﬁber volume fraction, the resonant conditions for the
selected mode changed continuously during the cure. The selected resonant heating
mode was maintained during cure by slightly tuning the cavity length and coupling
probe depth to compensate for these changes.

Once the measured temperature reached the control temperature during cure,
the input power was pulsed to maintain isothermal conditions at the control
temperature. Figure 4.5 shows the temperature/location/time proﬁles during
microwave processing of 24-crossply composite in various modes. The uniformity
and completeness of cure achieved during heating was strongly dependent upon the
resonant mode used for processing. In some cases, the samples were partially cured
in one region and burned in another. Uniform cure of the sample was achieved

using modes giving uniform temperature! location/ time proﬁles. The radial electric

86

ﬁeld pattern of the resonant heating mode along the cavity wall in the axial direction
was also measured during the processing. Figure 4.6 shows a typical ﬁeld pattern
obtained during processing of 24-ply crossply composite when a uniform heating
proﬁle presented. The strongest radial electric ﬁeld is at the laminate height. Similar
behavior was observed during microwave processing of unidirectional laminate.

In preparation for measuring the mechanical properties of the ﬁnal product, all
microwave cured composite samples were cut into four pieces, each about 15 mm
wide and 72 mm long. The ﬂexural strength and modulus of each testing coupon
were determined (ASTM D790 3-point ﬂexural test) using an MTS testing machine
with a support span of 64.22 mm, a crosshead rate of 0.06 in/min, and a support-to-
depth ratio of 16. The maximum ﬂexural strength and modulus were calculated

using

 

rs“ =25- (4-1)
2th1
3
m = L m (4-2)
4bd’
where F8.“ and FM“, = maximum ﬂexural strength at midspan and modulus of

elasticity
in bending, respectively, N/m2 (psi),
P = maximum load on the load-deﬂection curve, N (lbi),
L = support span, m (in.),
b and d = width and depth of the coupon tested, m (in.), and
m = slop of the initial straight-line of the load-deﬂection curve, N/m (lbs/in).

Figure 4.7 shows the typical load-deﬂection curves obtained during MTS
testing for the microwave cured crossply composite at L,=9.95 cm. The load
increased with the displacement linearly at ﬁrst. The coupon usually experienced
some local failures before the ﬁnal failure.

Team (°C)

Temperature (°C)

87

 

 

 

 

100

 

 

 

 

100
4 O n
1“ ‘ 0 1'2 Duonaaomoaonuauaauaaoaomao
d A T, U '
100 1 V 7‘ can a o
4 DD OOOOOOOOOOOOOOOOOOOOOOOO 0°
‘20‘ D 00°
. o oo 9
100 - a o 0666 0 000660
, U000 8“.¢.6.6 06° 6°
00 ' 90 vaXX
‘ vvaA
'0 ‘ °8 Vvv: A arson
V
,0 q va 4° who 15.22am
£82,344 Input Power: sow
20 in A l r l I T I
0 20 40 00 00
The (urinates)
Figure 4.5(a) PS mode at L,=9.l9 cm
100
a O n
100 ~ 13 1': 000000
1“ " 000°C anuﬂm:
. 0°00 000
120 ~ 000° Bananao
. o
100 q 00° 0000
o ODD
. 00° 000
so - 009 can
go - 0° DD
. 0°00” c1 mun
0° :1 Comm: 1820mm
‘9 ‘ 880:1“ Input Power: sow
20 . . . . . 1 . t
0 20 40 60 80
The (mlnutee)

Figure 4.5(b) UH mode at L,=9.40 cm

100

1'

Temperature (CC)

88

 

 

 

 

 

 

 

 

100
q 0 n 7
100 - a 1'2 vvvvvvvvvvvvavVvvvavVvav
- A 1'2 ,,I7‘7
300‘“ V T‘ vv
V
‘ V
120 . vv WEB! oeaeee
4 v 0000 Wagons“
1oo - V °
. V 8
v 8
so . v 8888
so : V 933
v 0.41m
v 38 Wtuﬂm
‘0 ‘ V938 Input Power. sow
20 4° . . . . . . . . r .
0 20 ‘0 20 20 100
'l'lmetmlnutee)
Figure 4.5(c) PS mode at L,=9.47 cm
200 0
. o 1,,
1.0 ' U 72 00°00
' A 1'3 0" °
100 r v 1'4 0 000000
1 00° 001200006
14o - 06 no“ chum“
‘ 0° DOD
120‘ 0° no
. 09 DD
100 7 00° 0° vvvvvaVvvvvvv
‘ O 0 If” A AAAA
00 ‘ o°° on v va AAAAAAAA AAA
. 000 D vVvvaAAAAA
5° ‘ a an vaVV VAAAAA
A 033cm
0° 983XXAMAAA who ﬂatworm
so 00,, Input Power: cow
5
20 . x . . . 4 . .
0 20 40 00 20 100
Tlme (minutes)

Figure 4.5(d) CH mode at L,=9.83 cm

Temperature (°C)

Temperature (°C)

89

 

 

 

 

 

 

 

 

100
. 0 T1
160 r U T2 0
. A 13 0° OOOOV
140 - ‘7 7‘ 00'300XXXXXXA
0° :83
J ooxxlbwnmﬂm
120 ~ .2339 53000
. ' DD
100 - Bglguﬂu
. 8 u
83 a°
so - gun”
. ,xgﬁ
‘03
00 r 0039
'3 8:..“11. 202mm
40 ‘ AA: Input Pngwenoow
. A8
20 ' I I I I I I I
o 20 40 so so too
Tlme(mlnutee)
Figure 4.5(e) CH—PM mode at L,=9.95 cm
1so F
. 0 T1
180 - a T2 AAAAAAAAAAAAAAAAAAAAAAAA
‘ A 73 AA“ antenna
140 - V 1'4 AAA DD DSSDODuggggaunnﬂﬂﬂ
A a
4 A U0
Cl
12o - 1“ on” W ooooooooo°°°
. A D0 vv 000000
A :1 v 00°
10° " D O o
. AA 0 Vv 0°00
:1
.0 - A D vv 000
. U v 00
so - 1 va 09°
. I3v 0 0. Length: 1 m
Dv 0° couvwlng Probe. w23.54mm
‘0 ‘ A1:1v o° putlnPower:oo
. V00
20 "'32 . l I I I
0 20 so 60 00 100
Tlme(mlnutee)

Figure 4.5(1) CH mode at L,=10.08 cm

(W

Temperature (°C)

Temperature (06)

 

 

 

 

100

 

 

 

100
. 0 n
100 - a T2 AAAAAAAAAAAAAA
4 A T: AA
140 . V T4 AAA
00
120 '1 AA 0
. AA 00°
1011 - AAA 0000 8
A 0° 8383
A 0
no « 1 03.4!“
. A o
A 030'
.0 ‘ AA 886 Imam
so 1 1A .938 lama-72mm
. A.|a mu“ "m' "I"
20 ‘M I I I r r
0 20 40 00 00
The(mlnutee)
Figure 4.5(g) CH mode at L,=l3.69 cm
100
100 ~ 1:) T2 A AAA AAA,“ 0°°°°°°°
4 3 3 AAA AAAAAAgoogAAAAAA A
140 - 0° 1:1
. AA 00°00 0000vaij
A 0 a v v WV
120 “ A DODvVV vva'
‘ . 09°68‘39“"
100 r A gaggovv
A 838
00 ~ A 088
A 0°“
so 4 Aoo°e°
o 99 Ca Length: 13.72cm
6 9 Inc Pmbcmemm
‘0 “£399 Input Power. sow
20 I I I I I 1 v
0 20 40 60 .0
Tlme (minutes)

Figure 4.5(h) CH mode at L,=13.72 cm

 

100

91

 

 

 

 

100 3,
100 ~ AAA A AAAAAAAAAAAAAAAAAAAAAAAAA
. AA AAAAAA
AAA on
1“ "‘ on W
v
0 v
1 on vvv
1W 7 V 000
. A a vvvv" oO°°°°°°°
r- lnv‘7 00°00
:1" v 00°
.0 -I A V O
V 0° 0 1'1
. a v 000 Immature
a ,8 A T2 InputPowerzoow
‘ :1 V T4
so 48" . . . . .
0 20 40 00 00 100
multitudes)

Figure 4.5(i) PS mode at L,=l4.13 cm
Figure 4.5 Temperature/position/ time Proﬁle during Microwave Processing
of 24 Crossply AS4/3501-6 Composites at Various Heating Modes.

 

I A l

 

T 1 f I I

 

1 r 1 r I r

coupllntutooehel hearsem J
Campos

 

.l J 1 I a 4 l l 4

 

00 0.00 0.12 0.10 0.2

0.30 0.36 0.42 0.40 0.54 0.60
DlspnotlePoweHmVl)

Figure 4.6 Typical Radial Electric Field Strength Distribution Along Axial Position

 

 

 

 

 

5. M
£2 ‘.-,_ ‘31
We
4. m P- . ,4”;
l /
0 3M L. ,/
a f
1 2.00 *-
g if: f,
ls m _ If”
0.“ l ' '
ON 0.50 LN 1.50 Elli“
displ. inch

Figure 4.7 Load Versus Deﬂection Curves for 3epoint Bending Test

For the purpose of comparison, 24—ply unidirectional and crossply laminates
were processed thermally. Because we were not able to apply pressure during
microwave processing, unpressurized thermal cures in the thermal oven (Fisher, 200
Series) were carried out for comparison. The oven temperature was dynamically
controlled so that a similar sample temperature proﬁle was obtained as compared to
that of microwave cure with ﬁnal temperatures of 160°C and 180°C at the end of 90
minutes. To compare our unpressurized microwave processed composites with the
thermally processed composite used in industry, the autoclave (United McGill) was
also used to process 24 ply unidirectional and crossply AS4/3501-6 composite using
the manufacturer’s suggested curing cycle. The mechanical properties of thermally
processed composite were determined using the same method as used for the

microwave processed samples.

In order to examine the bonding between the matrix and the ﬁber as function
of the extent of cure for microwave processed composite, 24-ply unidirectional

93

composite was processed under various conditions to obtain composites of various
levels of cure. Each processed sample was cut into a coupon with a length to
thickness ratio of 6 and width of 0.64 cm for short-beam interlaminar shear testing
(ASTM D2344-84). A support span to thickness ratio of 4 and a crosshead rate of
1.3 mm/min. were used in each test. The samples were subjected to 3-point bending
until delamination occurred. A small amount (10-20 mg) of each coupon was
removed from the delamination surface and tested for extent of cure in a differential
scanning calorimeter (DSC, DuPont 9900). The same test for an autoclave processed
composite using manufacture’s cure cycle was also conducted for comparison. The
failure surfaces of those samples, both thermally and microwave processed, were
photographed using a scanning electron microscope (SEM, JEOL ISM T330).

To demonstrate the ability to cure a relatively thick section graphite ﬁber/epoxy
composite laminate in the tunable resonant cavity, 72-ply, 7.62 x 7.62 cm
unidirectional and crossply composite laminates were processed. The modes were
selected using the criteria established based on the experimental results from
microwave processing of 24-ply laminates. The samples were heated for about 75
minutes with at input power of 75 W. Top surface and midplane temperatures were
measured during cure. Figure 4.8 shows the temperature/time proﬁles during
microwave processing of 72-ply unidirectional composite. The cured samples
appeared solid and uniform. No mechanical properties of the thick composites were
measured due to the size restriction. However, the extent of cure distribution across
the thickness of the composite was determined using DSC. As shown in Figure 4.9,
the microwave processed composite was uniform in cure extent across the sample.
Using the heating proﬁle obtained during microwave processing of 72-ply
unidirectional composite as oven temperature, a 72-ply unidirectional composite was
processed thermally. The heating cycle was a ramp to 170°C in 45 minutes then hold
at 170°C for 30 minutes. Both surface and midplane temperatures were measured

and the results are shown in Figure 4.10.

94

 

 

 

 

 

180 Cavity Length: 13.7 cm' '1._"_,..--.,
‘ Input power: 75 W rev
1601 xtxxxmﬂ‘“
+ .
140~ I”
a - 4’
g, 1204 '
E .
3 100- /
g 80-
% .. , 2’
a) 60- /
,_ .
404 ./
4 ’ _ H T1 Center
20 H 12 Top
a—a T3 TOP
0 [ 1 ' I r ﬂ '
o 20 40 6° 80

Time(minutes)
Figure 4. 8 Temperature! time Proﬁles during Microwave Processing of 72-ply
Unidirectional AS4/3501-6 Composite at a PM Mode.

 

 

 

 

 

 

 

 

 

 

8y DSC
0.5-1
X-0.927 . X-0.854 -
’3 0.4-4
9
8
c
C, 0.34
m x-0.875 . x-O.899
8
E 0.2-1
.2
E
0.1 4 X80373 . X-0.914
0.0 - 1 . I
0.0 1.0 2.0 3.0

Width (Inches)
Figure 4.9 Spatial Distribution of the Extent of Cure in the Cross Section of

the Microwave Cured 72-ply Unidirectional Composite.

95

 

   
 
 

 

 

 

20G

- o—o aurfoee Temp.

~ H Interior Temp.
A 150-
$) .
V
g
*6 1001
L-
0
D.
E .
t" 501

G [ I T j T ' j '
O 20 40 60 80

Time (minutes)
Figure 4.10 Temperature/time Proﬁles during Thermal Processing of 72-ply
Unidirectional AS4/ 3501-6 Composite

4.3 Results and Discussion

24-ply crossply AS4/3501-6 composites have been cured in nine different
resonant heating modes using 2.45 GHz radiation. Each resonant heating mode has
its own electric ﬁeld strength and pattern with a corresponding cavity length and
coupling probe insertion length. As shown in Figure 4.5, the temperature uniformity
is a strong function of the resonant heating mode. The temperature proﬁles in all PS
modes (a,c,i) are not uniform. Most temperature proﬁles in CH modes are not
uniform either except 4.6(b) and (e). Flexural properties of these microwave cured
crossply coupons are listed in Table 4.1. Table 4.1 listed the location of the
electromagnetic heating mode, thickness, ﬂexural strength and modulus of microwave
processed composites. In column 2 to 4, the number in front of the sign :1: is the

96

average value obtained from four coupons which were cut from the same composite
sample. The number after the sign :1; is the maximum deviation from the averaged
value. The composite thickness is ranging from 3.6 to 4.4 mm. The processed
composites is fairly uniform in thickness among the four coupons as the maximum

deviation is within 0.23mm.

As listed in Table 4.1, composites cured in a PS resonant heating mode (such
as resonant cavity lengths of 9.19 cm and 14.13 cm) had relatively low ﬂexural
properties due to non-uniform curing. Although samples showing the highest
flexural properties were cured in CH modes, not all crossply composites cured in CH
modes had high ﬂexural properties. The highest ﬂexural properties were measured
for samples cured in CH modes at 9.95 and 9.40 cm eavity lengths, while the lowest
measured values were recorded for a sample cured in a CH mode at 9.83 cm. Given
the fact that successful cure of crossply composites depends strongly on the mode
chosen, and that selection of a CH mode does not always result in a successful cure,
the CH modes resulting in high ﬂexural properties are designated as process modes
(PM) and the CH modes resulting in low ﬂexural properties are designated as non-
process modes (NPM). For example, the composite cured in the CH mode with a
resonant cavity length of LC =9.95 cm has relatively high ﬂexural prOperties and
therefore this mode is called a PM. The composite cured in the CH mode with
resonant cavity length of I.,=9.83 cm has relatively low ﬂexural properties so this
mode is called a NPM.

Figures 4.5(b) and 4.5(d-h) show the temperature/location/time proﬁles on the
surface of samples during cure in CH modes. Based on the above deﬁnition of PM
and NPM and ﬂexural properties listed in Table 4.1, Figure 4.5(b) and (d) show
proﬁles corresponding to PM and the others show proﬁles corresponding to NPM.
Clearly, the surface temperatures are more uniform in the PM than in the NPM.

This may due to a more uniform electric ﬁeld inside the composite in a PM than in a

NPM. The experimental measurement of electric ﬁeld during processing shows that

97

the identiﬁeation of PM and NPM can be made from the radial electric ﬁeld strength
distribution along the axial direction. If the radial electric ﬁeld strength distribution
along the axial position of the CH mode has the shape of Figure 4.6 and the
maximum radial electric ﬁeld strength is loeated at the axial position of the
composite, then the controlled-hybrid mode is a PM. Figure 4.5(a,c,i) show the
temperamre/loeation/time proﬁles on the surface of samples during cure in the three
PS modes. The temperature is nonuniform on the surface in all eases. The more
uniform temperature distribution during cure, the higha the flexural properties as
shown by relating Figure 4.5 to Table 4.1.

The rheology of the epoxy matrix and void formation mechanism during
microwave processing are very complicated. These complexity lead to a nonuniform
distribution of void, therefore, the flexural strength of microwave processed
composites, even in the composites processed in the PM modes. The max.
variation/averaged value ratios of flexural strength for composite processed in the PM
modes at L,=9.4O cm and L,=9.95 cm are 15% and 20%, respectively. To process
composites with a uniform mechanical property, a fully understanding of rheology
during microwave processing is required.

98
Table 4.1 Flexural Properties of Microwave Processed 24-crossply

   

 

 

 

AS4/3501-6 Composite
—
Cavity Length Composite Flexural Flexural
(cm) Thickness Strength ' Modulus
(mm) (MP8) (GPa)

I 9.19(PS) 4.09 i 0.04 125.34 i 18.08 33.93 i 1.75
9.40(CH-PM) 3.99 i 0.02 685.60 :1: 103.90 44.51 :1; 1.14
9.47(PS) 4.18 i 0.11 127.92 :1; 70.39 26.23 :1; 11.05

 

r 9.83(CH-NPM) 4.43 i 0.14 87.42 i 52.48 13.96 :1; 7.68
9.95(CH-PM) 4.35 i 0.18 724.91 i 143.73 40.46 i 4.29
10.08(CH-NPM) 3.64 i 0.03 269.30 1 239.06 43.85 a: 5.04
13.69(CH-NPM) 4.23 i 0.17 228.21 i 50.91 35.74 i 5.36
13.72(CH-NPM) 4.39 i 0.11 189.33 4; 27.15 29.46 i 5.01
14.13(PS) 4.09 i 0.23 135.54 i 234 31.80 5: 7.43 I

 

 

 

 

 

 

 

 

 

 

 

Unidirectional AS4/ 3501-6 composites of 24 plies were processed in the same
eavity at three fiber orientations, namely 0, 45, and 90 degrees, with respect to the
coupling probe. Figures 4.11, 4.12 and 4.13 show the temperature proﬁles of the
three orientations in their processing modes. Apparently, the rate of heating
depended on sample ﬁber orientation with respect to the coupling probe. Samples
oriented parallel to the probe heated the slowest, as shown in Figure 4.11. Parallel
oriented samples reached the set point temperature at times greater than 90 minutes.
Samples oriented 45 degrees with respect to the probe reached the set point within 45
minutes, as shown in Figure 4.12. Samples oriented perpendicular to the probe
heated the fastest, reaching the set point within 15 minutes, as shown in Figure 4.13.
A maximum curing rate for samples oriented perpendicular to the probe agrees with
previous studies. Lee and Springer” have found that unidirectional samples cured
fastest when oriented perpendicular to a plane polarized wave in a waveguide. Chen
and Lee37 have shown that heating rates in a resonant cavity for cured unidirectional

99

carbon ﬁber/epoxy samples were faster for samples heated perpendicular to the
coupling probe than for those oriented parallel. Temperature uniformity upon
reaching the set-point temperature also depended on the sample orientation. The 45
degree-oriented sample maintained the most uniform thermal proﬁle across the
specimen, as shown in Figure 4.12. The dependence of heating rate and
temperature uniformity on ﬁber orientation conﬁrms that the coupling of
electromagnetic energy into a carbon ﬁber composite is affected by the ﬁber
orientation as reported previously"""7. Coupling of microwave energy into the
anisotropic composite is dependent upon the relative orientations of the ﬁber axes and
electromagnetic ﬁeld, as reﬂected by heating rate. No attempt was made to optimize
the curing and to maximize the heating rate. The objective of this study was to
identify the important parameters for the processing of graphite ﬁber/epoxy

composites in a microwave cavity.

Table 4.2 lists the mechanieal properties of microwave processed 24 ply
unidirectional composites oriented at 0, 45 , and 90 degrees with respect to the
coupling probe. The highest flexural strength was obtained when the composite was
oriented at 45 degrees. The experimental results listed in Table 4.1 and 4.2 clearly
show that 24-p1y unidirectional and crossply graphite ﬁber/epoxy composites can be
processed using resonant standing microwaves in a tunable resonant cavity. The
ﬂexural properties of the cured sample are strongly dependent on the processing mode
and higher ﬂexural properties were achieved using a PM. The highest ﬂexural
properties measured for microwave cured 24-ply crossply composite are compared
with those of 24-ply unidirectional composites cured with ﬁber direction either
perpendicular or parallel to the coaxial coupling probe for the same heating time at a
lower input power density of 1.5 W/g, as listed in Table 4.3. Table 4.3 shows that
the ﬂexural properties of the microwave cured crossply sample fall between those of
the best unidirectional samples cured perpendicular and parallel to the probe. A
comparison of temperature/time proﬁles at the surface of unidirectional and crossply

composites during processing in their PM is shown in Figure 4.14. Figure 4.14

100

indieates that crossply composites require a higher input power density than
unidirectional composites to obtain a similar temperature/time proﬁle.

Table 4.2. Flexural Properties of Microwave Cured 24-ply Unidirectional
AS4/3501-6 Composites at Various Orientations and Modes

 

 

Sample Cavity length Cured Laminate Flexural Strength Flexural Modulus

Orientation (cm) Thickness (mm) (MPa) (GPa)

0° 9.46 4.58 :1; 0.09 615.47 1 72.10 30.37 :I: 3.71
9.61 3.87 :t 0.21 415.73 :1: 169.16 34.92 :1: 11.15
14.60 4.82 :1: 0.14 389.75 :1; 44.68 33.64 :1: 4.69

45° 14.05 3.65 :l: 0.07 1316.79 :1: 67.58 100.73 :1: 2.44
14.41 3.84 :1: 0.18 857.63 :1: 215.92 86.98 :1; 10.69
17.77 3.74 :1; 0.07 411.65 :1: 91.15 71.26 :1: 26.60
17.16 4.05 d: 0.41 970.50 :1: 342.19 83.47 :1; 19.48
17.10 3.83 i 0.12 763.14 :1: 158.48 83.30 :1: 10.16
17.08 3.61 i 0.08 1136.07 :1: 123.81 99.97 :1: 6.04

90° 8.66 3.42 :1; 0.14 851.28 :1; 208.78 59.30 :1: 3.93
9.01 3.17 :1: 0.04 944.56 1 183.14 74.53 i 11.97
13.60 3.30 :1; 0.13 769.72 :1; 76.22 60.89 :I: 10.57
9.03 4.31 :1: 0.10 838.64 1; 173.03 58.24 :1; 7.05
9.17 5.05 i 0.49 389.11 1 142.07 33.07 1 12.76

 

101

 

 

 

 

 

 

 

 

 

 

1
180-
. ”"’:;,:’
A ’,”’
8 1404 ”x; "
e/
m ,4;
S ,/
45 . ,/
‘- 100— ,4'
8 . 4’
E . 44’
.2 x,"
601 /’/, — T4 Right
‘ // / - rs Bock
' 3” -- T2 Front
V -— T1 Center
20 ...,.4.,...,...,...
20 40 60 80 100
Time (min)
Figure 4.11 Temperature/time Proﬁle During Microwave Cure of
24 ply Unidirectional AS4/ 3501-6 Composite at Fiber Orientation of
0° with respect to the Coupling Probe
-.-_ . .....__-_1,
A J
8 1204
D .
L.
= i
'H J
E
q, 80“
O. 4
E .
Q)
l- '1
40-1
J I - 73: Right
‘ -- T2: Center
0 -— T1: Left
0 ' ' 20 4‘0 6'0 ab 100

Time (minutes)
Figure 4.12 Temperature/ time Proﬁle During Microwave Cure of

24 ply Unidirectional AS4/ 3501-6 Composite at Fiber Orientation of
45° with respect to the Coupling Probe

 

Temperature (C)

 

 

 

 

- T3: Right
-- T2: Center
-— T1: Left
or,,.T...,f.rI...r...
O 20 . 4O 60 80 100

Time (minutes)
Figure 4.13 Temperature/time Proﬁle During Microwave Cure of

24 ply Unidirectional AS4/ 3501-6 Composite at Fiber Orientation of
90° with respect to the Coupling Probe

 

J o—e Perpendicular (Le-8.66 cm. 1 W/Q)
1 H parallel (Le-9.46 em. 1 w/Q)
150- H crossply (Lc=9.95 cm. 2 W/Q)

 
 
  
 

Temperature (°C)

Lc: Resonant cavity length

 

 

 

O I j— ' I ' 1 y I I I r I t I V I ' I

O 10 20 30 40 5O 60 7O 80 90
Time (minutes)

Figure 4.14 Comparison of the Temperature! time Proﬁles During Microwave
Cure of 24-ply AS4/3501-6 Composites

103

Table 4.3: Comparison of Flexural Properties between Microwave and Thermally

Cured 24-ply AS4/3501-6 Composites

 

 

Composites Lc (cm) Thickness Strength Modulus
for mw cure (mm) (MPa) (GPa)
lFinal Temp.(°C)
for th cure
Crossply
Microwave 9.40 3.99 :1: 0.02 685.6 :1: 103.9 44.51 :1: 1.14
9.95 4.35 i 0.18 724.9 :1: 143.73 40.46 :t 4.29
Thermal“ 160 --- ..... .....
180 3.53 i 0.05 313.7 1: 55.37 40.51 :1: 3.70
Autoclave" 2.99 :1; 0.03 817.4 1 70.3 62.17 :1; 1.36
Unidirectional
Microwave
Parallel 9.46 4.58 :1: 0.10 615.5 :1; 72.1 30.37 :1; 3.71
9.61 3.87 :1; 0.21 415.7 :1; 169.2 34.92 1 11.14
Perpendicular
8.66 3.42 1; 0.14 851.3 :1: 208.8 59.30 :1: 3.92
9.01 3.17 i 0.04 944.6 :1; 183.1 74.52 :1; 11.97
45 degree
14.05 3.65 :1; 0.07 1316.8 :1: 67.58 100.73 :1; 2.44
17.08 3.61 :1; 0.08 1136.1 :1; 123.81 99.97 i 6.04
Thermal" 160 3.33 :1; 0.08 165.6 :I: 27.8 39.8 :1: 20.23
180 3.74 :t 0.33 789.6 :1: 107.57 63.3 :1: 15.69
Autoclave" 2.80 :1; 0.05 1320.9 :1: 25.58 123.49 :1: 3.03

 

** Using manufacture suggested cure cycle (4 hours and pressure of 100 psig).

* Dynamic heating to the ﬁnal temperature in 90 minutes.

The unpressurized thermal processing of both unidirectional and crossply 24-

ply Hercules AS4/3501-6 composite was also conducted for comparison. Table 4.3

104

listed the ﬂexural properties of both microwave and thermal processed composites.
With a ﬁnal temperature of 160°C, the unpressurized thermally cured crossply
composites were too soft to test, the flexural strengths of thermally cured
unidirectional composites were much lower than those of microwave cured samples
(3-8 times); and the ﬂexural moduli of thermally cured unidirectional composites were
comparable to those of parallel (with respect to coupling probe) microwave cured
unidirectional samples but lower than the other two ﬁber orientations. This may be
due to the low extent of cure in the thermally cured composite. This result indirectly
implies that the polymerization rate of Hercules 3501—6 resin is faster during
microwave cure than thermal cure. In order to obtain a testable crossply composite,
the experiment was repeated with a ﬁnal temperature of 180°C instead of 160°C.
With a ﬁnal temperature of 180°C, the ﬂexural strength of thermally cured crossply
samples was about half of the value for the microwave cured samples and the flexural
moduli were comparable to those of microwave cured crossply samples. The
ﬂexural strengths of thermally cured unidirectional composites were between parallel
and perpendicular oriented microwave cured unidirectional samples and the ﬂexural
moduli were comparable to those of perpendicular microwave cured samples. The
ﬂexural properties of the microwave cured samples (without pressure) are also
compared to those of autoclave cured (with 100 psig pressure) samples. The ﬂexural
moduli of unpressurized microwave processed crossply and unidirectional composites
(0,45 , and 90 degrees) are lower than those of pressurized autoclave cured samples.
The ﬂexural strengths of unpressurized microwave cured 24 ply crossply, parallel
oriented, and perpendicular oriented unidirectional composites were lower than those
of autoclave cured samples. The unpressurized microwave cured unidirectional 24
ply composites had a comparable ﬂexural strength as pressurized autoclave cured
samples when they were cured at 45 degrees to the coupling probe. The lower
flexural properties of microwave cured samples may be due to the higher void content
in the unpressured microwave cured samples than pressured thermally cured samples.

105

Figure 4.15 shows the extent of cure effects on the apparent interlaminar shear
strength (ISS) of the microwave processed 24-ply unidirectional AS4/3501-6
composites. ISS increases with increasing extent of cure, the same trend as for the
microwave cured pure epoxy resins‘ ". The ISS of fully cured microwave processed
unidirectional composites is slightly lower than that of autoclave processed samples
because a signiﬁcant amount of voids existed in the microwave processed composites

due to the lack of pressure during processing.

 

 

 

 

ouwosomrmrmmwmmrm
extentofeure

 

Figure 4.15 Interlaminar Shear Strength of Microwave Processed 24-ply
Unidirectional AS4/3501-6 Composites at Various Extent of Cure

Figure 4.16 shows SEM photographs of the delaminated surfaces from the
short-beam shear test of both microwave and autoclave processed 24-ply
unidirectional AS4/3501-6 composites. Figures 4.16(a) and 4.16(b) are the SEM
photographs of the delaminated surface for microwave processed samples having high
and low extents of cure. For the sample having a high extent of cure, the
delamination was mainly due to matrix failure, as shown in Figure 4. 16(a). A close
look at the surface of the failed ﬁbers of this sample revealed that the pulled ﬁber was
coated by a layer of matrix about 2 microns thick as shown in Figure 4.16(c). The

106

shear failure modes for microwave processed samples of lower extent of cure were a
combination of both matrix failure and interfacial failure as revealed by Figure
4.16(b). For the autoclaved processed sample, the sample failed in a combination of
interfacial and matrix failure mode as indieated by matrix debris in Figure 4.16(d).
This result agrees with previous result‘ that a better bonding between graphite ﬁber
and epoxy can be achieved in microwave processing than in thermal processing.
Further more, the failure mode in the microwave processed samples of low extent of
cure is similar to that of thermally processed samples.

As a further demonstration of the feasibility of microwave processing of graphite
ﬁber/epoxy composite in a resonant cavity, 72-ply unidirectional and crossply
laminates were successfully processed in their PM. The PM was located using
selection criteria established based on the experimental results during microwave
processing of 24-ply laminates. The crossply sample was processed with the top ply
ﬁber direction perpendicular to the coupling probe at a resonant cavity length of 13.7
cm and the unidirectional sample was processed with its ﬁber direction perpendicular
to coupling probe at a resonant cavity length of 10.1 cm. It is interesting to note that
the midplane temperature T1 is always higher than the surface temperatures T2 and
T3 during microwave processing as shown in Figure 4.8. The uniform surface
temperature shown in Figure 4. 8 strongly suggests a uniform power absorption rate
around the surface area. The higher center temperature as compared to the surface
temperatures may be due to the surface heat loss or the combination of surface heat
loss and a higher power absorption rate in the center. Figure 4.8 suggests that a
signiﬁcant electromagnetic ﬁeld exists in the interior of the composite. As shown in
Figure 4.9, the cure distribution was quite uniform across the thickness of the
composite. Comparing Figure 4.10 to Figure 4.8, the advantage of microwave
processing of thermoset composites over thermal processing is quite clear. The
temperature excursion was eliminated during microwave processing by using pulsed
microwave power input and the temperature was maintained at the desired value. The
heat up time can be shortened by using higher input power if it is necessary.

107

*4: it: an ION ﬁne WQIQHAﬂ'ﬂ§,__

IOKU X1 508

(a) Fully cured (b) 58% cured
Microwave processed, ISS=7379 PSI Microwave processed, ISS= 1329 PSI

   

X5. 888

 

((1) Fully cured
Close look of pull-out ﬁber of (a) Autoclave processed, ISS=8430 PSI

Figure 4.16. SEM Pictures of Delaminated Surface from SBS Test for Microwave and
Autoclave Processed Unidirectional AS4/3501-6 Composites.

108

The advantages of tunable resonant cavity over other microwave applicator
were fully demonstrated in the processing of graphite ﬁber composites. While only
unidirectional graphite ﬁber/epoxy composite of 32 plies or less can be processed in
waveguide or multimode microwave oven, both unidirectional and crossply 72-ply
composites can be processed in the tunable cavity. When a graphite ﬁber/epoxy
composite is processed in a commercial microwave oven, a ﬁxed ﬁeld pattern and
strength distribution are obtained as the location of the composite inside the oven and
the processing frequency are selected. Usually the input power can not be
transferred into the composite efﬁciently due to impedance mismatch and the
composite can not be uniformly cured due to the nonuniform ﬁeld inside the
composite. Compared to commercial untunable microwave ovens, the tunable
resonant cavity offers several advantages in processing of composites, including (1)
the ability to obtain a resonant standing electromagnetic ﬁeld for unidirectional and
crossply graphite ﬁber/epoxy composite to transfer input energy efﬁciently into the
composite, (2) the ability to select an appropriate resonant heating mode to heat the
composite uniformly, and (3) the ability to maintain the selected resonant heating
mode to prevent loss of resonance as the material permittivity changes during the
heating.

4.4 Conclusion

This study demonstrates that it is feasible to process continuous graphite
ﬁber/epoxy composite materials in a tunable resonant microwave cavity.
Unidirectional and crossply graphite ﬁber/epoxy composite laminates consisting of 24
plies have been processed in a 17.78 cm tunable electromagnetic resonant cavity using
2.45 GHz microwave radiation. No pressure was applied during microwave
processing. Temperature distributions in the composite were measured during
processing using ﬂuoroptic sensors. The ﬂexural properties of the processed
composites were determined using a 3-point bending test. Temperature uniformity,

109

cure uniformity, and the ﬂexural properties of the processed samples were shown to
be strong functions of the processing mode. The ﬂexural properties of the
microwave processed unidirectional composites were also affected by the ﬁber
orientation of the composites with respect to the coupling probe. The maximum
ﬂexural properties of unidirectional composites were observed when the sample was
processed at an orientation of 45 degrees to the coupling probe. The ﬂexural
properties of microwave processed crossply samples have been found to be between
those values measured for microwave processed unidirectional laminates cured with
ﬁber orientations perpendicular and parallel to the coaxial coupling probe.
Unpressurized thermal cure in an ordinary thermal oven and pressurized thermal cure
in an autoclave of 24-ply composites have been carried out for comparison. The
ﬂexural strengths of the unpressurized thermally cured unidirectional and crossply
composites were much lower than those of unpressurized microwave cured samples
under similar heating conditions. The microwave processed 24-ply unidirectional
composites have comparable ﬂexural strength but lower ﬂexural modulus when
compared to that of thermally processed samples (with pressure) in the autoclave with
manufacturer’s suggested cure cycle. Both ﬂexural strength and ﬂexural modulus of
microwave processed 24 ply crossply samples were lower than those of autoclave
processed samples. The processing results show only a small decrease in ﬂexural
properties for unpressurized microwave cured composites as compared to those of
pressurized thermally cured samples although a short processing time (90 minutes)
and no pressure was applied in microwave cure while a long processing time (4
hours) and a pressure of 100 psig were applied during thermal processing in the
autoclave. Selection of the proper resonant made, however, has been shown to be a
critical factor in achieving high ﬂexural properties of microwave processed composite.
The concept of a processing mode (PM) is deﬁned based on the ﬂexural properties of
the processed composites. The criteria deﬁning a processing mode were obtained
from a database of types of processing modes, radial ﬁeld patterns during processing,
and the ﬂexural properties of processed composites. Using modes selected based on

l 10
the above criteria, 72-ply unidirectional and crossply composites have been
successfully cured with a uniform spatial distribution of the extent of cure.

The extent of cure effects on the mechanical properties of microwave
processed graphite ﬁber/epoxy composite were also studied using 24-p1y
unidirectional AS4/3501-6 composite. Apparent interlaminar shear strengths (ISS) of
microwave processed samples were determined using a short-beam method (ASTM
D2344-84). ISS of microwave processed composite increased with increasing extent
of cure. Scanning electron microscope analysis of the fracture surfaces of both
microwave and thermally processed composites revealed that graphite ﬁber/epoxy
bonding was higher in microwave processed composites than in thermally processed
samples. Microwave processed composites failed in the matrix failure mode for
samples of high extent of cure and in a combination of interfacial and matrix failure
modes for samples of low extent of cure. The autoclave processed samples failed in a

combination of interfacial and matrix failure modes.

CHAPTER 5
FIBER ORIENTATION EFFECTS ON THE MICROWAVE HEATING
OF CONTINUOUS GRAPHITE FIBER/EPOXY COMPOSITES

5 .1. Introduction

Chapter 4 demonstrated the feasibility of processing thin-section unidirectional
and crossply graphite ﬁber/epoxy composites using microwave energy in a tunable
resonant cavity. The results in chapter 4 showed that temperature uniformity is the
key parameter to obtain uniformly cured composites and therefore a composite of high
mechanical strength. This chapter discusses the microwave heating characteristics of
a fully cured thick-section crossply graphite ﬁber/epoxy composite at various ﬁber

orientations.

In the microwave processing of composites, the heating proﬁle and distribution
depend upon the electromagnetic ﬁeld pattern inside the sample. A uniform electric
ﬁeld inside the composite will result in a uniform temperature distribution and a high
electric ﬁeld strength will lead to a fast heating rate during processing. For a given
composite size and input microwave power of a given frequency, the electromagnetic
ﬁeld pattern depends on the cavity length, coupling probe length, ﬁber orientation,
and sample location. The key objective in the microwave processing of graphite
ﬁber/epoxy composite is, therefore, to experimentally ﬁnd a proper cavity length,
coupling probe length, composite location, and ﬁber orientation such that a uniform
electric ﬁeld pattern will be resulted inside the composite and all the input power will
be focused on the composite. In this study, a fully cured 3.8cm thick crossply
graphite ﬁber/epoxy composite was heated in a 17 .8cm tunable resonant cavity at ﬁve
ﬁber orientations. The effects of ﬁber orientation on microwave heating proﬁles were
studied.

111

112
5.2. Experiments

Microwave heating experiments were performed using a thermally cured
7 .6x7.6 x3.8cm crossply graphite ﬁber/epoxy composite (Hercules AS4/ 3501-6). A
teflon disk, 10.2cm in diameter and 3.8cm thick, was placed on both on the top and
bottom of the sample to prevent heat loss and to lift the sample from the bottom of
the cavity. The whole setup was placed at the center of the bottom plate in a 17.8cm
tunable microwave resonant cavity. A series of runs was conducted with top ﬁber
orientations of 0, 15, 45, 75, and 90 degrees with respect to the coupling probe. The
microwave applicator and accompanying diagnostic and control system were the same
as used in Chapter 4. Four surface and four midplane temperatures were measured.
The probe locations were the same for midplane and surface temperature
measurement, as shown in Figure 4.4. Five resonant heating modes were found for
each ﬁber orientation and each mode was used to heat the sample using an input
power of 60W. The type of each resonant heating mode was determined based on the
power absorption curve on the oscilloscope as discussed in Chapter 2. The heating
was allowed to continue at constant input power until one temperature, either on the
surface or in the midplane, reached 160°C. In addition, a fresh 7.6x7.6cm 200—ply
crossply sample was processed in the microwave cavity with a 15 ° ﬁber orientation at
a cavity length of 13.22cm and a coupling probe depth of 21.98mm.

5.3. Results and Discussion

A fully cured 3.8 cm thick crossply AS4/3501-6 composite was microwave

heated at ﬁve ﬁber orientations. Figures 5.1 to 5 .5 show the temperature proﬁles in

various modes for each ﬁber orientation.

'l'

T

113

 

 

    

 

 

170
1 o .11 g
. 0 I'll
1“ .l A ya ‘ 1-
v ‘ ‘.9* 'v-::."
1“ 1 . m (c, ‘ ‘ ‘3“???er egg-5:1
I out!) , - 5.5-7:52;" '3' ’ ‘
125 ‘ A U“: . 'I‘_‘i-!;{;‘-u .
.1 V .14 “3.32"- , .
110 - ‘ ' . . '
ee '7
i
no 4 amuse.-
. no 18.30-
.. . MOO-0W
.0 . ,ilf- ,w' slum-zu-
50'
u _‘(;Er’ !’
ea ' ‘ ,
m V V j T V I *V I V I V I V I V r f T T
0.0 u 0.0 1” 1” as 27.0 81.0 30.0 m

use (will...)

5.1(a) UH at L,=10.94 cm & L,=l6.30 mm

 

deOQDUO

    
  

 

 

 

 

u so so 42 48
mummies)

5.1(b) UH at L,=12.20 cm & 11:18.87 mm

r

I

 

1’

7mm ('0)

114

 

 

 

 

 

 

   
   

170
. o ring a
1“- D m -2 -
‘ A “B - - .11?
“°" x 3‘68 '
. - .n 1"- .; ;"
1”. ‘ mg I;;;;.‘
4 V min , «:4 .
110-
“a
”-1
.-
”a
86 _
a I I f v I 1 I v I I Tﬁ I Y I Tﬁ ' I i
0.0 4.8 0.0 1” 10.0 22.0 27 81.5 36.0 40.5
mommies)
5.1(c) CH at L,=13.91 cm & L,=17.02 mm
170 0 mg
4 D .1!
155‘ a 3g
v
ct I u
125' a :1!”
4 v mg
110-
“-

OCVIIY Le : 14.87500
Coup In em
In . WI

Ion: 0°
.6'

 

 

 

0.0 utrnrummsmsasssom
manhunt)

5.1(d) CH at L,=l4.38 cm & L,=9.69 mm

V

115

 

   

 

 

 

 

170
‘ ° as
. o _.
1“ d A :Tug - - , - ‘ I “.
V C _." _; : ‘ (1"
Ta“ . mm - 7' ‘,.A.“.."'
‘ a sun - ,'-,I:c‘et'"
125 - s are: ,rf'r'l
E 4 v are
110‘
u .
e04 Lem?!“
. 10.37.
I- - 4, 0W
.8 ‘ I 0'
in)" 5"
”-1 8,40%!
”-1 f’.‘
:0 I Y I T 1 I f I T W T ﬂ 1 T V fﬁ
0 B 1. 18 20 as 80 35 40 48 so

The (men)

5.1(c) PS at L,=17.65 cm & L,=18.37 mm

Figure 5 .1 Temperature/position/time Proﬁles during Microwave Heating of
3.8 cm Thick Crossply AS4/3501-6 Composite at 0° Fiber Orientation

 

 

 

 

 

110
4 ° $1:
. a
155‘ A =91: - 1:
1‘.‘ z m.” r-_’t”’
'1 . m 1,- " -‘ ---_ a
1281 A .135
0 1 v ere ._ ;
V ‘1.“ ’15:
”q 1:; .0. ‘ ‘
/’ ,;'
l‘ o ‘
e04 . 4.1L mmumrm
. 1" Oeupno 21m
g... u. hwmm
. ,0. Orlenhtlonzwo
. w, 812;?an
3' /.’44
J}..-
as 4.
2. l V I V I ' I V I V T I I ' I V I ' I V
0.0 u 0.0 18.5 10.0 m 21.0 313 see 40.! 48.0
The”)

5.2(a) UH at L,=11.68 cm & 1.,=21.89 mm

Tm (00)

'l'

116

 

 

QDIOCDUO

    

3333

$355

353%
8333

   

 

 

I 10 1' 10 25 30

I I I I j I I T v 1 ﬁ’ 7 y I

Is so es
minim-a)

5.2(b) UH at L,=12.31 cm & L,=17.l9 mm

 

 

4>IO<DUO

$353

    

   

21.9155

  

 

 

 

I V I V I V I

V T I 1' V 1' I

Tune (minutes)

5.2(c) CH at L,=l3.89 cm & L,=18.48 mm

117

 

   

110
ml 8 :38 .‘
V H(‘
1““ . m 4*“
'1 I .138 M
125- A In: 1'
E v are
110-

   
    

”-1
I -1
'° mm4m
.- “. MON
10'
.0. seem-int

 

 

 

” I V I V r V I V I V I V I 1 I 1 Y 1 I V

5.2(d) CH at L,=14.25 cm & L,=15.45 mm

 

 

170
. ° ea
. a
1“, A man
140‘ X .741?
I an
125‘ a In”
E I In:
110‘
“-1
”-1
i- u-
”-

 

 

 

” I ﬁ I V T V I V I V I V I V I V 1 V j V

'I'IIIO (W)

5.2(c) PS at L,=17.79 cm & I.,=17.01 mm
Figure 5.2 Temperature/position/ time Proﬁles during Microwave Heating of ,
3.8 cm Thick Crossply AS4/3501-6 Composite at 15° Fiber Orientation

118

 

«bleabno
$355
3333

i333

    

 

r
8

 

 

 

20 I ' F v I ' I I I V T v I V I Y I I I r
an IB 11Al 103 220 It! 330 SID 4&9 ﬁll 560
Tillﬂminlﬂn

5.3(a) PS at L,=10.71 cm & L,=13.36 mm

 

 

110
. 0 I11 8
. a in
155" A :gpgg
v 4
1“ q . .“ m . o '
‘ I cum — '
125- 3 en .- ,. _:,
E ' O" .,.,'I" :'_._e;f‘
110 4 44.24? 9"
a d
It
eo-
h- .5-
.0-
q
u .
20 d I I I I I I I I V I V

 

 

 

0 4 8 12 10 20 24 20 a: 86 40
Time (mules)

5.3(b) CH at L,=11.52 cm & L,=14.55 mm

119

 

‘
8
I
4>Io<>ao
£155
33”

3333
3333

 

      

 

 

 

 

 

 

 

.-
5.3(c) CH at L,=12.56 cm & L,=21.46 mm
170
- ° $3
- o
155 q A 3% , f
V Z -
1“ - . .u m ;
d . m
“B“ A dug:
g 110 « v gun '
i __.r
.5 ' ‘ .2345?” .
I0 ‘ .__;.; t J - ’
1 6.99;,“ 1 J
'- 05 ' 0:7” ’ .
' Q$" Lummﬁuaﬂbn
lo - 4.9,: . Coup Ing Probe 13.73am
, anlmunGMN
:5 ~ 0:me
- MM!
20 . . . t

 

0.0551101652202753MWQMDMMO

11m (minute)

5.3(d) UH at L¢=14.04 cm & 1.,=13.73 mm

120

 

170
. O ‘1‘! g
I“ _. D .72
‘ A 33
V
1‘0 ‘ o I“ (G)
I "I
18 . A ma ,p
E - V .1" ’p’
110 " "

‘l'

 

 

 

 

Tb. (HM)

5.3(e) PS at L¢=17.64 cm & L,=20.05 mm

Figure 5 .3 Temperature/position/time Proﬁles during Microwave Heating of
3.8 cm Thick Crossply AS4/3501-6 Composite at 45 ° Fiber Orientation

 

   
  
 

   

 

 

 

110
. ° :33
q 0
‘85; 0 mg
140‘ Z $.83
‘ u m
128- 9 .138
§ . v cum
" 110«
”1
..l
.— u:
cm L. 41.38%
IO- Oouplng 1m
In meow
II- Ion.”-
Hurtful"
a. I T I 7 I *I I V T V I I

 

5.4(a) UH at L¢=11.36 cm & L,=18.85 mm

121

 

 

 

 

110
. o :22 a
. a _ - -
‘5‘ . A mm ; ‘ -
140 - 2 {ﬁg '
‘ I an
125 - A In:
E . ~ v mm
110-
“ .
n q 'l
d '4 1"}
I- “ - ' " 1:1 I: 1
d ' (i a 2m
80 i " {(27 ' couplm 11m
J "-;-;1' mums"
u ' o , e W“’;
so . .

 

MUM1OJ1MMNJNJMMM
nun-(mun)

5.4(b) UH at Lc=12.29 cm & L,=l7.09 mm

 

      
 

17o
* 8 $8
155: A :38
V
I“ ‘ . .n‘c,
I on
125 - A mg
E « V cm!)
110 -

T
8

 

 

 

m I ' I r I ' I I I 7 f I I 1 I ' I V I Y
0.0 4.5 0.0 13.5 10.0 22.5 27.0 81.5 33.0 40.3 48.0
'I'IIIIO (M)

5.4(c) CH at Lc=14.04 cm & L,=12.02 mm

122

 

w. 0 I11
155 - '2 $8
' v dug;
I“ ‘ . .11 (q
I In
”5* A dug:
E V out!)
11. ‘

    
  

 

1'
8

Cu}, 14.1“
con Inc ”with”

 

 

 

 

 

’. ' I I I I I I I I I I T I I I I I I I
0.0 uIIJIumqumuocum
man“)

5.4(d) CH at Lc=14.19 cm & L,=16.41 mm
170
1554

Iblooboo
3359
3333

1mm (00)
8 8

 

 

 

1 I I 1 v I I I I I v v I I 1

0 G 12 1O 24 3. .6 42 48 54 .0
11m. (mm...)

5.4(c) PS at Lc=l7.64 cm & L,=18.09 mm
Figure 5 .4 Temperature/position/time Proﬁles during Microwave Heating of
3.8 cm Thick Crossply AS4/3501-6 Composite at 75° Fiber Orientation

123

 

<OIO£ODO

T

 

 

 

 

o . I r I v I v I I r I I I I I I I I v
00 25’ an 'Z5 1.0! 125 I‘D TE! 2‘0 225 250
Iillﬂmlwiun

5.5(a) UH at Lc=ll.07 cm & L,=19.37 mm

 

{-355

I
I
I
1'
I)
I
V

E
l
dOIOOODO

- :38
g 126: mg
"0:

T

 

 

 

 

 

10 18 20 28 30 35 do 45 lo
TIIII (nIlnuIII)

5.5(b) UH at Lc=12.45 cm & L,=l7.37 mm

124

 

$353

40I0<000

$353
3333

'I’

gamma

hput Power. ”W
Won: 00-
ms“

 

 
  

 

I T

I I

I I I

I

III. (M)

5.5(c) CH at L,=l3.86 cm & L,=14.47mm

 

«Oncoooo
I355
3333

i353
3333

 

T

 

     

 

h+

5.5(d) CH at Lc=14.32 cm & L,=20.89 mm

 

 

125

 

O .‘I‘lg
1“ _, C! II!
t ~° :33
V
I“ "‘ . mm
‘ I “In
125 ‘ 0 CI.”
‘ V “‘n

TWIN)
I__l 14 I 1
g\
I .
\
. ,
\
l .
\
:
.‘~ -
Q .

 

 

 

If

”I f 1!

J II}!

/"

”1 "1‘ gum'?‘

4 ,5’ m5.

1
m j I I I I I I I r T I I

 

0.0 un1u1u1umammmm
mailman)

5.5(e) PS at L,=17.64 cm & L,=l7.72 mm

Figure 5 .5 Temperature/position/ time Proﬁles during Miérowave Heating of
3.8 cm Thick Crossply AS4/3501-6 Composite at 90° Fiber Orientation

For processing purposes, the more uniform the spatial temperature distribution
and the smaller temperature gradient across the thickness of the composite, the better
the mode. Therefore, the preference of processing mode selection can be based on
the temperature uniformity. From Figure 5 .1, the sequence of preference for the 0°
ﬁber orientation was the CH mode at L,=14.38 cm, PS mode at L,=l7.65 cm, CH
mode at L,=13.91 cm, UH mode at L,=12.20 cm, and UH mode at L=lZ.20 cm.
From Figure 5.2, the sequence of preference for the 15° ﬁber orientation was the CH
mode at L,=13.89 cm, UH mode at L,=12.31 cm, CH mode at L,=l4.25 cm, PS
mode at L,=l7.79 cm, and UH mode at L,=ll.68 cm. From Figure 5.3, the
sequence of preference for the 45 ° ﬁber orientation was the PS mode at I.‘ = 10.71
cm, UH mode at L,=l4.04 cm, CH mode at L,=ll.52 cm, CH mode at L,=12.56
cm, and PS mode at L,=17.64 cm. From Figure 5.4, the sequence of preference for

126

the 75° ﬁber orientation was the CH mode at L,=l4.l9 cm, PS mode at L,=l7.64
cm, UH mode at L,=12.29 cm, UH mode at L,=ll.36 cm, and CH mode at
L,=14.04 cm. From Figure 5.5, the sequence of preference for the 90° ﬁber
orientation was the UH mode at L,=12.45 cm, CH mode at L,=l4.32 cm, PS mode
at L,=l7.64 cm, CH mode at Lc=13.86 cm, and UH mode at L,=11.07 cm. If the
allowable temperature gradient is 30°C, all of the heating modes at the 0° and 15°
ﬁber orientations could be used for processing while none of the modes at the 90°
orientation were qualiﬁed for processing. There were two modes having overall
temperature gradient less than 30°C for the 45 ° and 75 ° orientations. The mode
having the most uniform spatial temperature distribution was the CH mode with
L,=l3.89 cm at the 15° ﬁber orientation with AT=10°C. The mode having the
most uniform overall temperature distribution was the CH mode at L,= 14.19 cm at
the 75° orientation with AT=10°C.

For easy comparison, the results in Figures 5 .1 to 5 .5 were converted into a
tabular form. Tables 5.1 to 5.5 list the conditions and results of the microwave
heating of a 3.8cm thick fully thermally cured crossply graphite ﬁber/epoxy
composite at ﬁve ﬁber orientations. The ﬁrst column in the tables shows the mode
type along with the cavity length L, and the coupling probe depth LP. Comparing
Table 5.1 to Table 5.5, the types of resonant heating modes are ordered in the same
way for 0° and 90° ﬁber orientations. The sequence is two uncontrolled-hybrid
modes (UH), followed by two coupled controlled-hybrid modes (CH), and a pseudo-
single mode (PS). The corresponding modes are also located at similar cavity lengths
for both orientations. Comparing Table 5 .2 to Table 5.4, the same similarity of
modes (type, sequence, and LC) is found between the 15° and 75° orientations. The
sequence is again two UH, two CH, and a PS. The mode sequence and locations for
the 45 ° ﬁber orientation are unique as listed in Table 5.3. The mode sequence is a
PS, two CH, a UH, and a PS. These results suggest that a symmetry of mode
characteristics (mode type and L,) exists with regard to ﬁber orientation. The
existence of a symmetry of mode type do not necessarily imply the symmetry of

127

electric ﬁeld pattern. The symmetry of electric ﬁeld pattern ean be determined from
the comparison of the temperature/time/position proﬁles of corresponding modes.
Comparing Figure 5.1, heating results, at the 0° ﬁber orientation, to Figure 5.5,
heating results at the 90° ﬁber orientation, only one similar heating pattern, case (b),
was found. This may be because these two modes not only share the same mode
type (UH) and cavity length, but also similar coupling probe depth. However, the
symmetry of heating pattern is not found for PS modes even though they share the
same mode type, cavity length, and coupling probe depth, such as ease (e). The
same symmetry phenomenon of heating pattern was observed for 15 ° and 75 ° .
Comparing Figure 5.2 to Figure 5.4, the (b)’s shared the same mode type (UH), L,,
and L, and the heating patterns were the same. Though the PS mode in the (e)’s
shared the same L6 and LP, the heating pattern was different for the 15° and 75° ﬁber
orientations.

128

Table 5.1. Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrieal Cavity for 0° Fiber Orientation.

 

 

 

 

 

 

 

 

 

 

 

Mode Type Time' Overall Overall

& to reach AT, AT. AT Processing
Lc(cm) 160 °C (°C) (°C) (°C) Index
Lp(mm) (min.)

UH
Lc=10.94 42-M 21 22 . 30 8.1
Lp=16.30

UH
Lc=12.20 60-M 15 17 30 7.6
Lp=18.87

CH
I.c=l3.9l 44—M 17 14 25 6.5
Lp= 17.02

CH
Lc=14.38 51-M 10 16 17 5.1
Lp= 9.69

2(Ts>Tm)

PS
Lc= 17.65 46-M 13 15 22 5.9
Lp=18.37

129

Table 5 .2. Conditions and Results for Microwave Heating of 7.8 X 7. 8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 15 ° Fiber Orientation.

 

 

 

 

 

Mode Type Time’ Overall Overall

& to reach AT, ATIII AT Processing
Lc(cm) 160 °C (°C) (°C) (°C) Index
Lp(mm) (min.)

I UH
Lc=11.68 42-S 18 20 30 7.6
Lp=21.89 ‘
3(Ts>Tm)

UH
Lc= 12.31 49-M 6 16 22 5.4
Lp=17.19

CH
Lc= 13.89 49-M 5 10 23 4.8
Lp= 18.48

CH
Lc= 14.25 47-M 8 19 19 5.5
Lp= 15.45

PS
Lc= 17.79 39-M 8 18 25 5.9
Lp= 17.01

 

 

 

 

 

 

130

Table 5.3. Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 45 ° Fiber Orientation.

 

 

 

 

 

 

Mode Type Time' Overall Overall
& to reach AT, AT. AT Processing
Lc(cm) 160 °C (°C) (°C) (°C) Index
Lp(mm) (mitt)
. PS
Lc= 10.71 55-S 10 12 20 5.3
Lp= 13.36
3(Ts> Tm)
CH
Lc=11.52 39-S 30 12 45 9.5
Lp= 14.55
4(Ts >Tm)
CH
Lc= 12.56 37-8 43 16 43 10.9
Lp=21.46
2(Ts > Tm)
UH
Lc= 14.04 54-M 13 10 30 6.6
Lp= 13.73
PS
Lc= 17.64 29-M 26 48 65 14.5
Lp=20.05

 

 

 

 

 

131

Table 5.4. Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501-6 Composite in a 17.8cm
Cylindrical Cavity for 75 ° Fiber Orientation.

 

 

 

 

 

 

Mode Type Time‘ Overall Overall

& to reach AT, AT.l AT Processing
Lc(cm) 160 °C (°C) (°C) (°C) Index
Lp(mm) (min.)

UH
Lc= 11.36 408 30 4 32 7.4
Lp= 18.85

4(Ts>Tm)

UH
Lc= 12.29 44-M 10 19 38 7.5
Lp= 17.09

CH
I.c=14.04 44-M 21 23 35 8.8
Lp= 12.02

CH .
Lc=14.19 52-8 15 6 15 4.4
Lp= 16.41

2(Ts>Tm)

PS

Lc=17.64 55-M 13 4 20 4.8

 

 

 

 

 

Lp=18.09

132
Table 5.5. Conditions and Results for Microwave Heating of 7.8 X 7.8 X 3.8cm
Fully Thermally Cured Crossply AS4/3 501—6 Composite in a 17.8cm
Cylindrical Cavity for 90° Fiber Orientation.

 

 

 

 

 

 

Mode Type Time' Overall Overall
& to reach AT, AT. AT Processing
I.c(cm) 160 °C (°C) (°C) (°C) Index
Lp(mm) (min.)
UH
Lc=11.07 23-M 43 53 88 18.9
Lp= 19.37
UH
Lc=12.45 49-M 15 17 39 8.1
Lp= 17.37
CH
Lc= 13.86 39-M 35 25 50 8.8
Lp= 14.47
CH
Lc= 14.32 40-M 15 26 35 8.4
Lp=20.89
PS
Lc= 17.64 34-M 22 29 51 10.9
Lp= 17.72

 

 

 

 

 

 

 

 

* The time required to reach the control temperature 160°C. The M or S
indicate that the location of the temperature probe that reached 160°C was at

midplane or surface.

133

In Table 5.1 to 5.5, the value in column 2 is a qualitative index of the heating
rate for the modes. Columns 3, 4, and 5 list the maximum surface, midplane, and
overall temperature differences respectively. As shown in Figures 5 .1 to 5.5, the
midplane temperatures were usually higher than the corresponding surface
temperatures. The number of loeations was listed using T,>T_ as indicator in
Column 5 when the surface temperature was higher than the corresponding midplane
amnenmne

Columns 2-5 in Tables 5.1 to 5.5 show that the microwave heating rates and
temperature uniformity vary with ﬁber orientation and cavity length of the resonant
heating mode. Figure 5.6 shows the heating time as function of eavity length (i.e.
mode) for various ﬁber orientations. The time ranges from 20 minutes to 60 minutes.
Figure 5 .7 shows the overall temperature difference as function of cavity length (i.e.
mode) for various ﬁber orientations. From Figures 5 .6 and 5 .7, it is clear that time
and the temperature difference are related. If the temperature difference was low, the
microwave energy was evenly distributed inside the composite and it took a longer
time to heat the whole composite to the control temperature. If the temperature
difference was large, the microwave energy was focused on one spot and the time

required to heat one spot to the control temperature was short.

To quantify the quality of each heating mode, the overall processing index (I)

was introduced. The I is a dimensionless value and was calculated by:

_ ttATJ+bATﬂI +AT+dt

' 10 (5-1)
where a,b,c,d are relative weight factors. a,b, and c have units of 1/°C while d has

units of 1/min. The smaller the value of I, the higher the quality of the heating

 

mode. In the processing of composites, the temperature uniformity, AT,, AT,,, and
AT,,, is critieal to the ﬁnal mechanical properties. The time, t, required to reach
control-temperature ean be reduced, however, by increasing input power. Therefore,
the weight factors a, b, and c are much more important than (1. For the calculation in
Tables 5.1 to 5.5, a, b, and c were set equal to 1, and d was set equal to 0.2.

Time to reach 160°C (min.)

134

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

901 o -o 0° fiber orientation
701 a
50-I O I ' ' ~~~~~ 9,9 """"" '0
301
10 I r ' l ' I ' '
1O 12 14 16 18 20
901, 4: 15° fiber orientation
70:1
50:1 B o a" " " ' ”BB. ---------- a
301
10 . T ' I ' ' ' I
10 12 14 16 18 20
90‘: A -A 45° fiber orientation
70%
50+ A~
. s A. - -A“ _. .. :- 'é' .........
301 "'4
10 I I r l ' I ' '
10 12 14 16 18 20
90': + -+ 75° fiber orientation
70-
4 .......... +
50+. ,--...----.a--
SO-j
10 I I ' I ' ' ' ' r
10 12 14 ‘5 ‘8 2°
90': at» as 90° fiber orientation
70:1
50-1 , 3K- - - - at * _ _
:so-j ,, , ’ ’ ------ 3"
1G . r ' I r ' ' I
10 12 14 16 18 2°

Cavity Length (cm)

Figure 5.6 Heating Time for Various Resonant Heating Mode
at Various Fiber Orientation

135

 

 

 

 

 

 

 

 

 

 

 

 

 

Overall Temp. Difference (°C)

 

 

 

 

 

 

 

90"} o-o 0° fiber orientation
70~
50f
30:4 G-"e'I-“a ........ e
10 I i ' ﬁll-I. l ' I '
10 12 14 16 18 20
90‘1 a-a 15° fiber orientation
70:
50-1
I153: E‘&----EE}. -------- 8
1o I1I2'1I4ﬁ1'6I1'8'20
901 a -e 45° fiber orientation
701 ,4
:31 A"-b ........
.1 ,’ “A"
104 Ar 1 ' I ' I ' I I
10 12 14 16 18 20
901 + -1- 75° fiber orientation
70-4
501
30; +“"""““;
‘10‘ I l ' ‘1??---I---:-+I '
10 12 14 16 18 20
90': 316‘
70': \\\
50- x x ....... 3"
so 3" 3‘
":4 air-11: 90° fiber orientation
1o 12 1'4 1'6 1'8 2°

Cavity Length (cm)

Figure 5.7 Temperature Difference for Various Resonant Heating Mode
at Various Fiber Orientation

136

Figure 5.8 shows the overall processing index I versus eavity length for
various ﬁber orientations. The cavity length of each heating mode is similar for 0°,
15°, 75°, and 90° ﬁber orientations. The similarity increases with cavity length.

For the resonant heating modes between I.c = 17cm and L, = 18cm, the cavity length
is essentially the same, I.c =17.6cm, even for 45° ﬁber orientation. The overall
processing index I is, however, very different and is a strong function of ﬁber
orientation and cavity length. If the maximum allowable index value is set equal to 6,
(the maximum temperature difference is 25 °c), the number of allowable resonant
heating modes for 15°, 75°, 0°, 45°, and 90° ﬁber orientations were found to be 4,
2, 2, 1, and 0, respectively. The best ﬁber orientation for microwave processing of
3.8cm thick crossply composite is 15°. As listed in Tables 5.1 to 5.5, there are 3
modes having I lower than 5. They are the CH mode of 75° ﬁber orientation at L =
14.19cm, L, = 16.41 mm, the PS mode of 75° ﬁber orientation at L, = 17.64am, 1.,
=18.09mm, and the CH mode of 15° ﬁber orientation at L, =13.89cm,
L,=18.48mm. The temperature differences listed in columns 3, 4, and 5 of Tables

5 .1 to 5 .5 indicate that the temperature proﬁles are the most uniform in these modes.

Quality Index

137

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

18:1 o-o 0° fiber orientation
1“.
10-
. e--- -__
6': 9- “9°- ---------- 0
2 I I r I I r I I I
. 1O 12 14- 16 18 20
18f a -a 15° fiber orientation
144
10-1 9
6': ‘15----351- """""" B
2 ' I a I ' I ' I '
10 12 14 16 18 20
18': b-A 45° fiber orientation
144 .—IA
101 g-“‘~\ ,.—"’
61 a” ‘11"
2 I I I I ﬂ I I I I
10 12 14 16 18 20
18% +4 75° fiber orientation
141
104.
SJ +--+.--"“: .
1 + ----------- +
2 ' I ' I I j? I I rt
10 12 14 16 18 20
184 5“.
141 I.
10- ‘~ ------- 9“
. ---aK ---
6-J he *-
2- all-111 90° fiber orientation '
1o 13 1'4 16 18 20

Cavity Length (cm)

Figure 5 . 8 Quality Index for Various Resonant Heating Mode
at Various Fiber Orientation

138

Column 5 in Table 5.1 to Table 5.5 also shows that there were only two eases
where all surface temperatures were higher than the corresponding midplane
temperatures. This result implies that there is a signiﬁcant amount of the electric
ﬁeld at the midplane. According to the skin depth theory, the electric ﬁeld along the
ﬁber direction of AS4/3501-6 composite has a sln'n depth of 9.8mm. The low skin
depth implies that the electric ﬁeld in this direction will not contribute to the heating
in the midplane. The skin depth for the electric ﬁeld perpendicular to the ﬁber
direction of AS4/3501-6 composite, however, has a value of 3.2m. The qualitative
description of penetration depth for various conﬁgurations between electric ﬁeld and
ﬁber directions for individual ﬁbers, unidirectional composites, and crossply
composites are shown in Figures 5.9, 5.10, and 5.11 respectively. For individual
ﬁbers and unidirectional composites, there are three directions of the incident electric
ﬁeld which will have a large penetration depth. They are electric ﬁelds perpendicular
to the ﬁbers, one from the top and two from the sides, as shown in Figure 5.9 and
5 .10. For crossply composites, only the electric ﬁeld from the side has a large
penetration depth as shown in Figure 5.11. Therefore, the high midplane
temperatures suggest that the electric ﬁeld inside the composite was from the side for
the majority of the heating modes. The data in Tables 5.1 to 5.5 reveal that a 3.8cm
thick crossply graphite ﬁber/epoxy composite can be uniformly heated in a 17.8cm

resonant eavity by 2.4SGHz microwave radiation.

139

 

 

E-field direction fiber direction qualitative penetration depth
I or X 1 1 I I j Large
xvvvvvv

 

Oil)

>< @ .....

Figure 5 .9 Qualitative Description of Penetration Depth for Individual Fibers

E-fieid direction fiber direction qualitative penetration depth

Large

’1 5......

 

 

 

 

 

 

 

 

 

 

 

X Large

 

 

 

 

 

 

 

 

Figure 5 .10 Qualitative Description of Penetration Depth for
Unidirectional Composites

140

E-fieid direction fiber direction qualitative penetration depth

Large

 

 

 

 

 

 

 

 

m.‘..I..°.L.°.1

Figure 5.11 Qualitative Description of Penetration Depth for Crossply Composites

Based on the microwave heating experiments of thick-section graphite ﬁber
composite, a fresh 7.6x7.6x3.3 cm crossply AS4/3501-6 laminate (200 plies) was
microwave processed using a resonant mode at the 15° ﬁber orientation with L, =
13.22cm and L, = 21.98mm. This mode is the analog to the best heating mode for
the fully thermally cured 3.8cm thick crossply sample. The difference in the cavity
length is due to the difference of thickness and loss factor between the fresh and fully
cured samples. Figure 5 . 13 shows the temperature/position] time proﬁles for the fresh
sample. As shown in Figure 5.13, both surface and midplane temperatures in the
fresh composite are uniform by themselves during microwave processing, similar to
those of the fully cured composite. Comparing Figure 5.13 and Figure 5.3(c), the
temperature difference between surface and midplane of the fresh sample is 50°C
higher than that of the fully cured composite at the time when the midplane reached
controlled temperature. The heating rate of the midplane in the fresh sample is,
however, twice as fast as in the fully cured sample. Both differences are due to the

141

exothermic reaction in the fresh sample. With the midplane temperature controlled at
160°C, the surface/midplane temperature difference in the fresh composite decreased
with time, and reached 40°C after 65 minutes. To reduce the temperature difference
between the surface and midplane, a resonant heating mode with uniform planar
temperature proﬁles but a higher surface heating rate should used for the processing
of fresh composite. The best eandidate for achieving uniform temperature distribution
across thick-section composites during microwave prowssing is the CH mode at the
75° ﬁber orientation at L, = 14.19cm as shown in Figure 5.4(d).

 

1 80 . - . , 1 . . . . . . , . T .
764 I ’7" ‘3: IWW'WJM ~
.. .'.e“e:~::' :‘w:"" e e"" e," e, 'e,’ " .
1 48 1' 0 go." e "V” '
,. O Q ' V
O
A ‘| 32 b . .. ' eeeeeee
L) __ 0 g ......... .. 1 .
2’ . ‘ ........... , ,.-.~. .,-_,;.;,;...-....-... .'-'-’-'-'
e l 1 5 ” I‘ 1
L- “. M ‘ . o oee' .f. .......... ' c1
3 ...... m9 .
6 1 OO - 6'“. WWW". 0 TS C
b °‘ v"' ‘ Ts e ‘
Q) _1\' v" _
Q vV'" ' Ts (e)
:3, . I' o T, (e) ‘
I— 433." ° ;m Ecg
‘ m e
' Tr11 (6)
° Tan (9) ..

 

 

 

 

18 27 36 45 54 63 72 81 90
Time (min.)
Figure 5.13 Temperature/position/time Proﬁles during Microwave Processing of

ZOO-crossply Fresh AS4/3501-6 Composite.

5.4. Conclusion

Fiber orientation effects in the microwave processing of graphite ﬁber/epoxy
composite were studied in a 17.8cm cylindrieal tunable resonant cavity using 2.45

142

GHz microwave radiations. A thick-section fully autoclave cured crossply
(7.8x7.8x3.8 cm) AS4/3501-6 composite, supported by a 3.8 cm thick teﬂon disk,
was located at, center of the bottom plate and heated at ﬁve ﬁber orientations. Five
resonant heating modes were found for each ﬁber orientation. The heating rate and
the temperature uniformity were strong functions of the ﬁber orientation and the
cavity length. The composite ean be uniformly heated in several resonant heating
modes at 0°, 15°, 45°, and 75° ﬁber orientations. All the modes at 0° and 15° ﬁber
orientations have overall temperature differences of less than 30°C during microwave
heating. Among all the modes, the planar temperature uniformity is the highest in the
CH mode at the 15 ° ﬁber orientation at L, = 13.89 cm. The most uniform overall
temperature distribution was found in the CH mode at the 75 ° ﬁber orientation at I,
= 14.19 cm. To quantify the processing qualities of the heating modes, a concept of
processing index (I)was introduced. It was a function of the heating rate and
temperature uniformity, and was a strong function of the ﬁber orientation. The
processing quality was the highest in 15° and 75° ﬁber orientations.

A fresh ZOO—crossply AS4/3501-6 composite (7.8x7.8x3.8 cm) was processed
at a 15° ﬁber orientation with L, = 13.22cm, L, = 21.98 mm. The temperature
proﬁles of both surface and midplane were uniform with themselves. The
temperature difference between surface and midplane in the fresh composite was
higher than that in fully cured sample.

CHAPTER 6
SCALE—UP STUDY OF MICROWAVE HEATING IN TUNABLE CAVITIES

6.1. Introduction

Microwave heating of graphite ﬁber/epoxy composite was discussed in
Chapters 4 and 5 . However, all the previous experiments were conducted in the
small cavities, such as 15.24cm and 17.78 cm. In order to process large composites,
the scale-up of microwave, heating needs to be studied. This Chapter investigates the
scale-up effects by comparison of temperature proﬁles in fully cured epoxy and
graphite ﬁber/epoxy squares heated in a 17.78cm cylindrical cavity using 2.456112
microwave radiation and a 45.72cm inner diameter cavity using 915MHz microwave
radiation.

6.2. Seale-up Frequency

The heating of a material by microwave radiation depends not only on the
complex permittivity and permeability, but also on the microwave frequency and
electromagnetic (EM) ﬁeld strength incident on the material. Since polymers and
composites are non-magnetic materials, it is the electric ﬁeld strength rather than the
magnetic ﬁeld strength that determines the heating of these materials. A resonant EM
ﬁeld pattern or mode must be established inside a cavity to conﬁne microwave power
inside the cavity and to heat a lossy material in the eavity. For an empty eavity, the
EM ﬁeld of each mode can be obtained analytically through Maxwell’s equations and
the application of the proper boundary conditions. In order to obtain a resonance
condition in a circular cylindrical cavity, the source frequency must be higher than a
speciﬁc cutoff frequency. The speciﬁc cutoff frequency, f,, is the lowest frequency
required for establishing the resonance mode. The f, for TM and TE modes

respectively is given by‘2
143

144

«pm = X, (6'1)

(fe) = _.L_ (6-2)

"I" 21" e011,

where r is the radius of the cavity, x, and x,’ are the ordered zeros of the Bessel
function J,(x) and its derivative J.’(x’), and e, and u, are the permittivity and
permeability of free space.

Two commercial frequencies are being investigated, 2.456Hz and 915MHz.
A eavity of 17.78cm (7") inner diameter was used for 2.4SGHz system. The
corresponding cavity size for 915MHz system is 45.72cm (18") as determined from
Equations (6-1) and (6-2).

6.3. Experiments

Two microwave systems, one operating at 2.4SGHz with a 17.78cm inner
diameter cavity and one operating at 915MHz with a 45.72cm inner diameter cavity,
were used in this study. The microwave circuit setup was described in Chapter 2.

Parallel experiments in the 17 .78cm and 45.72cm cavities were earried out for
two materials, thermally cured diglycidyl ether of bisphenol-A
(DGEBA)/diaminodiphenyl sulfone (DDS) epoxy and thermally cured unidirectional
Hercules AS4/3501-6 graphite ﬁber/epoxy composite squares. The midplane of the
epoxy squares was located 2.94 cm and 7.29 cm above the bottom plate in the
17 .78cm and the 45.72cm cavities, respectively. The midplane of composite squares
was located 2.80 cm and 6.91 cm above the bottom plate in the 17.78cm and the

145

45.72cm cavities, respectively. The empty cavities were used for mode reference.
For the empty cavities, each resonant mode was located by the power absorption
curve on the oscilloscope. Two additional measurements were made for mode name
assignment - the existence of axial E-ﬁeld was determined from the top of the cavity
for differentiation between TE and TM modes and the radial E-ﬁeld pattern in the
axial direction was measured for determination of q, the number of half waves in the
axial direction. For the epoxy loaded cavities, modes were located with the swept
frequency microwave sources and samples were heated in each mode using the single
frequency microwave energy at an input power density of 0.327 W/cm’. Four
midplane temperatures were measured during microwave heating of the epoxy
squares. The probe locations are shown in Figure 6.1(a). T1 was located at the
center of the epoxy square, T2 was aligned with coupling probe, and T3 and T4 were
on the diagonals opposite the coupling probe. T2, T3, and T4 were loeated the same
distance (b) away from TI. This distance was 2.5 cm and 6.4 cm for samples heated
in the 17.78cm and the 45.72cm cavities, respectively. For the composite loaded
cavities, modes were located with the swept frequency microwave sources and
samples were heated in each mode using the single frequency microwave energy at an
input power density of 0.423 W/cm’. Eight surface temperatures were measured
during microwave heating of the composite squares. The probe locations are shown
in Figure 6.1(b). T1 was located at the center of the square, T2, T3, and T4 were
evenly distributed around T1 at radius bl, and T5 to T8 were located at the four
corners of the square with the same distance (b2) away from T1. bl and b2 were
2.54 cm and 5.08 cm for the sample heated in the 17.78cm cavity. For the sample
heated in the 45.72cm cavity, b1 and b2 were 6.35 cm and 12.7 cm. Table 6.1
summarizes the sample size and operating conditions for the heating experiments in
both 17.780m and 45 .72cm cavities.

146

Table 1. Heating Conditions for Scale-up Experiments

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.1 Coupling Probe

Figure 6.1 Location of the temperature probes

6.4. Results and Discussion

1.1 Coupling Probe

( b )

 

Matenal Cavr Source Sample Dlmenslons Input
Radius(cm) Frequency (cm) Power (W)
Cured 8.89 2.45 GHz 6.92x6.92x0.56 8.8
Epoxy 22.86 915 MHz l7.78x17.78x1.45 150
Graphite/epoxy 8.89 2.45 GHz 10.16x10.16x0.28(24-ply) 12
Composite 22.86 915 MHz 25.4x25.4x0.68(60-p1y) 185
oT “rap
0 T1 1° /
T T1
b T 2 0T 3 OT I-l
-I- ° °T 6 T 7°

The resonance conditions of an empty cavity can be calculated from Maxwell’s
equations and the boundary conditions for the circular cylindrical cavity“. Equations
63 and 6-4 are used to calculate the cavity length of resonant modes for a given
cavity radius and operating frequency. Table 6.2 lists the theoretically calculated and

 

147

measured cavity length of a given resonant mode for both cavities and the scale-up

factor for various modes. The scale-up factor is deﬁned as the ratio of the cavity
length of the 45.72cm cavity to that of the 17.78cm eavity for the same resonant
mode. The calculations of Le assume a perfect cavity. The measured cavity lengths
of the resonant modes in the empty eavity are very close to the theoretical values in
every case. The measured scale-up factors are fairly close to the ealculated values at
each mode. With a constant ratio of cavity radii (2.57), the scale-up factor for cavity
length is different for each resonant mode, ranging from 2.7 to 3.7.

Table 6.2. Theoretical and Measured Results of Modes for Empty Cavity

 

 

 

 

 

 

 

 

 

 

Mode L, in small (cm) L, in large Scale-up Factor
cavity' eavity” (cm)
measured theoretical measured theoretical measured theoretical
TE", 6.73 6.69 18.10 18.07 2.69 2.70
l mm, 7.31 7.20 19.20 19.62 2.63 2.72
Tia,ll 8.33 8.24 22.50 22.88 2.70 2.78
I TM“, 11.35 11.28 33.70 33.87 2.97 3.00
1 TE", 13.43 13.38 3570 36.14 2.66 2.70
mm, 14.52 14.41 39.00 39.23 2.69 2.72
[ T5,,l 15.83 15.71 58.13 --- 3.70
I Tam, 16.58 16.48 45.60 45.75 2.78

 

 

 

' Inner diameter 17.78cm
” Inner diameter 45.72cm

6.4.1. Scale-up of Epoxy Loaded Cavities

 

 

 

 

Table 6.3 summarizes the LC of the resonant modes and the heating results for
the cavities loaded with epoxy. Modes are named based on the radial E-ﬁeld strength

148

pattern along the axial direction, the existence of axial E-ﬁeld, and the heating
proﬁles. Although the same names were used for the loaded cavities as for the empty
cavities, the electric ﬁeld patterns were not the exactly same as those of the empty
cavity. Since the cavity was loaded with a lossy material, the actual EM ﬁelds inside
the cavity were disturbed and not exactly the same as those of the theoretical modes.
The naming procedures of modes are described as follows. A resonant mode was
found at Le = 9.24cm in 17.78cm cavity. The epoxy square was heated in this mode
and the resulting temperature proﬁles are shown in Figure 6.2(a). The radial E—ﬁeld
strength was also measured along the axial direction during the heating as shown in
Figure 6.2(b). The axial E-ﬁeld strength at the center point was measured through an
opening at the top of the sliding short of the eavity. The existence of axial E-ﬁeld
implies that the resonant mode at Lc = 9.24cm is a TM mode. The radial E—ﬁeld
pattern shown in Figure 6.2(b) suggests that the number of the half waves in the axial
direction, q, is 1. The possible names for this resonant mode are TMou, mm, and
TM,“ and so on. The temperature proﬁles in Figure 6.2(a) shows that the heating is
strongest in the center, T1, while the edges, T2, T3, and T4 are lower. As the local
heating rate reﬂects the E—ﬁeld strength at that location, TMm is the only possible
TM mode that will result this heating pattem‘m. Figures 6.2 and 6.3 show the typical
temperature proﬁles and the electric ﬁeld pattern distributions in the axial direction
during microwave heating of epoxy squares in both cavities for TM", and TEm
modes respectively.

For the 45.72cm inner diameter cavity, the corresponding 'I'Mm mode was
found at Le = 26.4cm using the same procedure as described for the 17.78cm cavity.
Figure 6.2(c) shows the temperature proﬁles of the scaled-up epoxy square during
microwave heating in 45.720m cavity in the TM“l mode. Figure 6.2(d) shows the
radial E-ﬁeld strength along the axial direction.

Similar radial E—ﬁeld patterns are observed for TM“l mode for both cavities as
shown in Figures 6.2(b) and 6.2(d). With the same input power density, the heating

149

rate in the epoxy square is similar in the 17.78cm cavity operating 2.456Hz to the
45.72cm cavity operating at 915 MHz. The microwave heating of epoxy in the TM“l
mode in a 17.78cm cavity can be scaled up in three dimensions to a 45.72cm cavity.
The scale-up factor of the TMm mode in epoxy loaded cavities is 2.86, which is
similar to that of the empty cavities, 2.97.

Figure 6.3 shows the comparison of the TB“, mode in both cavities. The non-
existence of axial E-ﬁeld in both eavities proved both resonant modes were TE.
From Figure 6.3(b) and Figure 6.3(d), it was determined that q equaled 2 in both
modes. Both the radial E—ﬁeld along axial direction and the temperature proﬁles in
the samples are similar. TB“ is the only ﬁeld pattern“ which will eause the
temperature proﬁles seen in Figure 63(8) and 6.3(c). From Figure 6.3, it is apparent
that the microwave heating of epoxy at TE"2 mode in a 17.78cm cavity operating at
2.4SGHz can be also scaled up in three dimensions to the 45.72cm eavity operating at
915MHz. The scale-up factor of the TEm mode in epoxy loaded eavities is 2.76,
which is similar to that of the empty cavities, 2.66.

The Lc of various modes, the seale-up factors, and the comparison of the
heating patterns in 17.78cm and 45.72cm cavities are listed in Table 6.3. All the
heating patterns are similar for the two cavities using the same modes. This suggests
that the results of the microwave heating of epoxy squares or other low to medium
loss materials can be scaled up from the 17.78cm cavity to the 45.72cm cavity.
Similar heating patterns can be achieved for the same resonant mode regardless of the
cavity size if the same sample/cavity size geometry is maintained. A scale-up rule for
scale up in three dimensions can be applied to the microwave processing of low to

medium loss materials.

150

Table 6.3. Seale-up Results for Epoxy Loaded Cavity

 

 

 

 

 

 

 

 

 

 

Mode Cavity Cavity Scale- Temperature Temperature
Length length up Proﬁle‘ in Proﬁle' in
Name Small Large Factor Small Cavity Large Cavity
I Cavity Cavity
TEm 6.67 20.2 3.03 T2=T3 =T4 >Tl T2=T4=T3 >Tl
TM"l 9.24 26.4 2.86 T1>T32T4=T2 T1>T22T3=T4
TE", 11.83 32.7 2.76 Tl 2T3=T2=T4 T1212=T4 2T3
[mm 13.38 37.2 2.78 T4=T2=T3>Tl r4=13=12>r1
I T5511 14.45 m” --- T2 =T4 > T2 =T1 -------
; TEm 15.06 43.2 2.87 T3 2T4=T2 >T1 T4=T3=T2 >T1

 

 

' The symbols =, 2 , and > are used to represent temperature difference AT at a
heating time of 2 minutes. = means AT less than 5°C, 2 means between 6-10°C,
and > means 11-20°C of difference.

” The TE," mode can not be loeated in the large eavity because of the limit of the

cavity length.

Figure 6.4 shows the scale-up factors of various modes for the eases of empty
cavity (calculated and measured) and cavity loaded with epoxy. The measured seale-
up factors of the empty cavities are slightly lower than those of the calculated values.
The seale-up factors of the epoxy loaded cavities are very close to those of the
calculated values for empty cavities. This suggests that the processing of epoxy
samples or other materials with similar dielectric properties follows the empty cavity
scale-up rule.

Tumnnmnﬂol

 

.2;
.1',

0120400700101112

 

 

 

 

C+DO

23:3

 

 

151

 

 

 

 

 

 

 

 

run
14 Milt”!- :12!
« E11!
1:3 arm
a are
10‘ 1am
g . g E7311
0* :7”
I . 8 tan
OI i915
. >4.
4« 18371
I 124‘
2 1m
mania-n
(0)
II- u
“3 met-nan
: call-1mm ’15
11: Inuit-lum- ‘
3 1“
a; an
SI ’
115 till
I "3 use
111 , -
113 mm
73 175
4‘1 ,
1 an
nun-lune
(0

Figure 6.2 Temperature Proﬁles and Radial Electric Field Pattern Along Axial
Direction during Microwave Heating of Epoxy Squares at TMm Mode

(a) Temperature proﬁles in 17.78 cm cavity

(0) Radial electric ﬁeld pattern in 17 .78 cm cavity

(c) Temperature proﬁles in 45 .72 cm eavity

(d) Radial electric ﬁeld pattern in 45.72 cm cavity

 

 

 

 

1.
”Mill"
lﬂmm 00°
hull-unste- 000
U 000
0o
0
9 0° ++
s 00 +44»
0 + A
0 ++ AAA
I 0° +4.“: As I“
oo +++:IAI‘
vVV
i- 1 °++I°A ""Wv
°+Z°A 7' W" o 11
Sipv + n
b v ii
a I I r T T V I T I Y I
0 i 2 t 4 0 0 7 0 0 10 ii 12
mm

 

Temperature (‘0)

 

«+90
tau:

 

 

 

152

 

 

 

 

 

 

 

 

1:11

14‘ alert-unnu- .11,
a ~11a

1:4 11m
. :05

10‘ »uu
. E >1”

II am
I 5 I019

s« 16215
- tun

4< 1371
. 1M

2 1511

when”
(0)

a, an
1 mnem- r

“3 measure- 15

8': auburn-I ’

,3 I“

151 a II!"

a; .

II-j ’1“

u§ >151!

"i 1

104: Ill!

7} 475

4: 1

1 ‘ as

wanna-p

Figure 6.3 Temperature Proﬁles and Radial Electric Field Pattern Along Axial
Direction during Microwave Heating of Epoxy Squares at TE", Made
(a) Temperature proﬁles in 17.78 cm cavity
(b) Radial electric ﬁeld pattern in 17 .78 cm cavity
(c) Temperature proﬁles in 45.72 cm cavity
(d) Radial electric ﬁeld pattern in 45.72 cm cavity

153

Figures 6.5 and 6.6 show the shift of cavity length of each resonant mode for
the empty cavities (calculated and measured) and cavities loaded with epoxy squares.
The cavity length of the epoxy loaded cavities shifted down from that of the empty
cavities for all resonant modes. For the 45.72cm cavity, the shifts of the cavity
length are similar for all the modes (1.8 to 3cm) except the TM“, mode which shifted
down 7.3cm. The same shift pattern of the cavity length was observed for 17.78 cm
cavity. The shifts of the cavity length are similar for all modes (1.1 to 1.7cm)
except for the TMm mode which shifted down 2.11cm. The percentage of cavity
length shift for all the heating mode is within 20% in the range studied.

3.25

 

 

0 Calculated
+ Measured
A Epoxy Loaded
g 3.00 a
0
.2
g 2.75 —
2.50 4 T

 

 

 

T6211 T001" T6112 T0401: T5212

Roeonant Mode

Figure 6.4 Shift of the Scale-up Factor during Microwave Heating of Epoxy Squares

154

 

Epoxy
Loaded

Moaaured _____________

CIlcuhtOd _____________

 

“‘31

 

 

 

 

1 8 24
Cavity Length (cm)

Figure 6.5 Shift of Cavity length during Microwave Heating of
Epoxy Square in a 45 .72 cm Cavity

 

E x
ngdzd

Measured

Calculated

 

 

 

 

a 7 a a 1'0 11 {2 1'8 1'4 15 1'8 17
Cavity Length (cm)
Figure 6.6 Shift of Cavity Length during Microwave Heating of
Epoxy Square in a 17.78 cm Cavity

155
6.4.2 Seale-up Study of Graphite Fiber Composite Loaded Cavities

For unidirectional graphite ﬁber/epoxy composite loaded cavities, the
electromagnetic ﬁeld is greatly disturbed and the single-mode nomenclature is no
longer applieable. However, the electric ﬁeld pattern along axial direction and the
existence of z direction ﬁeld can still be measured. Tables 6.4 and 6.5 list the
heating modes and the results for microwave heating of a 10.16 cm x 10.16 cm 24-
ply unidirectional composite square in the 17.78 cm cavity and a 25.40 cm x 25.40
cm 60—ply unidirectional composite square in the 45.72 cm cavity, respectively.

Table 6.4 Heating Modes and Results for 60-ply Unidirectional Composite

 

 

 

 

 

 

 

in a 45.72 cm Cavity
_ - 1
Mode location q 2 ﬁeld AT“, T,., Temperature
distribution
L,=l7.2 cm 1 No 54 90 T5 =T7=T6=T8
L,=50.0 mm > > >T4=T2=T3=T
1
L,=22.5 cm 1 No 49 92 T5 =T7=T6=T8> >
I.,=25.3 mm >T4=T2=T3=Tl
L,=29.4 cm 2 Yes 32 80 T7=T6=T4=T2=T5
L,=39.8 mm =T8>T3=T1
L,=32.4 cm 2 Yes 53 97 T7=T6=T5=T8> >
L,=27.0 mm >T4=T2=T3=T1
L,=52.8 cm 2 No 35 84 T2=T4=T72T5 =T8
11:43.2 mm =T6> >T3=T1
L,=57.7 cm 1 Yes 18 65 T5=T7=T6=T8=T1
L,=11.8 mm =T2=T4=T3
L,=65.7 cm 1 Yes 22 70 T5 =T7=T6=T82T4
Lp= 19.0 mm =T2=T12T3

 

 

 

 

 

 

’ The meaning of the symbols =, 2 , and > are the same as in Table 6.3. Each
additional > represents additional 10°C in temperature difference.

156

Table 6.5 Heating Modes and Results for 24-ply Unidirectional Composite
in a 17.78 cm Cavity

    
  
   
   
   
   
      

  
  
    
 

Mode location q 2 ﬁeld AT.” Temperature distribution

 

 

 

 

= 7.56 cm 1 No 18 63 T8=T72T6=T2=T4=
= 4.24 mm T3=T5=Tl
L,= 7.91 cm 1 Yes 10 52 T8=T6=T5=T7=T2=
L,=ll.62 mm T4=T3=T1
L,=10.60 cm 2 Yes 26 68 T6>T5=T8=T7=T2=
L,= 8.11 mm T3=T1=T4
L,=15.24 cm 2 No 28 66 T2>T42T1=T8=T5=

L,=l$.83 mm T6=T3=T7

L,=15.83 cm 2 Yes 9 51 T3=T6=T8=T2=T5=
L,=10.90 mm . T1=T4=T7

L,=16.39 cm 2 Yes 14 60 T2=T6=T8=T4=T7=
Lp= 8.41 mm T52T3=Tl

 

 

 

 

 

 

 

‘ The meaning of the symbols =, 2, and > are the same as in Table 6.3.

In Tables 6.4 and 6.5, the ﬁrst column lists the cavity length, L,, and the
coupling probe length, L,” for the various resonant heating modes. The second
column lists the measured number of half waves in the axial direction. The third
column lists the measured results for the existence of z-ﬁeld in the center of the top
of the cavity. The fourth column lists the maximum temperature difference at the
composite surface at the end of nine minutes heating. The ﬁfth and sixth columns
list the maximum temperature and the temperature distribution in the composite in
descending order for various heating modes after 9 minutes heating. As shown in
columns 4 to 6 of Tables 6.4 and 6.5, the heating patterns are strong functions of the
resonant heating modes for both cavities. With the same input power density, the
temperature difference during microwave heating of graphite ﬁber/epoxy composite

is, in general, lower in the 10.16 cm square sample in the 17.78 cm cavity with 2.45

157

GHz radiation than in the 45.4 cm square sample in the 45.78 cm cavity with 915
MHz radiation. This may be due to non-uniform electric ﬁeld inside the composite
and the ﬁnite thermal conductivity of the graphite ﬁber composite. Heat is more
easily transferred from hot spots to cold spots across the composite in the small
sample than in the large sample. Comparing the composite heating results between
the small sample in 2.45 GHz system and the large sample in 915 MHz system, no
one-to-one correspondence can be found as in the epoxy heating results.

The maximum temperature, T,,“, is closely related to the maximum
temperature difference, AT,,, during microwave heating, as shown in columns four
and ﬁve of Tables 6.4 and 6.5. With the same input power density, the higher the
maximum temperature, the higher the temperature difference was regardless the
sample size. Figure 6.7 shows the relationship between AT.“ and T,., for
microwave heating of graphite ﬁber/epoxy composite in both the 915 MHz and 2.45
GHz systems. A linear relationship is found between the ATM and Tm.
Extrapolating to ATm=0, T,,, is 44.6°C at the end of 9 minutes heating. From the
energy balance,

p. =sh(T-Ta) +vpcpg (63)

the energy required to heat the composite uniformly up to T in t minutes is

_ ska-T.)
1"» ‘ —_ .1. (6-4)
l-e ”c'

where s and v are the surface area and volume of the composite, p and Cp are the
density and the heat capacity of the composite, h is the heat transfer coefﬁcient at the
surface, T, T, , tare the composite temperature, environment temperature, and time,
and P, is the microwave power dissipated in the composite. With the parameters in
Chapter 9, the microwave power required to heat a 10.16 x 10.16 x 0.28 cm and a

158
25.4 x 25.4 x 0.68 cm composite to 44.6°C in 9 minutes are 7.14 W and 89.7 W
respectively. As the input powers used for composite heating are 12 and 185 W for
the 17 .78 cm and 45.72 cm cavities, the percentage of input power dissipated in the
composite is 59.5 and 48.5 for the 17.78 cm and the 45.72 cm cavity respectively.

 

  
    

10o............f....
a4 ‘
ee
82
78
7o I
84 I
58
52 I
48
‘0 I . 1 l I l r
o e 12 18 24 so so 42 48 54 50
Maximum temperature difference

0 email cavity
X . big cavity

Maximum temperature

L l a l A l a l a l a l a

 

 

 

Figure 6.7 Heating Rate and Heating Uniformity Relationship for Microwave Heating
of Graphite Fiber/epoxy Composite

6.5. Conclusions

The microwave heating of DGEBA/DDS epoxy and Hercules AS4/3501-6
composite squares in various resonant heating modes was studied in 17.78cm inner
diameter and 45 .72cm inner diameter tuneable cylindrieal resonant cavities. A
microwave source operating at 2.45GHz was used for the 17.78cm cavity and
915MHz was used for the 45.72cm cavity in order to reproduce the same empty
cavity resonant mode for the same cavity geometry ratio. The scale-up factor is
deﬁned as the ratio of the cavity length in the 45 .72cm cavity to that in the 17.780m

cavity at the same resonant mode. For a given ratio of the cavity radii, the scale-up

159

factor varies with resonant mode. For microwave heating of epoxy squares, the
electromagnetic modes were identiﬁed using the empty cavities as a reference. The
same heating proﬁles were observed for the same resonant modes in the 17.78 cm and
45.72 cm cavities with the same cavity geometry ratios, the same sample/cavity size
ratios, and the same power densities. The scale-up of microwave heating of low to
medium loss materials, such as an epoxy square, can be approximated by the scale-up
of the empty cavities. For the microwave heating of graphite ﬁber/epoxy composites,
no similar heating proﬁles were found when the samples were scale-up in three
dimensions according to the scale-up of the cavity. The maximum temperature was
linearly related to the maximum temperature difference during microwave heating of
graphite ﬁber/epoxy composite. This relationship was the same for both 2.45 GHz
system and 915 MHz system. The power absorption efﬁciencies for the composite
loaded cavity are 59.5% and 48.5% in the 17.78 cm and 45.72 cm cavity,

respectively.

CHAPTER 7
CONTINUOUS PROCESSING OF GRAPHITE FIBER/EPOXY TAPE
IN A MICROWAVE APPLICATOR

7.1 Introduction

Microwave radiation effects on the kinetics of epoxy polymerization
and the glass transition temperature of cured epoxy resins were discussed in Chapter
3. Chapters 4 and 5 discussed batch microwave processing of epoxy based graphite
ﬁber composite in a 17.78 cm cavity. Chapter 6 discussed the scale up of
microwave heating from a 17.78 cm cavity to a 45.72 cm cavity. This chapter
addresses the feasibility of continuous processing of graphite ﬁber/epoxy prepreg tape

using microwave energy.

A critical problem in continuous processing of materials using microwave
energy is control of the microwave leakage through the entry and exit ports of the
cavity. The threshold limit value (TLV) at 2.4SGHz is 10 mW/cm2 I”. The TLV
refers to the maximum radiation level to which workers may be repeatedly exposed
without adverse health effects. The quoted TLV value is based on an average whole
body speciﬁc absorption rate of 0.4 W/kg in a six-minute period.

Doubly corrugated reactive chokes have been studied and used for reducing the
microwave leakage at the entry points during the continuous processing of
polystyrenem. Many rubbers, polar and non-polar, have been processed
continuously using microwave heating systemsm'm. Non-polar rubbers, such as
natural rubber, were usually mixed with carbon black or blended with polar rubbers
for better heating during microwave processing. Continuous processing of rubbers in
the microwave environment is, however, fundamentally different from the continuous
processing of graphite ﬁber/epoxy composites. The continuous conductor in the latter

case would cause tremendous microwave leakage in microwave heating systems

160

161

designed for the continuous processing of rubber. With proper modiﬁcation of the
batch tunable resonant applicator, continuous processing of graphite ﬁber/epoxy
prepreg tapes using microwave energy has been demonstrated". The preliminary
results of the continuous processing of graphite ﬁber/epoxy prepreg tape in a
microwave applicator are reported here.

7 .2 Experiments

The microwave circuit of the continuous processing system is similar to that of
the batch processing system as described in Chapter 2. A swept frequency oscillator
was used to locate resonant modes. A single frequency (2.45 GHz) power source was
used for heating. The input power and reflected power were measured on-line during
processing. The microwave applicator and the temperature measurement system for
the continuous system are, however, different from those of the batch processing
system. The 17.78cm inner diameter tunable cylindrical batch microwave cavity was
modiﬁed for continuous processing of conductor reinforced materials. Figure 7.1(a)
shows the modiﬁcation of the cavity. Two rectangular slots were cut in opposite
sides of the cavity wall perpendicular to the coupling probe. A jacket was put
around each slot to protect against microwave leakage. The input and output ports of
the cavity and the jackets were speciﬁcally designed to eliminate leakage due to both
the opening and the conducting graphite ﬁbers. Figures 7.1(b) and (c) show the
design of the cavity wall and the jacket dies. As shown in Figure 7.1(b), the ﬁns are
movable along the cavity wall and their distance from the material can be adjusted.
As these ﬁns are grounded to the cavity wall, the majority of the induced current in
the conducting ﬁber will be shorted by these ﬁns if the distance between the
conducting material and the ﬁn is properly adjusted. This design ensures that the
induced current in the conducting ﬁber will not be signiﬁcant outside the cavity. Also
these ﬁns reﬂect some of the microwave leakage back into the cavity. Figure 7 . 1(c)
shows the details of the design of the jacket dies. The dies are designed to be

162

adjustable so that prepregs of different thickness and width can be processed. The
ﬁnger stock touches the material to eliminate the remaining induced current in the

conducting ﬁber and conﬁne the microwaves inside the jacket.

 

 

 

 

cam 'AIL cam '00].

/m

Dem... - m...

(b) (C)

Figure 7.1 Modiﬁed 17.78 cm Tunable Cavity for Continuous Processing

The material used in this study was continuous graphite ﬁber/epoxy prepreg
(Hercules AS4/3501-6). The microwave circuit for the continuous processing system
is the 2.45 GHz system described in Chapter 2, except that no feed-back temperature
control mechanism is present. Figure 7.2 shows the continuous processing system.
The modiﬁed applicator allowed the material to pass through the applicator
continuously and controlled microwave leakage to well below the TLV.

163

   
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

AS4/ 3501-6
Prepreg Tape
0.6.
Power Sippiy
D/A .
- Guiding Roller
. Directional
Circulator Courier Coaxial
Switch Continuous Processing
O m Microrave Applicator
Sweep
Microwave T Oscillator .
5”“ I - Pulling Roller
Durrrny p:
Load Oscilloscope
1
pc 11.—
A/D lnfraed

 

 

 

 

 

 

 

 

 

 

 

Thermometer

 

Figure 7.2 Microwave System for Continuous Processing

An infrared thermometer (Omega 081 100) was used to measure the surface
temperature on the center line of the prepreg tape during processing. The
temperature measurement was accomplished through an opening in the top of the
cavity. The temperature is taken at ﬁxed spot, the center point of the tape inside the
cavity, while the continuous tape is pulling through the cavity. The pulling roller was
controlled by a variable speed stepper motor which controlled the speed of
transportation. The prepreg tapes, 3.81cm wide and 0.68mm thick, were processed
continuously in all available resonant modes using a single frequency source at an

input power of 60W. The prepreg feed rate was 0.508 cm/min. and the maximum

164

microwave leakage was controlled to within 0.3 mW/cm’. The reﬂected power was
minimized to be less than 0.1 W during the process by continuous ﬁne tuning of the
cavity length and coupling probe length. Differential Scanning Calorimetry (DSC)

was used to determine the extent of cure of the processed tape.

7.3 Results and Conclusion

There were ﬁve resonant heating modes available in the loaded cylindrical
tunable cavity in this study, two controlled-hybrid (CH) modes and three pseudo-
single (PS) modes. Table 7.1 lists the heating conditions and results for the various
modes. Both CH modes were able to heat the prepreg tape up to 200°C while the PS

modes could not heat the samples higher than 130°C with the same input power.

Table 7.1. Heating Conditions and Results

 

 

 

 

 

 

 

Mode No. Mode Type Cavity Length(cm) Maximum Temp. (°C) I
1 PS 7.41 126 ‘ I
2 CH 10.45 202 I
3 CH 12.45 212
4 PS 14.77 110
I 5 ps 15.85 50 __I

 

 

 

 

Figure 7.3 shows the center surface temperature versus residence time during

continuous processing of prepreg tapes in four resonant modes. The temperature
reached steady state in 20 seconds for both CH modes. In the PS modes, it took one

and three minutes to establish steady state for mode 1 and mode 5, respectively.

Clearly, the center line surface temperatures of the prepregs in the CH modes are

much higher than in the PS modes.

165

Figure 7 .4 shows the extent of cure distribution along the edge of the
microwave processed composite tapes at various residence times for the two CH
modes and one PS mode. In both CH modes, the prepreg achieved 90 percent cure in
15 minutes. In the PS mode, the prepreg achieved 90 percent cure in 35 minutes.
The faster curing on the edge of the prepreg tape in the CH mode than that in the PS
mode implies a higher cure temperature at the prepreg edge in the CH mode.

220 *
200 Mode 8

 

180
180
140
120
100
80
so
40
20.......4.ﬁ,-......,.

o 4 8 12 18 20 24 28 32 86 4o

Tepmerature (°C)

 

 

Time (minutes)

Figure 7.3 Center Surface Temperature during Continuous Processing of
Hercules AS4/ 3501-6 Prepreg using Microwave Energy

166

 

 

 

 

 

1.0 e
1 1
0.9 -
0.8 J
0.7 -
a «l
3 0.8 -«
E 0.5 J
5?, °-‘ j 0 Mode 3
0.3 _, X m 2
A Mode 5
0.2
0.1
0.0 I I I I ﬂ j T f
O 5 10 ‘i 5 20 25 30 35 40

Time (minutes)

Figure 7 .4 Extent of Cure at Various Resident Time in Three Modes

Figures 7.5 and 7.6 show the extent of cure distribution across the processed
composite tape at various residence times for the CH mode at Lc =12.45cm and the
PS mode at Lc=15.85cm respectively. Although the extent of cure is higher in the
center than the edge at the beginning of the processing in the CH mode, the prepreg '
reached almost full cure across the tape in 15 minutes. Figure 7.5 also implies that
the cure temperature is higher in the center than at the edge. Since the prepreg was
uniformly fully cured in 20 minutes, the prepreg feed rate can be increased to 0.89
cm per minute if the input power and applicator size remain the same. For the PS
mode, the extent of cure is higher at the edge than in the center as shown in Figure
7 .6. This implies that the cure temperature must be higher at the edges than in the
center in this PS mode. The laminate was also not fully cured at the end of 30
minutes residence time. Higher input power is therefore required in order to fully
cure the composite in this PS mode.

 

EMotCuto
5 0‘5 1.“

 

Figure 7 .5 Extent of Cure Distribution Across the Tape as Function of
Resident Time for a CH mode at 12.45 cm

 

Figure 7.6 Extent of Cure Distribution Across the Tape as Function of
Resident Time for 8 PS mode at 15.85 cm

168

Comparing Figure 7.5 to 7.6, the curing was much faster and more uniform in
the CH mode than in the PS made across the prepreg tape during continuous
processing. As the curing rate is controlled by heating efﬁciency which, in turn,
depends on the energy coupling efﬁciency of the resonant mode, the coupling of
microwave energy was more effective in the CH modes than in the PS modes.
Clearly, it is more efﬁcient to select CH modes for the continuous processing of
graphite ﬁber/epoxy prepreg in the microwave environment under the current
conditions. The microwave heating is not only a function of the electromagnetic
resonant modes, but also a function of the dielectric properties and the location of the
load in the cavity. The PS modes may also be able to heat the prepreg efﬁciently and
fully cure the prepreg in other sample locations. More detailed studies of sample
location effects on the continuous processing of graphite ﬁber/epoxy prepreg in the
microwave environment are required for optimum heating results. Other experimental
modiﬁcations may be required to fully explore the advantages of continuous

processing in the microwave environment.

With the modiﬁed microwave cavity, continuous processing of graphite
ﬁber/epoxy composite was realized using microwave energy. This breakthrough
brings the microwave processing technique to a new stage of commercialization
potential. The success of this study will enable us to process high performance
composite parts, such as long pipes, cost-effectively. This technique also makes
possible many other continuous processes for conducting materials using microwave
energy, such as pultrusion, ﬁlament winding, and production of carbon] graphite
ﬁbers.

7.4 Conclusion

It was demonstrated that graphite ﬁber reinforced epoxy prepreg tapes can be
processed continuously using 2.456Hz microwave radiation in a modiﬁed 17 .78 cm

169

tunable resonant applicator. The microwave radiation leakage was controlled to under
0.3 mW/cm2 during processing, which is much lower than the safety threshold limit
value (10 mW/cm’). The absorption and distribution of the input power in the
prepreg tapes were a strong function of the resonant heating mode. The continuous
prepreg tape effectively absorbed the input power and was fully cured in the
controlled-hybrid modes. The prepreg tapes were not fully cured in the pseudo-single
modes. The CH modes heat more effectively than the PS modes during continuous
processing of the graphite ﬁber/epoxy prepreg under these experimental conditions.

CHAPTER 8
POWER ABSORPTION MODEL FOR MICROWAVE PROCESSING OF
COMPOSITES IN A TUNABLE RESONANT CAVITY

8.1 Introduction

Graphite ﬁber/epoxy composites have been processed using microwave energy
in tunable resonant cavities. The temperature uniformity and heating rate during
processing were controlled by the electric ﬁeld pattern and strength inside the
composite. In order to take full advantage of microwave processing, the interaction
between microwave radiation and composites needs to be understood. For an empty
cavity, the electric ﬁeld inside the cavity can be calculated based on Maxwell’s
equations and the boundary conditions. For a cavity loaded with a small object
which only perturbs the resonant frequency by a few percent, a cavity perturbation
technique is usually used to calculate the electric ﬁeld inside the cavity". This
technique has been used to measure the dielectric properties of polymers in the TM”;
mode". For a coaxially loaded cavity with a homogeneous, isotropic lossy rod, the
electric ﬁeld inside cavity was calculated using a mode-matching techniquem”.
However, the electric ﬁeld inside an anisotropic composite plate loaded cavity is very
complicated. In order to approximate the microwave power absorption rate during

processing, a simpliﬁed ﬁve-parameter model is presented in this chapter.

8.2 Problem Simpliﬁcation

The electromagnetic ﬁeld inside a cavity loaded with an anisotropic composite,
especially a graphite ﬁber reinforced composite, is rather complicated. Figure 8.1
shows the conﬁguration of the composite loaded cavity and interactions between the

composite and the electromagnetic standing waves inside the cavity. As shown in

170

171
Figure 8.1(a), the composite is lifted up from the bottom where the electric ﬁeld is
very small. The electric waves interact with the composite in every surface as shown
in Figure 8.1(b). As a ﬁrst approximation, the incident wave on each composite
surface was considered to be a linearly polarized transverse electromagnetic (TEM)
wave. The arrow in the Figure 8.1(b) represents the direction of the propagating
TEM wave. The interaction between travelling TEM waves and graphite ﬁber/epoxy
composites have been studied previously in the areas of material

sciencesazonnnzs and mmmnao.

Since the incident TEM waves from the four edges are perpendicular to the
incident TEM waves from the top and bottom, the power dissipated in the composite,
P, can be decomposed into two terms, P, and P,. P, is the power dissipated due to
the incident TEM waves from the edges while P, is the power dissipated due to the
incident TEM waves from the top and the bottom. To further simplify the problem,
P, is considered to be constant through the thickness of the composite.

Based on the above assumptions, the electric ﬁeld distribution across the
thickness of the composite only depends upon the propagating behavior of the incident

TEM waves from the top and bottom. Once E is known through the thickness, the

power dissipation due to TEM waves from top and bottom can be calculated from
Poynting’s theorem.

1», = lei-328’ (8-1)
2

where P2 is time average power dissipated in the composite due to electric ﬁeld of the

incident TEM waves from the top and bottom of the composite (ir- and jg, in W/m’), 3’

and 5'" are the electric ﬁeld vector and its conjugate in V/m, and 3f is the effective

dyadic loss factor of the composite in F/m.

172

 

 

 

 

 

 

 

 

J
‘-—>

\Top 9 I ate

 

Lp C0upl l ng probe

 

Compose l to
\ ‘ 7
I I 1 ‘""

 

 

 

 

TOﬂOﬁ plat. Bottom plate

 

 

 

 

 

(a) Conﬁguration of the Composite Loaded Cavity

/E.

 

 

 

 

/ E

/

Es

 

 

 

(b) Interaction Between Composite and EM Waves
(The Arrow Shows the Propagating Direction of the TEM Waves.)

Figure 8.1 Incident Waves on the Composites During Processing.

173
8.3 Electromagnetic Model

Figure 8.2 shows the conﬁguration for the simpliﬁed electromagnetic model.
The composite, consisting of N laminate plies, is exposed to top and bottom incident
TEM waves. To calculate the power absorption rate due to the top and bottom TEM
waves in each ply, the wave behavior at each interface must be derived ﬁrst.

Let’s consider +z travelling waves at the 11" interface. As shown in Figure

8.3, part of the incident wave, 135,, will reﬂect back as a reﬂected wave, 6,. and the

rest of the incident wave will go through the interface as a transmitted wave, E,- The E,

can be decomposed into two parts, Ex and By. Using new coordinates p and q to
represent the composite’s principle direction, the angle from the x-y coordinate to the
p—q coordinate is 6 as shown in Figure 8.4. We assume that the ﬁber direction is the
principle direction of the complex permittivity tensor for the composite and let p and
q be parallel and perpendicular to the ﬁber direction, respectively. Therefore, the
dyadic complex permittivity in p-q—z coordinates is a diagonal tensor.

is l
8,00

8 = 0 e; 0 (8’2)

 

 

K0 0 8;,

Since there are no off-diagonal terms, the calculation in this coordinate system
will be much easier than in x-y-z coordinates where off-diagonal terms exist. Once
E, and E, are known, B, and E, can be easily obtained by projection of E, and E,I in

x-y coordinates.

1%

 

 

 

 

 

 

 

interface
'1 2 3 n N-1 rm N+4
o I y
_._____). [I 2 ............................................. N— ’1 N ‘—_
top bottom
2:0 Z=L

Figure 8.2. One Dimensional Conﬁguration for Power Absorption Model

 

 

Figure 8.3. TEM Wave at Two Isotropic Media a and b

 

 

 

 

 

 

Figure 8.4. Base and Principle Coordinates

Maxwell’s equations are used to relate the electric ﬁeld vector to the magnetic
ﬁeld vector“. Maxwell’s equations in an anisotropic, non-magnetic composite are

in = -jw [117 (8'3)

vxn qua-E (84)
Taking the curl of Equation (8-3), we have

VxVxE = -jmpVxI? (3'5)
Combining Equations (8-4) and (8-5), we obtain

V(V-E) -vzi = -j0 [1003‘ 1'?) (8-6)
As if is a linearly polarized TEM wave propagating in the z direction, we have
vi = 0 (3'7)
Put Equation (8-7) into (8-6), the wave equation in the anisotropic medium becomes

As if is a planar TEM wave travelling in z direction, we obtain

176
v12+ 01 p a, -E = o (8-8)
3 = E", +qu. (8'9)
vcggﬂ (3-10)
dz2
Substituting Equations (8-9) and (8-10) into (8-8), we obtain
£5 +szP = O

2
dz (8-11)

LE +135 = 0

dz; 9 1

where k,2=w2ue,° and k,’=w’ueq°, u is the permeability of free space, and e,’ and e,’
are the absolute complex permittivity of the composite in the p and q principle
directions, respectively.

The solutions for E, and EI are
5353'” +53"? (8-12)
Eq=15;e""v‘+£;e”v‘
where superscript + and - represent +2 and -z propagating waves, respectively, and
k, and lrq are the wave numbers in the p-axis and q-axis directions respectively.
Rewriting Equation (8-3) and combining with Equation (8-9), we obtain

H=—j—in
”F (8-13)
-.-_L[-_a.§1¢ +952!)
up 62 ' az '
Deﬁning

177

j 6E
H,--— a:
M (8-14)
H =_La_E£
' on 32

and putting Equation (8-12) into (8-14), we obtain

k + - - .
H's—Ll-E'eﬁéq-E‘eﬁal
«on
(8-15)
k . - - .
H'=:;ﬁ[E'e”-E'e”]

Deﬁning

O

=_1_ "F" o E; (8-16)
‘ Ji 0 (a: E

O

P

a: ﬁt

and

H =-_L - 6‘ 0 E‘ (8-17)

ﬁlo/3:3;

and combining Equations (8-15), (8-16), and (8-17), we obtain

'Q

If

I.

H =H‘e“¢‘+H'e"f
p r p (8-18)

H,=H;e"r‘+H;e ”‘r‘

The E and H of the incident TEM waves in the x-y coordinates can be

expressed in the p-q coordinates by

sinb cash

1 - l
E) E!

and

178

r , r
H: - 0MB -srnb Hp (8'20)

sinb 0036 H:
For incident waves, the relationship between i and I? in the x-y coordinates

can be obtained from Equations (8-16), (8-19), and (8-20).

‘ i
H" . M. 5* (8-21)
11‘ E‘

y
where i represents the incident wave, M. is the characteristic matrix for an anisotropic

homogeneous medium a, and 6 is the angle from the x-y coordinate to the p—q

coordinate in medium a.

Ma=;[°f’“ m] ° J? (a... w] (.22)
ﬁsrnb cosb R O —srnocoso

For reﬂected waves, the relationship between E and I? in the x-y coordinate
can be obtained from Equations (8-17), (8-19), and (8-20).

r f
”x g -u 5* (8-23)
r ‘ r
H, Er
where r represents the reﬂected wave.
Similarly, the relationship for a transmitted wave is
t t
”x g M 5* (8-24)
I b t
H E

where t represents the transmitted wave, and M, is the characteristic matrix for an

anisotropic homogeneous medium b.

179

_ 1 cosb’ -sino’ 0 'y/e—t; cosb’ sinb’ ($25)
" . -sin6’ cosb’

l1 e, 0
where 6' is the angle from the x-y coordinate to the p—q coordinate in medium b.

sinb’ 0085’

 

At the interface, the continuity of the tangential component of the electric and

magnetic ﬁeld leads to
E 8 E + 3' (8-26)
[7' = H‘ + 17'
Writing out each component in Equation (8-26), we obtain
(8-27)

A E‘

Vl’!‘ km
“a. .m

t‘ i r
H, H, H, (8-28)

 

H; H; H;
Putting Equations (8-21), (8-23), and (8-24) into Equation (8-28), we obtain

5: E: E:
M, = M. - M, (3'29)
5‘ E; E;

y
Solving Equations (8-27) and (8-29), we obtain

1 Er
x = T x (8.30)
t i
E)
and
E" g 5 5* (8-31)
r t
E? E!

where T and ﬁ are the transmission and reﬂection coefﬁcients at the interface

respectively.

180

- -r
7 " m"- m")! M- (8-32)
R = (M, +M,)' (M, -M,)
The above derivation is for incident TEM waves propagating in the +2
direction only. If incident TEM waves come from both sides, as shown in Figure

8.5 , the continuity condition at the interface becomes

5"; = if +5; (8-33)
5!. = id. +3;

with
a; 42'; (3-34)
is;- = are;

where if is the reﬂection coefﬁcient when the TEM wave is travelling from medium

a to medium b as deﬁned previously.

 

 

a b

ET E:

> ———>

E
__ E:<—- ——> E;
E
<———
Et El
~<——— <

 

 

Figure 8.5. Waves at Interface with TEM Waves Propagating at Both Directions

181
Deﬁning the +2 and -z direction effective transmitted waves at the interface as

E":- EHWE' (8-35)
3 =5 +5
and putting Equations (8-33) and (84%) into (8- -35), we obtain
E" = (1‘ +R)E,* 42; (8-36)

5' . (f—EMZ‘,’ +33:

For a multi-ply composite, each interface, 1,2,...N +1, ean be treated in the
same way as above. From the wave behavior at each interface, the electric ﬁeld
inside each ply can be derived. Consider the electric ﬁeld inside the n‘II ply as shown
in Figure 8.6. The electric ﬁeld inside the n‘ll ply can be obtained from the +2

effective transmitted waves at the n" interface, the -z effective transmitted waves at

the (n+1)"I interface, and the attenuation tensor of the n‘l ply, In . The -z incident
travelling wave at the n‘” interface, (3"), , and the +2 incident travelling wave at the

(n+1)" interface, (3’)”! , ean be calculated as

(59.4“ ”IE;
(8-37)
(3;): = Xe ELI

If X , g +, and E 1- are known, the average electric ﬁeld in the n“ ply ean be
3 n 8*
approximated by

(E), = éa‘s‘; +1.29 +i;,,+A 11'3"“) (8-38)

For TEM waves in a homogeneous isotropic material with permittivity e‘, the

electric ﬁeld is attenuated according to following equation.

5(2) =E(0)exp( -yz) (8-39)

where 133(0) is the electric ﬁeld vector entering material, 2 is the propagating

182

direction, and 7 is the propagating constant:

 

 

 

 

7 =17: =jm/e‘ u “‘40)
h h+1
CEIDn
——9 'ﬁ
_ > Eh> CEi 39:1
(Erna
CET’ jn+1
— _ —- <—
< <__— <
CEt Bria-’1
L— e—

 

 

 

Figure 8.6. Electric. Field Inside n‘II Ply

For TEM waves in the composite material, the electric ﬁeld can be
decomposed into its principle direction, parallel and perpendicular to the ﬁber. The
attenuation of the electric ﬁeld in each direction can then be treated separately.

Ep(z) = E,(0) exp( 1,7.) (8 41)
E,(z) = 54°) exp( -v,z)
where
7p =11!
Y,=ik,
Rewriting Equation (8-41) in matrix form

183

Fa”) T“ ° ]

- 5’0) (8-42)
E,(z)

0 exp( -v,z) E,(0)
From Equation (8-19)

E,(0) 3 [costs sins] 540) (843)
1250) as could 1:;(0)

5,.(2) [cost sins) EA!)
552) -sinb coca E,(z)

From Equations (8-42), (8-43), and (8-44)

] (3-44)

 

[E,(z)] a I. E,(0)] (8 _4 5)
E,(z) E,(0)
where
X =(cosb -sinb][¢xP('sz) 0 “cash sins] (8.46)
' sinb cash 0 exp(-y‘z) «ind coed

The effective transmitted +z travelling wave at the n‘ interface, 2" , can be
expressed by the effective transmitted +2 travelling wave at the (n-l)m interface,
EH‘ , and the effective transmitted -z travelling wave at the (n+ l)‘II interface, EM“ ,
using Equations (8-36) and (8-37). Similar relations can be obtained for the effective

transmitted -z travelling wave at n“ interface, E".
E; = (i+§n)Xu-l ﬁve-1 - RI In E-Ml

in” = (i-ﬁnﬂ. 8"”, + R X 131““

a 3-1

(8-47)

184

In addition, we look at two special interfaces, the top and bottom interfaces

where n=1 and N +1. As shown in Figure 8.7, the incident TEM waves are assumed
to be known.

(531 = Er (8-48)
(Edam =3:
,1 N-o-‘l
CE| 31=ET CEt)
——>
En CErDN+4=AerIS CE:DN+1
-o- — ——_’ ——.
CE? )1 CEVD’l E... ‘1
- .— —'- _~ _ ——'1;
Eq (Er DN-o“ (Er-3N-“
CE: 1 (5731:;4E; E;*“{
‘—- +—— <———-
E E_ =E
C tDN*“ C l DN-‘ﬂ B

 

 

Figure 8.7 TEM Waves at Top and Bottom Plies

Rearranging Equation (8-47), the general equations for travelling waves
become

(i+§ u)Aa-1E n-l 2.2: KARE n+1 =0

(8-49)
ill” ,1?“ 45’; +(i 4,91, E”; = o
For n=1, Equation (849) becomes
-5; 45,1, 5; = -(i+§,)i, (3-50)
'Er- +(i’ﬁr)zriz- 3 -§IET
For n=N+ 1, Equation (8-49) becomes
0.4-R ,1)Xl‘,i§N+ “Em; = Ennis (8-51)

ﬁmrXNE; -EN+1- = -(i-§N+I)EB
Therefore, 2N+2 equations can be generated from (N + l) interfaces to solve for

185

2N+2 unknowns, E; and Ei', i=1,2,...N+l. Based on the above derivation, A"
and E depend upon the properties of the composite, incident wavelength, and the

angles from x-y coordinate to ﬁber orientation (6) of each laminate.

Putting 2N +2 equations generated based on Equations (8-49), (8-50), and (8-
51) into a matrix form,

Gist, (s-sz)
and
0, G,
= (8-53)
‘5 [6. a]

where 6 is a (2N+2) by (2N+2) matrix. The Q’s are (N+1) by (N+l) matrices

with i=1,2,3,4. E and E0 are (2N+2) by 1 matrices. The expressions for the Q’s,

E, and E0 are
’ -i o o 0‘
(13491, -i o o
,, o o
0‘ g o o (1315,)?” -I‘ 0 (8-54)
0 o 0
K o o o o (i+§,,,,).i’, -

 

 

WANNA)

 

 

 

 

 

(0 ’ﬁr “l O 0 O O O
o 0 452A: 0 0 o 0
o o 0 ° ° °
62: 0 0 0 0 '5. “a 0 O
o o o o 0 0
0 O o 0 O O 'RNXN
{0 0 O o O O 0 lawman)
o o 0 ° 0‘
a“. o o 0 ° °
0 .. o 0 O
63: 0 0 £1“ 0 0 0
o o o 0
o 0 0 0 ENE" 01w.1)xav+r)
{—i (i- E) I, 0 O 0 O 0 \
o J (Tl-EH} 0 O O 0
o 0 ° ° 0
O 0 o —i (i-RJX. 0 °
0 0 0 ° 0
O 0 0 0 0 J (i'ﬁﬁAN
\0 0 0 0 0 O J I

 

w+l)x(N+1)

(8-55)

(8-56)

(8-57)

187

 

 

RN” EB
-RIET

 

' -(i+§,)i5',‘

(2N4)x1

 

((i-ﬁ +05»;

(2N+2)x1

(8-58)

(8-59)

188

E,“ and E; in the x-y coordinates are

a . 042, (8460)
where
I I
6.12 61 62 (8'61)
0; 61
and

c: = (6, -0.0."0.>“
61 - (0. -O,é;‘62)"
c5; = -6;‘6,61
(i; = -6."é,6{

(8-62)

Once E; and EJ, i=1,2,...N+l, are known, the electric ﬁeld inside each ply

ean be estimated using Equation (8-38). For unidirectional composites, there are only
two interfaces, n=1 and N+l. The only parameters that need to be calculated are

E,’, gm, El', and ERA. The reﬂection coefﬁcients at the n=1 and N+l

interfaces have the following relationship

EN,

vi

1 1

For a unidirectional composite, Equation (8-52) can be simpliﬁed to

’ -i o 0 45,1, ”if ‘ '-(i+R,)E,‘

(f -§,)Z, —i o 0 5,3,, -§,iv:',
o o -i (If-RM, E,’ -R,i:’,

, 4,1, 0 o -i Asia) r6491?”

(8-63)

 

 

 

 

 

 

189

Once E12 gm , El', and EL, are known, the electric ﬁeld strength at any

location inside the composite can be calculated based on the attenuation of the ﬁeld.
To simplify the calculation, the x-y coordinate is taken to be the same as the p-q

coordinate with the ﬁber direction along the x or p axis.
E,(z) = (new -v,z) + (antennae-L»
£,(z) = (Ef),exp( -t,z) + (5,}.1),exp(r,(z -L»

(8.64)

To simplify the calculation of the microwave power absorption inside the
composite, i.e. to eliminate the off-diagonal terms in e,’, the electric ﬁeld strengths
are needed in p—q coordinates. The electric ﬁeld in p-q coordinates can be ealculated
from the in x-y coordinates.

(5.). = (mean, +<E,>.sin6.
(3')" = -(E,)nsin5. +(E,).coso,,

(8-65)

6,, in Equation (8-65) for unidirectional composite equals zero. The microwave power
dissipation rate due to top and bottom incident waves in the n‘ll ply, P2, ean be
determined using Equations (8-1) and (8-2).

F. = nﬂeﬁ’ I (12,), I2 +e,’,’ I (Ly), I”) (866)

The total time-average microwave power dissipated in the composite due to the
TEM waves from the top, bottom, and sides is

N .
P =82 h.(P.).. (347)
u-l

where s is the area of the composite surface, h, is the thickness of the n“ ply, and
Fe = P1 +P2.

190
8.4 Parameter Estimation

In Equation (8-60), the 2N +2 unknowns are E" and Ex, i=1,2,...N+l.
For a given composite, A" and E are ﬁxed. Therefore, E" and E; are only

functions of £1 and E, . During microwave processing, the cavity is in a resonant
condition and the microwave energy is conﬁned inside the eavity and coupled to the

composite. For a ﬁxed composite location, £1 and E, are only functions of the

resonant mode and the input power level. £1 and E. ean be expressed as
i, =A,(cos6,5‘+sin0,s)exp(jwt+yz)
(8-68)

3, = A,,(cosﬂ,._e‘,I +sin0,5)exp(jtot - yz)
where AT and A, are the magnitudes of the incident TEM waves at top and bottom,

respectively and 0, and 0, are the polarization angles of the incident TEM waves at
the top and bottom, respectively.

The total microwave power dissipated in the composite per unit volume per
time, P,, is a function of P,, Er and E3. From Equation (8-68), Pc can be expressed
as

P, = P, +P,(A,,A,,e,,e,) (8-69)
The values of P,, AT, AB, 0,, and 0,, for a given mode in microwave processing of

composites can be obtained from the temperature measurement of fully cured
composites in that mode.

191
8.4.1 Energy Balance

Assuming that the temperature is uniform across each laminate, the one-

dimensional energy balance for a fully cured composite during microwave cure is

ar 6‘1
pc—at- = lg? +P,(p,,A,,o,,A,,o,) (8-70)

with the initial condition
mm - T,
and the boundary conditions

81'
—lz-L=O

62

37'
"E '52- luo = MT, - 7101))

The heat loss occurs at the top, z=0, with free convection while the other side, z=L,
is insulated. Once P,, A,, A,,, 0,, and 0, are known, T(z,t,P,,A,,A.,,0,,0,) can be
ealculated numerieally From Equation (8-70). To simplify the notation, B, , 32, B3, 8,,
and B, are used to represent P,, A,, A,, 0,, and 0,, respectively.

8.4.2 Optimization of Parameters

Ordinary least squares minimization is used to optimize 81.5. The optimization

scheme is

I

5 =2 2 (Y,-r,(a,,,))2 (8-71)
8' I

where i represents time, j represents the measurement point, I is the number of points

measured, Y,- is the measured temperature at pointj and time i, and T,,(B,,,) is the
calculated temperature using Equation (8-70) at pointj and time i.

192

To estimate B’s, we ﬁrst need to ealculate the sensitivity coefﬁcients, X”. The
sensitivity coefﬁcient is deﬁned as the ﬁrst derivative of a dependent variable, the
temperature in this case, with respect to an unlmown parameter, 81 to BS.

67!: )
X,(z,.t,) 2%, n=l,2,...,5 (8-72)

If the sensitivity coefﬁcients are either small or correlated with one another, the
estimation problem is difﬁcult and very sensitive to measurement errors, and the
solution will not be unique. Therefore, the Xh’s are required to be large and
uncorrelated. X, is considered to be large if 3.x, is on the order of the temperature
rise. Xj’s are uncorrelated (linearly independent) if B,X,, versus time curves have
different shapes for different n. Furthermore, the estimation problem is linear if the
sensitivity coefﬁcients are not functions of the parameters.

One way to calculate the sensitivity coefﬁcients is using a ﬁnite difference
method131

3 37194.3”)

 

X

 

 

1" an”
(8-73)
= RZJJ.B..B.+_53.) -nzjstsﬂ]-5)
op,
where n=l,2,...,5, m=1,2,...,5, but m¢n, and 58.=0.000IB,.
Taking derivatives of Equation (8-71) with respect to B’s, we obtain
’ 6T
as = X 2 2(Y,-T,)(-J), n=l,2,...,5 (8-74)
35.. I I an.
Minimizing S with respect to 8,, we obtain
’ 0T
2: 2: arm-6?") = 0. “1,2,...: - (345)
‘ I I

As T is a function of the Bs, the temperature of the (k+ l)“ iteration can be related to
the temperature of lr'II iteration.

193

 

 

a]- 0:) 3T 0:) 31- (k)
100M) =Tua°+—-9—AB,+ 0 AB,+...+ V A0,
331 3B2 635 (8-76)

= T,,“) 41,000)”, «fans, +... +x,,<*>(nAp,

Putting Equation (8-76) into (8-75), we obtain

Bf)“, wuwapgafapgaﬁap, +3.9)”, =4,“

(8-77)
It - 1,2,...,5.
where
1
Ana) 3 z 2 (Y0 _. 16(1)) Elma)
i J
and
J
B...” = 2: 2: thto lift». a =12.....s.
‘ I
Putting Equation (8-77) into matrix form, we obtain
Eng 3 10:) (8-78)

where

{511(k) and!) 3,30" 3140‘) 3150:”
3210:) 3220:) 3230:) Bum 3250:)
Ea) ‘ 3310:) 332a) 3330:) 834m 335m
341a) 342“) 3‘30» Bu“) 845a)

@5100 3520:) 3530:) 8540‘) Essa) ;

 

 

(All,l
AB,
B= AB.
AB,
\A $5)

 

 

3x1
and

 

 

5x1

Clearly, E‘” is a symmetric matrix since 13.,“ equals 3,“). The increment of 81,,
between the k“ and (l(+1)‘ll iteration can be obtained by solving Equation (8-78).

The next set of B’s is the values of previous set plus the difference ealculated

based on the least squares optimization, §. The iteration is terminated when change

in S is within 0.1 percent and the Bl.5 values of the last iteration are the ﬁve
parameters for the electromagnetic heating mode. The electric ﬁeld inside the
composite can be generated based on these ﬁve parameters.

8.5 Measurement of Five Parameters for Microwave Power Absorption Model

A ﬁve-parameter microwave power absorption model has been developed in
section 8.3. The ﬁve parameters are Pl (the power dissipated due to the TEM waves
from sides), A, and 0, (the magnitude and the polarization angle with respect to the
ﬁber direction of the top TEM wave), and AB and 0B (the magnitude and the
polarization angle with respect to the ﬁber direction of the bottom TEM wave). The
methodology to generate the ﬁve parameters based on the temperature proﬁles
obtained from microwave heating of a fully cured composite were described in section
8.4. The microwave power absorption model and the optimization methodology

were applied to the microwave heating of a fully cured 72-ply unidirectional Hercules

195

AS4/3501-6 composite in a resonant mode at Lc=l6.03 cm and a 0° ﬁber
orientation. The dielectric properties of the unidirectional AS4/ 3501-6 composite is
available elsewhere ‘. The computer code for the optimization is attached in
Appendix II.

The experiments were conducted at three input power levels, 70W, 80W, and
150W. Temperatures were measured at ﬁve locations with T1, T2, T3, T4, and T5
loeating at z=0, 0.177, 0.278, 0.532, and 0.785 cm respectively. The temperature
proﬁles across the thickness during microwave heating at 70W, 80W, and 150W are
shown in Figures 8.8 to 8.10, respectively. The points are experimental values and
the lines are the calculated values from the optimization. In general, the two are in
good agreement. The slightly higher calculated values than the experimental data at
the beginning are due to the lack of critical coupling at the beginning of the
microwave heating. Figure 8.11 shows the ﬁve parameters versus input power for
the resonant mode at Lc= 16.03 cm. Clearly, P,, A,, and A, are functions of input
power and the electromagnetic mode while 0, and 0, are only functions of mode for a
given composite. 0, and 0, are equal to 1.556 and 0.0201 radian respectively for the
given mode. A, and An increase linearly with the square root of input power as
shown in Figure 8.12 and Figure 8.13. Their relations are A,=9.75x10t2 P” and
AB=5 .867 P“. Pl increases linearly with P“3 as shown in Figure 8.14 with
P,=0.104 P”. With the relationship of the ﬁve parameters as a function of input
power for the given mode, the temperature proﬁles during microwave cure can be

readily simulated.

Temperature (°C)

Temperature (°C)

951)
815
BOT)
72i5
SST)
515
SCI)
42£5
35f)
27£5
20f)

 

 

 

 

 

#- + q
L + _
+ f
1' 0 T1 + . f A "
f ‘ T2 . + 9 . ‘ -.
_ V T3 + - 2 ‘ q
_ 0 T4 - , .
_ + T5 .
L ‘ . S
O
. /// _
L g 3 Input Power : 7OW _
/ 8 8 .
O 24 48 72 96 120 144 168 192 216 240

Time (sec)

Figure 8. 8 Temperature Distribution Across the Fully Cured 72-ply
Unidirectional Hercules AS4/3501-6 Composite during Microwave Heating

130
120
110
100
90
80
7O
60
50
4O
30

at L,=16.03 cm with 70W Input Power

 

 

 

 

 

L r i
+ O
P o O J
1- V .-
a V A
P v . -r
. ° 0 J
r "’ ° ..
LI 3/;/” q
///° ° ° Input Power : 80W 1
O
O 24 48 72 96 120 144 168 192 216 240

Time (sec)

Figure 8.9 Temperature Distribution Across the Fully Cured 72-ply
Unidirectional Hercules AS4/3501-6 composite during Microwave Heating

at L,= 16.03 cm with 80W Input Power

197

 

 

 

 

130 V I ' l V I I I v I v 1 r T 7 I ' 1 f
* 4
119 r o -
* 0 T1 , v -
108 : ‘ T2 . Z A j
‘7 T3 . v a o d
8 97 . 0 T4 , v - 0 ° 4
-3 86 - + T5 . ' _ ° —
E t' '°' . v/ _,
.3 75 P + °/. 1 7
S P + + 0/0/ ' q
8. 64 t '* /g/. 0 ° ..
E r + /V' O -1
,2 53 r + T ,/< 3 ° 4
‘ ’ X o .l
42 - 2 g 3 ° —
F / 3 ° Input Power: 150W ‘
31 /. - , 4
1

20 . r . r a r r r r 1 . l r

O 24 48 72 96 120 144 168 192 216 240

Time (see)

Figure 8.10 Temperature Distribution Across the Fully Cured 72-ply
Unidirectional Hercules AS4/3501-6 Composite during Microwave Heating
at Lc= 16.03 cm with 150W Input Power

 

 

 

13.0 ' r Y T T r I 1 T T T 1 v I r r v I
.+
11.6 _ u
_ ° ongIeT(rod) _
10'2 _ A on IeB(rod) . q
(0 8 8 P 7 AB V/m) X 8-1 + _
“3’ ' _ 0 P1 W/cm3) x e+1 + q
:3 7.4 _+ AT(V/m) xe+1 _
3 6 0 ~ _
qé . ‘
E 4.6 b ‘1
E 3.2 - ’ .
1'8 h e e e 7
O4 ' I. a j -1
_1O 4 r 1 Q r r r r g r a 1 1 gr % 1 1 1

 

 

 

O 16 32 48 64 80 96 112 128 144 160
Input Power (W)

Figure 8.11 Input Power Effect on the Parameters in Microwave Power Absorption
Model during Microwave Heating of Composite at Resonant Mode with Le =16.03 cm

198

 

 

 

 

1.20 . 1 1 . . 4 , . , . . . , . , . , . 4 4
1.08 - 1
0.96 L o l
0.84 i I
0.72 l i

E 0.60: I

E 0.48 - .

’2 0.38; i
0.24 l 3
0.12 l :
000L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ‘

0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0

[:05 (w0.5)

Figure 8.12 Linear Relationship Between AT and P“ During Microwave Heating
of AS4/3501-6 Composite at Resonant Mode with L,=16.03 cm

72,0 .
64.8
57.6
50.4
43.2
36.0
28.8
21.6
14.4
7.2

 

AB (V/m)

I
llllLlLllllllllllll

' I

 

 

O O 1 l 1 1 1 l 1 l L l l L 4 1 1 1 J.

0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.411.71.10
p05 (web)

 

Figure 8.13 Linear Relationship Between A,, and P” During Microwave Heating
of AS4/3501-6 Composite at Resonant Mode with 1.516.013 cm

 

0.50 . T . T . T T T T T T T T 4 . T T T

0.45 i .1
0.40 L S
0.35 l o 3
0.30 T i
0.25 l i
0.20 1
0.15 -
0.10 r —
0.05 i i

0.00 1 I 1 l 1 L 1 l 1 l 1 1 1 L 1 i L l .a
0.00 0.45 0.90 1.35 1.80 2.25 2.70 3.15 3.60 4.05 4.50

p03 (W03)

P1 (W/cm3)

 

 

 

Figure 8.14 Linear Relationship Between P1 and P” During Microwave
Heating of AS4/3501-6 Composite at Resonant Mode with I.,=16.03 cm

8.6 Conclusion

A one-dimensional ﬁve-parameter microwave power absorption model was
developed for microwave power dissipation inside the composite. A FORTRAN code
combining the energy balance equation, the microwave power absorption model, and
the least squares optimization was developed to generate the ﬁve parameters based on
the temperatureltime/position proﬁles obtained during microwave heating of a fully
cured composite. A set of ﬁve parameters was generated for microwave heating of
72-p1y unidirectional AS4/3501-6 composite at a resonant mode at L,=16.03 cm.
Among the ﬁve parameters, Pl increases linearly with input power to the power of
0.3, AT and A,, increase linearly with the square root of the input power, and 0, and
0,, are not functions of the input power. With the relationship of the ﬁve parameters
as functions of input power for the given mode, the temperature proﬁles during

microwave cure can be readily simulated.

CHAPTER 9
PROCESSING MODEL FOR MICROWAVE AND THERMAL
PROCESSING OF COMPOSITES

9.1 Introduction

In the previous chapters, the microwave and thermal cure of epoxy resins and
composites was studied. Various processing techniques were investigated, and a
microwave power absorption model was developed. This chapter will focus on the
prediction and control of the temperature proﬁle and extent of cure during microwave
and thermal processing of Hercules AS4/3501-6 composites.

Optimum material properties and reliable processing parameters are often
required for the processing of high performance thermoset composites. These are
usually obtained through extensive testing of various processing conditions
experimentally. However, the result for one geometry is not applicable to others.
Therefore, this procedure of locating the optimum processing conditions is very
costly. A fast and cost—effective way to ﬁnd the optimum processing condition for
various processing requirement is through simulation of processing using a processing
model‘”. The model should include the chemical, physical, and mechanical
behavior of the composite during processing. The chemical behavior and the
temperature/time/position proﬁles during both microwave and thermal processing
were discussed in previous chapters. A processing model that consists of coupled
kinetic and energy submodels is developed in this chapter. The kinetic submodel
describes the reaction rates and predicts the processing time. The energy submodel
consists of the power absorbed from microwave radiation and the exotherm released
from reaction. A computer code was written to simulate the processing under
various conditions. The code can be used to predict the various properties of the

composite on-line during microwave and thermal processing. It can be also used to

200

201
design an optimum processing condition based on the required properties of the ﬁnal
products.

9.2 Background

The review of kinetics can be found in Chapter 3. A general review on
rheology is provided here. Rheology deals with viscosity and resin ﬂow in the
composite during cure. Unlike thermoplastics, the viscosity of thermoset is not only a
function of temperature and flow geometry but also a function of the extent of cure.
The effect of increasing temperature is complieated. On one hand, increasing
temperature will lower the viscosity. On the other hand, increasing temperature will
increase the reaction rate, and thus the extent of cure, and increase the viscosity.
Resin ﬂow is inﬂuenced by ﬁber compaction, ﬂow velocity, actual resin pressure, and
void formation. Because the ”apparent” rheologieal data generated on the ﬁber-
reinforced prepreg material closely resembles data generated on the neat resin as

133

shown experimentally , the viscosity data for pure resins are usually used for

processing modeling of composites.
9.2.1 Viscosity

There are two types of model for the prediction of the viscosity of epoxy resin
during processing, WLF form and Arrhenius forms. Based on the free volume
argument, the WLF equation was derived”

103(_‘L(D_) .-. - M (9-1)
MT) c2 + T-T,

where c1=26.8, c2=13.4, T, and 11(1‘.) are functions of temperature and time. To ﬁt

the chemoviscosity data of epoxy system, a modiﬁed WLF equation was proposed by

Tajima et. a1. ‘3’

The modiﬁed WLF equation was used to ﬁt the experimental data of non-isothermal

cure of Hercules 3501-6‘“. The ﬁtted parameters are c1=29.667, c2=36.926,

202

ln(ﬂ) -.- _ 29:32 (9-2)

11(1) car-r,
Tg=283.42 + 196.5 a - 925.4 a2 + 3435 83 - 4715 a‘ + 2197 015, and

ln(n('l‘g))=20.72+8.56a-9.69a2 + 41.17013.

A theoretical dependence of the viscosity 1; on molecular weight and
temperature has been proposed for epoxy-amine systems using WLF-type of
expressions‘”.
c1(T, - T EB]

M = g [M] CZ + T'-I£'_. (9-3)
11(7'0) M“ e c1(T, - Tp(a))
c2 + T, - T‘(a)
Where 1; is the viscosity, M, is the weight average molecular weight of the polymer,
g is the ratio between the radii of gyration of a branched chain and a linear chain of
the same molecular weight, T, is the reference temperature, T,,, is the glass transition

temperature of the unreacted system, T, is the glass transition temperature, a is the

34 exPI

 

extent of cure, and cl, c2 are constants. This model requires knowledge of the
composition and structure of the epoxy and the amine molecules and a knowledge of
the reaction mechanism. The validity of this equation has been veriﬁed for
TGDDM/DDS epoxy system at temperatures of low reactivity.

Another type of model is in Arrhenius form which is based on the assumption

that ﬂow is an activated process. The general form is ”3"”

11(7) aroma/Rn (94)
where A and n are constants, and AB is the activation energy. For thermoset

polymers, an exponential function was proposed as the chemoviscosity equation

n(T.t)=n.¢xp(kt) (9-5)
where no and k obey the Arrhenius relation. For non-isothermal cure, the above

equation was expressed as W

203

11mm) =1nn,+ AEJRT+f‘k.exp(aE./R1)dt (945)
0
However, this type of model earries no information about the chemical conversion-

time relationship. Higher orders of t and the introduction of an entanglement factor
into k have been used for better ﬁt to the experimental data‘“. The viscosity of
Hercules 3501-6 resin under isothermal cure was modeled using equation “°

11 .. n.exp(U/RT+K¢) (9-7)
where n... is a constant, U is the activation energy for viscosity, and K is a constant
independent of temperature. This equation is only valid before gelation, that is
a<0.3. For this speciﬁc system, K=14.1, U=9.08 x 10‘ J/mol, and n.= 7.93 x
10“ Pa s. ,

The viscosity 1; was also assumed to be a function of the initial viscosity 1;, of
unreacted monomer and a term involving the reaction dependence of the bulk
viscosity”. ‘

n = no * reaction term (9‘8)
The initial viscosity can be expressed as either a WLF or Arrhenius-type equation.
The reaction term of the viscosity is dependent upon the extent of cure. For 1st order

kinetics under isothermal curing

1110.) = ln(n) +EJRT+ okxexp( -E,lR1)t (99)

where 17, is initial viscosity at inﬁnite temperature, E. is the activation energy, R is
the ideal gas constant, T is the absolute temperature, <l> is entanglement factor, k, is
the kinetic pre—exponential factor, B, is the kinetic activation energy, and t is time.
For nth-order kinetics under isothermal curing

111m) 41101,) +EJRT+ —¢—‘-l-1n[1+(n-r)k,exp( -E,/Rr):] (9-10)

"—
The dielectric properties of thermoset and thermoset composites have been

used to monitor the viscosity. Based on Stoke’s law and an assumption that the
molecules are spherical, the viscosity of the resin 17 is found to be proportional to the

204

inverse of the ionic conductivity 0‘“. For resin systems with R) >1, l<e' <25,
and 12< <e" < <R, the ionic conductivity ean be expressed as‘“

o =oe.¢”s21rfeoe” (9‘11)
Where 111 is the angular frequency, f is the frequency in Hz, e. is the permittivity of
free space, e" is the relative loss factor, 15' is the relative permittivity, and R is a
constant which depends on the geometry of the sensor and the material”. Several
experimental results have been reported on the comparison of viscosity and the
inverse of the ionic conductivity for TGMDA/DDS based epoxy and prepreg.
Approximate agreement was found for Fibredux HT/ 6376 earbon ﬁber/epoxy prepreg
from CIBA-GEIGY“5 and good agreement was reported for Hercules 3501—6
epoxy‘“. However, differences between viscosity prediction and inverse ionic

147

conductivity measurements were reported on Hercules AS/3501—6 prepreg .

9.2.2 Resin Flow

There are two main approaches to predict resin ﬂow behaviors in the
composite. One approach describes the resin ﬂow in the composite in terms of
Darcy’s Iaw‘“-“""°. This approach requires knowledge of ﬁber network
permeability, porosity, and resin viscosity, and predicts the consolidation of the plies
across the thickness starting from the bleeder surface. The second approach uses a
lubrication theory approximation to calculate the components of "squeezing flow"
created by compaction of plies‘”. The second approach assumes a plane of
symmetry at the horizontal midplane of the laminate and no ﬁber-to-ﬁber interaction
which means the ﬁbers do not touch each other and do not carry any load during
compaction.

Two ﬂow models exist in the ﬁrst approach. One was proposed by Springer148
assuming that the ﬁber bed carries no load, and that there is a compaction wavefront

205

travelling through the laminate which starts from the bleeder interface and eventually
reaches the other end of laminate. This model predicts the resin flow as sequential,
the layer closest to the bleeder becomes totally compacted, then the next layer, and so
on. This model is over simpliﬁed and does not agree with experimental data‘”.
Using Chlorine and Bromine as a tag to follow resin mixing, Pourstartip et a1.
observed considerable local resin mixing such that the excess resin of the tagged ply
ean go as far as 8 plies in the ﬂow direction during processing. The second model
was developed by Dave et al."’ and Gutowski et a1. "° independently with identieal
models. This model assumes that the ﬁber bed is fully saturated with resin and
allows for three-dimensional resin flow and one-dimensional consolidation of the
composite. The ﬁber bed is considered to behave like a spring. This model can
predict local resin mixing and agrees with the experimental data. A uniﬁed resin
ﬂow model in terms of Darcy’s Law has been proposed which can be readily applied
for most processing, including bleeder ply molding or autoclave processing,

pultrusion, and resin transfer molding‘”.

Assuming no resin ﬂow in the horizontal direction and resin ﬂow out of
composite as flow through porous media, Springer” proposed an expression for the
quasi-steady, seeping ﬂow using Darcy’s law. According to this model, the resin

flow rate out of the composite is

 

 

d(M) = '1 (Sf)
d: ms, h,‘ 1111, (9-12)
Sb hl

where h is the height of the liquid surface at time t, A is the cross sectional area of
the composite perpendicular to the direction of the ﬂow, Sc and S. are the
permeability of the composite and bleeder, h. and hl are thickness of resin in the
bleeder and compacted prepreg layer, ,1 is the resin viscosity, F is the applied force,
and n is the number of compacted layers. The change in the composite thickness

during processing is‘“

 

 

.41. t'
1., .398 (943)
Sbhl
S
('5 (F t
1111111014

where h, is the initial thickness of the composite and t’ is
dimensionless time. This model was tested by placing porous
plates and rod layers in the oil having constant viscosities of
16000 cps and 1257600 cps and a good agreement was found.

Based on the second approach, the resin ﬂow was modeled as
a one-dimensional conﬁned compression (no boundary motion in
x, y directions) and a three-dimensional seepage ﬂow in a
porous medium‘”. The governing equation is
man: -p__)_ _[_ a(_ T 6p,.6 _(fz $1,352), (9-14)

at n 81: ayn 6y 62(11
where k,, k,, k, are stress dependant speciﬁc permeabilities 1n

the x, y, and z directionsm is the resin viscosity, 0 and p are the applied pressure and
the hydraulic resin pressure, and m, is the coefﬁcient of volume change (the absolute

change in porosity e per unit increase in the axial (normal) stress).

The porosity here refers to the non-ﬁber portion of the medium. The
coefﬁcient of volume change, 111,, is equal to a,/(l +e,), where e, is the initial void
ratio and a, is the coefﬁcient of compressibility, a,,=-de/dp. The porosity can be
calculated from the void ratio, e=e/(l +e). The simplest case of resin ﬂow is one
dimensional consolidation under a constant load with one dimensional seepage ﬂow in
the vertical direction. Once p is known, the resin velocity at any point and time in

any direction can be evaluated using Darcy’s law.

207

k
u.(x.y.z.t) = 7‘5 13%;?)

where i=x,-y,z. The motion of the moving boundary layer in the thickness direction
of the laminate has to be accounted for by changing the computatioml grid size after
each time step during the consolidation process. This model ean be simpliﬁed to
Springer’s model by assuming: 1) no ﬂow in the y direction (perpendicular to both
ﬁber and thickness directions), 2) spring-like ﬁber-to-ﬁber interaction is negligible,
i.e. a=p, 3) flow in the horizontal and vertical directions ean be decoupled.

Three partial differential equations were proposed to model vertical and
horizontal ﬂows‘“. Assuming vertical resin ﬂow to be a ﬁltration compaction
process through prepreg and bleeder, the Kozeny-Carman equation is used to describe
the flow. Both vertical and horizontal resin ﬂow in the bleeder is governed by the
Kozeny-Carman equation.

_d_P=K”(1")2”~ (9-15)
4L D’gTE’

 

where P is the pressure, L is the bed length, '1 is the viscosity, 6 is the void fraction,
u,, is the superﬁcial velocity, D is the mean effective ﬁber diameter, g, is the
gravitational constant, and K is the bed characterization constant.

For horizontal flow within ﬁber layers, a flow ﬁeld equation is used to
describe the velocity proﬁle as a function of ﬁber spacing and diameter. Assuming
ﬁbers are uniformly distributed in a triangular pitch, the ﬂow ﬁeld equation becomes

2”.“ .ﬂ = -2ch (946)
Ely2 622 11 dx
where u is velocity and x is the length.

 

208

The horizontal ﬂow between ﬁber layers is modeled as ﬂow between parallel
plates.

-£.lz_’"‘_~

dx 8:23
Where 2,, is distance between parallel plates.

(9-17)

The solution of these three partial differential equations can be calculated using
the ﬁnite difference method.

-1P=1w (9-18)
Where AP is the pressure difference between two nodes, R is the ﬂow resistance, and

U is the mass/area ﬂow velocity. The ﬂow resistance R varies with the type of flow.
For vertical ﬂow,

1% = Kn (1 -e)’1L
018.3 9
Where p is the resin density. For horizontal flow within ﬁber layers,

 

 

12,4: “‘3‘
28.9

Where K is the flow coefﬁcient, K =f(e). For horizontal ﬂow between layers,

= 121] 81
3.439

 

Rm

9.2.3 Voids

Void formation is a complicated problem. It is affected by factors such as
prepreg variations, environmental history, layup variation, and cure cycles. It would
be ideal to model the void formation, growth, transport during cure as well as the

209
relationship between applied pressure, resin hydrostatic pressure, and void formation.
There are two possible sources for void formation, the moisture dissolved in the resin
during prepregging and layup and the entrapped pockets of air. In order to prevent
the potential for water void growth by diffusion, the resin pressure at any point within
the curing laminate must satisfy the following inequality)“

Pain :4. 962x10’exp(—)(RH) (9-19)

Where P... is the minimum resin pressure required to prevent water-vapor void
growth by diffusion. (RH), is the relative humidity (in %) exposure to which the
prepreg is equilibrated prior to processing. A value of 60% is suggested.

Voids will become mobile when the viscous ﬂow forces are large enough to
overcome the surface tension forces. By considering the void transportation in a
laminate as oil ganglia mobilization by water-ﬂooding, a void mobilization criteria
was suggested “6

41’ Law 920
1%171 1= d—T— < 1

Where dp/dl is the resin pressure dlgmdient 1n the direction of flow, 7“ is the resin-
void surface tension, 0 is the apparent contact angle, d, is the diameter of the
narrowest constriction perpendicular to the ﬂow, and L, is the void length projected in
the flow direction. Contact angles between epoxy resins and graphite ﬁbers, with and
without surface oxiding treatment, are roughly 14° and 23°, respectively'”. The
effect of cure pressure on void content was examined. Two characteristics of voids
during processing are similar size across the laminate and no migration through the

resin”.

The void formation will directly affect the compaction of the composite. For
graphite ﬁber/epoxy laminate, experiments showed that the total compaction of the
thin laminate was independent of cure temperature and pressure. However, the rate

of compaction was dependent upon the cure temperature and pressure‘”. For

210

laminate consisting of 16 plies, the total relative compaction (the ratio of measured
compaction and uncompacted laminate thickness) is about 22% . This is because the
resin gels after - the completion of compaction. For thick-section laminate, it may not
be the case.

9.2.4 Composite Properties

Glass transition temperature is an important parameter for composite
applieations. To obtain the highest possible T,, it is necessary to cure the composite
as complete as possible. However, to obtain the highest possible ﬂexural and tensile
strength and the best resistance to compressive creep and water absorption, the cure
has to be stopped after an optimum conversion is reached‘”. For thermal cure of
several epoxy/amine systems, this conversion is located in the postgel stage at 0.10 to
0.15 higher than a”. “°.

For Hercules 35016 resin, a tetraglycidyl 4,4’-diaminodiphenylmethane (TGDDM)
with 33 phr curing agent 4,4’—diaminodiphenyl sulfone(DDS) epoxy resin, the
DiBenedetto equation has been used to ﬁt the T, (°K) versus a data‘“.

 

where a is the extent of cure. The maximum extent of cure at cure temperature
177°C was 0.873.

The residual stress, either due to thermal gradient or volume shrinkage during
the cure, will reduce the mechanical properties of the composite. Although the
volume of epoxy resin decreases during curing, the contractive stresses in the
composite are only due to the volume contraction after gelation”. Tensile stresses
were found to be present in the interior and compressive stresses near the surface in
isothermal cured epoxy polymer. Higher stress magnitudes were obtained when

higher curing temperatures were used‘“.

211

The thermal conductivity of the composite may be ealculated combining the
thermal conductivities of the matrix, k_, and the ﬁber, k,. The thermal conductivity
of AS4/EPON 828/ mPDA composite with 70% ﬁber weight percentage was measured
experimentally. The measured values are 0.73 (W/m/K) and 4.18+0.032*T
(W/m/K) for directions perpendicular to and along the ﬁber orientation of
unidirectional composite, respectively”.

9.3 Microwave and Thermal Processing Model

The microwave and thermal processing of composites is described as follow.
lamina are exposed to a thermal and/or microwave environment as shown in Figure
9.1. The edges of the lamina are surrounded by an adhesive cork dam during the
processing to prevent resin ﬂow out from the sides. The tap is exposed to the
environment directly and the bottom is covered by the bleeder and the support plate,
either a steel plate in the case of thermal processing or a Teﬂon plate in the case of
microwave processing. The heat ﬂow at the interface only includes heat transferred
by thermal conduction and convection. Heat ﬂow caused by thermal radiation is
neglected because the temperature of the system is much lower than 400°C. In this
study, a one dimensional model was developed for the prediction of distributions of
the extent of cure (a) and temperature ('1') across the thickness of the composite as
function of time (t).

212

release film

 

 

 

 

 

 

 

 

 

 

 

 

- z=0
laminate
, — z=L
release f 11m
bleeder layer 1 7
support plate
Figure 9.1. Composite Conﬁguration for Processing Simulation
From an energy balance, the one-dimensional heat transfer equation was
derived as
61' 8 6T -
C _=— K— + H+P (9’21)
9 , a: ax( ax) p ..
with generalized temperature boundary conditions
a 615%,) + b n3,» 4’ C To“) =0 (9_22)

s = 0,L
where p and C, are the density and speciﬁc heat of the composite, K is the thermal

conductivity of composite in the thickness direction, 1'] is the rate of heat generation

per unit weight by the chemical reaction, and P, is the rate of heat generation per unit
volume by the absorbed microwave energy. The power absorption P, can be
calculated using models from Chapter 8. In this study, the model simulation is
focused on the microwave and thermal processing of unidirectional AS4/3501-6
composite. For cured Hercules AS4/3501-6 composite, K=4.457x103 (W/cm/°K),
C,=0.942 (J/g/°K), and p=l.52 (g/cm’) “5. The microwave power absorption
model for unidirectional composite was described by Equations (8-63), (8-64), (8-66),

213
(8—68), and (8-69). For boundary conditions, T,(t) and T(s,t) are the environment and
the composite surface temperatures, respectively. n is the outward unit vector
normal to the top or bottom surfaces. The coefﬁcients a,b, and c control the type of
the boundary conditions. For the Dirichlet(prescribed) boundary condition, their
values are a=0, b=-c= 1. For the Neumann(insulated) boundary condition, their
values are a= l, b=c=0. For the Robin(convective) boundary condition, a=l and
b =-c=(hlk),,,, where (II/k)“ is the effective heat transfer coefﬁcient. When the
effective heat transfer coefﬁcient is very large, the Robin boundary condition becomes
the Dirichlet boundary condition. For autoclave processing, the effective heat transfer
coefﬁcient was due to forced convection and was usually very large. In general, the
Robin boundary condition was used for the top surface and Neumann boundary
condition was used for the bottom surface. The effective heat transfer coefﬁcient for
microwave processing was essentially due to free convection and needed to be

measured experimentally. Therefore, the boundary conditions for the energy balance

 

equation are
inf—11..., = «£1,111»- 1110.4»
' (9-23)
U h
a"; ’1... 42141.14) 41114»
The initial condition for Equation (9-21) is
211404.12) (944)

To solve for T(z,t) in an AS4/3501-6 composite during processing, the
reaction kinetics for Hercules 3501-6 resin, the effective heat transfer coefﬁcient of
top surface, and the ﬁve parameters for microwave power absorption P, are required.
To simulate the temperature proﬁle and extent of cure proﬁles across the thickness of
composite as function of time, implicit method of Lee’s‘“ was used to code the
non-linear parabolic equation, Equation (9-21). The use of Lee’s algorithm gives a
truncation error similar to that of the Crank-Nicolson method but leads to a higher

214

computational rate. The algorithm for Lee’s method is based on the central-
differencc approximation. The ﬁnite-difference expressions for the partial derivatives

are

37 Tw'Tu-l (9-25)

at 261

K015211141 ’ 71) "(7.1-.;.)(Tu ' 73-1)
812

where T,1,,,J=(T,J+T,1,J)/2, TU=(T,},+T,J+T,J+,)/3, i represents nodes in space, j

represents nodes in time, i,j=1,2,3,... The Crank-Nicolson method‘” was used

(9-26)

 

a -
a“?-

to generate the ﬁrst set of temperatures, T11. i=0,1,2,...

9. 3.1 Reaction Kinetics

Hercules 3501-6 resin is a catalyzed tetraglycidyl ether of methylenedianiline
(TGMDA)/ diaminodiphenyl sulphone (DDS). The reaction kinetics for both
tetraglycidyl ether of methylenedianiline (TGMDA)/ diaminodiphenyl sulphone (DDS)
neat resins and TGMDA/DDS based carbon ﬁber reinforced prepreg have been
studied'“. The reaction kinetics of Hercules 3501-6 resin was modeled by the

following reaction kinetics”.

da -{(k,+k1a)(l-a)(B-a) a<¢~ (9,27)

dt-k,(l-a) one",

where k,, k2, and k, are the reaction rate constants, B is the ratio of initial hardener
equivalents to epoxide equivalents, and a”. is the extent of cure at the gelation point.
Their values are a,,,=0.3, B=0.47, k,=2.101x109 exp(-8.07x10‘/(RT)), k2=-
2.014x109 exp(-7.78x10‘/(RT)), and k3=l.960x105 exp(-5.66xlO‘/(RT)). R is the
gas constant R=8.314 (J/mol/K).

215

The rate of heat released by the polymerization reaction is

11:91” <9-28)
d:

where f] is the rate of heat generated by polymerization reaction. H.l is the total heat

of reaction that ean be generated during cure. For Hercules AS4/3501-6 prepreg,
HR=175.2 J/g as determined by DSC. The extent of cure during processing ean be
ealculated by

Ff‘(%)dl (9-29)

The closed, multi-step, fourth-order Milne method‘“ was used to code the
reaction kinetic expression. The ﬁrst order ordinary differential equation can be

shortened to .d_“ =1“). The multistep method is chosen beeause it requires

considerably less computation than the one-step method for results of comparable
accuracy. The closed method is preferred because its loeal truncation error is
considerably smaller than that of the open method. The Milne’s algorithm is a
predictor-corrector method. The predictor is

a,,,=a,_,+i:-:—t(21}-j}-+2j;_2) (9-30)

and the corrector is

“1+1=“1%(fr+1*4f1+f11) (9-31)
The choice of (it must satisfy the convergence condition

3
Marl) (9‘32)

da
The single-step Runge-Kutta method was used to generate the values of the ﬁrst four

8r<

points.

216

9. 3.2 The Measurement of the Effective Heat Transfer Coefﬁcient

The effective heat transfer coefﬁcients for the top surface were different for
microwave and thermal processing. For thermal processing, (It/k)... was mainly due
to forced convection and the value is very large. The top surface temperature of the
composite was equal to the environment temperature during thermal processing. For
microwave processing, (Ir/k)“ was mainly due to free convection and the value
needed to be measured experimentally. A two-ply fully cured AS4/3501—6 composite
was heated up and the temperature was measured as a function of time during
cooling. The thickness of the composite was 0.058cm. Assuming all the heat lost is
due to the heat loss at the surfaces, the effective heat transfer coefﬁcient ean be
calculated from the temperature versus time data.

(11'
" Cry? (9-33)

h": 2.4051(2))

 

where T, is the ambient temperature and A and V are the surface area and the volume
of the composite respectively. Figure 9.2 shows the measured temperature during
cooling. Assuming the temperature difference between the composite surface and the
ambient temperature (T, =23), T-T,, decays exponentially with time:

T - T“ = cc“ (9'34)

With this assumption, ln(T-T,) is a linear function of time, t, with slope h.
This assumption holds very well within a certain temperature range as shown in
Figure 9.3. Three straight lines were obtained, one before 60 sec, one between 60
and 110 sec, and one after 110 sec. In other words, the effective heat transfer
coefﬁcient is a step function of T-T,. Once )1 was obtained from Figure 9.3, the
effective heat transfer coefﬁcient for the straight line temperature range can be easily
calculated.

11 = -£_z_;, (9-35)
1’ 2A

The slopes are )t=—0.0168, h=-0.00944, and x=000150 for the lines before 60 sec,
between 60 and 110 sec, and after 110 sec respectively. The experimentally
determined effective heat transfer coefﬁcients are h¢=0.000698 (W/cm’IK) if T-T,

> 43°C and h,,,=0.000392 (W/cm’lK) if 27°C < T-TII s 43°C , and
M=0.0000623 (W/cm’lK) if T-T, s 27°C.

 

 

 

 

 

150
1377
124
/\111.
*3 85:
E:- 72.
1— 59
46
33:
20’ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 J 1 1 1
0 130 260 390 520 650 780 910104011701300
Time(sec)

Figure 9.2 Temperature/time Proﬁle During Cooling of 2-ply
Hercules AS4/3501-6 composite

218

 

5.0
4.6
4.2 ’1 heff= 6.98 x10-4
3.8 -
3.4 i
To;
2.6 ~
2.2 ~
1.8 l
1.4 -

   
  
   

h.«=3.92x10-4

ln(T—Ta)

h." = 6.23 x10 '5

 

 

10 4 L4 1 J l 14 g; a l a l 4 l 4 l
e

O 130 260 390 520 650 780 910104011701300
Time (sec)

Figure 9.3 Ln(T-T,) Versus Time Curve

9.4 Simulation

With the expression for P, and i1 , the temperature and extent of cure proﬁles

inside the composites can be readily calculated. The computer simulation was
carried out in two parts, thermal processing and microwave processing. The
FORTRAN code for the simulation is attached in Appendix III. The composite

thickness and effective heat transfer coefﬁcient effects on the processing were studied.

9.4.1 Thermal Processing

In thermal processing, P, equals 0. The temperature and extent of cure

proﬁles as a function of time at the midplane and the bottom surface were simulated

219

during processing of 2.0 cm thick AS4/3501-6 laminate using the manufacturer
suggested cure cycle. A Dirichlet and Neumann boundary conditions were used for
the top and bottom surfaces respectively in this simulation. The cure cycle is a ramp
from room temperature(25°C) to 116°C in 40 minutes, then hold at 116°C for 60
minutes, then ramp to 177°C in 30 minutes, then hold at 177°C for 120 minutes, and
cool down to 95°C in 50 minutes. Figure 9.4 and Figure 9.5 show the temperature
and extent of cure proﬁles respectively. In Figure 9.4, the straight line represents the
cure cycle, i.e. the oven temperature. The solid and dot-dash curves are the
temperatures at the bottom surface, z/L= l , and the midplane, z/L=O.5, respectively.
In Figure 9.5, the solid and dot-dash curves are the extent of cure at the bottom
surface, z/L=l, and the midplane, z/L=0.5, respectively.

As shown in Figure 9.4, the bottom surface temperature was higher than the
midplane temperature except during the ﬁrst 50 minutes. Both the midplane and the
bottom surface have two temperature excursions, a small excursion during the 116°C
isotherm and a large excursion at the beginning of the 177°C isotherm. Since the
bottom surface temperature is only slightly higher than midplane temperature, the
cure is only slightly faster at the bottom surface than the midplane as shown in Figure
9.5.

220

 

 

o...’

1001- cam”

Temperature (C) - ->

 

 

 

 

 

50 IN 150 ace 150 3”
Tb” —>

Figure 9.4 Temperature! time Proﬁles for Thermal Processing of
AS4/3501—6 Laminate

 

0.9 1-

0.8 1-

0.7%

->

Cm:

0.4 1-

0.3+

it

0.21-

0.11-

 

 

 

 

0 so too 150 an 7.10 3O
11.1”) ->

Figure 9.5 Extent of Cure Proﬁles for Thermal Processing of AS4/3501-6 laminate

221

' The effect of two parameters, the laminate thickness and the effective heat
transfer coefﬁcient at the top surface, on the temperature and extent of cure at the
bottom surface were studied during thermal processing simulation. Using the same
composite properties(K, C,, p, and (h/K),) and processing cycle, the temperature and
extent of cure proﬁles at the bottom surface were simulated for four laminate
thicknesses, 1.0, 2.0, 3.0, and 4.0 cm. Figure 9.6 and 9.7 show the temperature and
the extent of cure proﬁles for the four different thicknesses. The numbers beside the
curves represent the thickness of the laminate in centimeters used in the simulation.
As shown in Figure 9.6, there are two temperature excursions during processing, a
small one during the 116°C isotherm and a large one during the 177°C isotherm.

The time needed for the occurrence of the ﬁrst excursion is a strong function of
laminate thickness and increases with the laminate thickness. For example, the time
needed for ﬁrst excursion to occur is 45, 60, 90, and 105 minutes for 1, 2, 3, and 4
cm thick laminates, respectively. However, the time required for occurrence of the
second excursion is relatively independent of the laminate thickness within the
thickness range studied. For the 1 cm thick laminate, the second excursion occurred
at 140 minutes while for the other three thicknesses, the second excursion occurred at
145 minutes. As the laminate thickness increases, the occurrence of second excursion
may start before completion of the ﬁrst excursion. The two excursions become
indistinguishable for thick laminates and the ﬁrst excursion peak can be barely seen,
such as in the simulation for 4 cm thick laminate. The magnitudes of both excursions
increase with laminate thickness. The maximum temperature at the bottom surface
reached 260°C for the 4 cm thick laminate. Since the thermal degradation
temperature of Hercules AS4/3501-6 prepreg is 290°C as determined by
thermogravimetric analysis (TGA), the maximum thickness of laminate that can be
processed using the manufacture’s cycle is 4 cm if the operation margin is set to
30°C. As shown in Figure 9.7, The laminate thickness has two effects on the extent
of cure at the bottom surface, a delay of the initiation of the reaction and a shortening
of the reaction time. For the 1 cm thick laminate, the reaction started at 35 minutes
and ended at 200 minutes, total reaction time of 165 minutes. For the 4 cm thick

222

laminate, the reaction did not start until 60 minutes and ended at 140 minutes, total

reaction time of only 80 minutes.

The effect of the effective heat transfer coefﬁcient, (tr/K)“, on the bottom
surface temperature during thermal processing was studied while maintaining the other
composite parameters(K, C,, p, and L) constant. The laminate thickness, L, used in
this simulation was 2.0 cm. Figure 9.8 shows the bottom surface temperature for
(h/K),,> >1, (h/K),,=l, (h/K),,=0.5, and (ll/K),=0.l. The simulation results
show that once (II/K)“ is larger than 10 (cm"), the temperature proﬁles are very close
to those for an inﬁnite heat transfer coefﬁcient. Comparing Figure 9.8 to Figure 9.6,
the effect of (h/K),,, on the bottom surface temperature is similar to the effect of
laminate thickness on the temperature proﬁles. A decrease in (II/K)“ has the same '
effect on the temperature proﬁles as an increase in laminate thickness. As shown in
Figure 9.8, the maximum bottom temperature reached 280°C if (h/K),,,=0.l cm",
though the laminate thickness was only 2.0 cm. Thus, the thickest laminate that ean
be processed by the manufacture’s cycle is 2.0 cm if the effective heat transfer
coefﬁcient is 0.1 cm". This value is close to that of the boundary condition with free
convection. Therefore, it is important to keep the effective heat transfer coefﬁcient
as high as possible during thermal processing of thick AS4/3501—6 laminate.

->

Cure - -

‘l'emperamrc (C) - ->

 

223

 

 

 

 

 

150 200 2.50 300
Thu (mhl —>

Figure 9.6 Temperature Proﬁles for Various Laminate Thickness

 

 

 

 

 

Figure 9.7 Extent of Cure Proﬁles for Various Laminate Thickness

224

 

 

Tangent!!! (C) "->

 

 

 

:0 It!) 130 an 230 am
Thu.) ->

Figure 9. 8 Temperature Proﬁles for Various Effective Heat Transfer Coefﬁcient

9.4.2 Microwave Processing

Microwave processing of a 72-ply unidirectional AS4/3501-6 composite at a '
resonant mode, Lc= 16.03 cm, and 70 W input power was simulated using the
parameters generated in the Chapter 8. The ﬁve parameters for the microwave
power absorption model in the given resonant mode and 70W input power are 0.347
W/m’, 0.797 V/m, 46.79 V/m, 1.54 rad., and 0.0201 rad. for P,, AT,, A,,, OT, and 0,
respectively. A free convection and insulated heat transfer boundary conditions are
used at the top and bottom surfaces respectively. The measured effective heat
transfer coefﬁcients for free convection are (h/K),,=0.014 [l/cm] if T-T, s 27°C,
(h/K),,=0.088 [l/cm] if 43°C 2 T-T, > 27°C, and (h/K),,=0. 157 [l/cm] if T-T,
2 43°C. The control temperature was set to 177°C during the simulation. Figure
9.9 and 9.10 show the simulated temperature and the extent of cure proﬁles at the top

surface, midplane, and bottom surface during microwave processing.

225

 

' zJL- l
1 , zlL- .5
1501- ! zJL-o .

TempetalurdC) 81 Input Power (W)

J)!‘)'.'.‘.‘.‘.'.‘. .;

 

mmmmsmeay
O

0 10 20 30 40 50 60 70
Tim «rum

Figure 9.9 Temperature/ time Proﬁles during Microwave Processing of 72-p1y
Unidirectional AS4/350L6 Composite at L,= 16.03 cm and 70 W

l

 

 

0.9 1-

0.8 P

0.7 -

0.6 1-

0.51-

Cure

0.41-
0.3%
0.21-

0.1 1-

 

 

 

 

 

0 10 20 30 40 50 60 70
Tim (m:

Figure 9.10 Extent of Cure Proﬁles during Microwave Processing of 72-ply
Unidirectional AS4/3501-6 Composite at L,= 16.03 cm and 70 W

226

In Figure 9.9, the zlL=0, 0.5, and 1 curves represent the temperature proﬁle
at the bottom, midplane, and top surfaces of the composite, respectively. The dash
line is the power input curve. The laminate can be heated directly to the desired
temperature in 8 minutes at an input power of 70W. Though the power is turned off
as the temperature reached 177°C, there still have small temperature excursion during
the processing. This temperature excursion is due to the reaction exotherm and the
small heat transfer coefﬁcient at the composite surface and ean be eliminated by
increasing the heat transfer coefﬁcients. As shown in Figure 9.10, the extent of cure
reached 0.3 when the temperature reached 177°C at the bottom surface. The
exotherm from the reaction was sufﬁcient in maintaining the temperature once the
reaction started until the composite was almost fully cured at the simulation condition.

Since the temperature excursion can be eliminated, the processing temperature
can be raised to 250°C without any risk of composite degradation. Figure 9.11 shows

the temperature proﬁles during microwave processing of the 72-ply AS4/3501-6
composite at the resonant mode L,= 16.03 cm and 500 W.

 

 

   
   

 

 

      

 

 

 

sooT—f
1
1 2111-0: .2
.. 35°. 211.111
2 am
a :
g 2001- f
=- : 1
E'- 1
3 1501-
3 ~¢ cs
‘8; 1 2 : 2 .i: 3'3
5 1001- ‘- : :~= 1 1
2 ; ' . ‘5 - ‘ 1.
§ ' *3 it: P
9 .
l.- .
' i : 1 1
5° 3:11
1131
£1 ,
o 7, . .
o 2 4 6 s 10 12 14

Tim «mm

Figure 9.11 Temperature! time Proﬁles during Microwave Processing of 72-ply
Unidirectional AS4/3501-6 Composite at L,= 16.03 cm and 500 W

227

The heat transfer coefﬁcients used in Figure 9.11 are 1.51003 and 2.011103
(W/cm’lK) for the top and bottom surfaces. As shown in Figure 9.11, the
temperature of bottom surface reached 250°C in 5 minutes and no temperature
excursion was observed. The time required to heat the composite up to the control
temperature can be shortened with higher input power. Comparing Figure 9.11 to
9.4, the advantages of microwave processing over thermal processing can be
summarized as: l) The temperature excursion during microwave processing is
eliminated. 2) High temperatures ean be used during microwave processing. 3) The
ideal heating proﬁle is realized during microwave processing, that is directly heating
the laminate to the desired temperature as quickly as possible then maintaining the
laminate at the desired temperature.

9.5 Conclusion

A one-dimensional process model was developed for both microwave and
thermal processing. The ﬁve-parameter microwave power absorption model was
applied to calculate the microwave power dissipated inside the composite. The
process model was coded numerically using Lee’s algorithm and Milne’s method.

The computer code was used to simulate the temperature and extent of cure proﬁles
across the composite thiclmess during autoclave and microwave processing. With the
effective heat transfer coefﬁcients for the top and bottom surfaces as inﬁnite and zero,
the thickest AS4/3501-6 composite that can be processed is 4.0 cm. A decrease in
the effective heat transfer coefﬁcient will increase the magnitude of the temperature
excursion and decrease the thickness of composite that can be processed. The
temperature excursion can be eliminated during microwave processing. The ideal
heating proﬁle is realized during microwave processing, that is directly heating the
laminate to the desired temperature as rapidly as possible then maintaining the
laminate at the desired temperature.

CHAPTER 10
SUMMARY OF RESULTS

The feasibility of using microwave energy to processing polymers and
composites was systematically explored. Microwave radiation effects on epoxide
polymerization, the properties of the cured epoxy resins and composites, and various
microwave processing techniques were studied. This study shows that microwave
processing offers a fast heating rate, fast polymerization rate, and better property
products as compared to thermal processing.

In the study of the neat epoxy resins, stoichiometric DGEBA/DDS and
DGEBA/mPDA epoxy systems were chosen. The main results are summarized in the

following.

* Faster reaction rates were observed in the microwave cure when compared to
those of thermal cure at the same cure temperature for both systems.
Microwave radiation has stronger effects on the DGEBA/DDS system than the
DGEBA/mPDA system.

* Autocatalytic kinetics model was successfully used to analyze the cure
behavior of DGEBA/mPDA and DGEBA/DDS systems for both microwave
and thermal cure.

* The reaction rate constants of microwave cure were higher than those of
thermal cure at the same cure temperature for both DGEBA/DDS and
DGEBA/mPDA systems.

228

229
* For the stoichiometric DGEBA/mPDA system,

a) Microwave radiation increases the values of k1 Ikz at low cure
temperatures, but decrease the values of k,/k2 at high cure
temperatures.

b) In the model, the reaction rate constants of primary amine-epoxy are
equal to secondary amine-epoxy and the etheriﬁcation reaction is
negligible for both microwave and thermal cure.

* For stoichiometric DGEBA/DDS system,

a) Microwave radiation decreases the values of kllk2

b) The reaction rate constants of primary amine-epoxy are greater than
those of secondary amine-epoxy and the etheriﬁeation reaction is
negligible at low cure temperatures for both microwave and thermal

cure.

c) Microwave radiation decreases the reaction rate constant ratio of
primary amine-epoxy to secondary amine-epoxy and the ratio of
primary amine-epoxy to the etheriﬁeation reaction.

* Microwave radiation increases the activation energies of both kl and k2 in the
DGEBA/DDS system while it only increases the activation energies of k, in
the DGEBA/mPDA system.

* Microwave radiation increases the glass transition temperature of cured epoxy
after the extent of cure reaches or“. The increase was much more signiﬁcant
in the DGEBA/DDS system than the DGEBA/mPDA system. The
DiBenedetto model was used to ﬁt the experimental Tg data.

230

* The full-cure epoxy-monomer lattice energy is lower in microwave cure than

1 in thermal cure and the full-cure epoxy-monomer segmental mobility is lower
in microwave cure than in thermal cure for both DGEBA/DDS and
DGEBA/mPDA.

* The TTI‘ diagrams for both microwave and thermal cure were generated using
data and models. A good agreement was found between the two.

* The vitriﬁcation time is shorter in microwave cure than in thermal cure,
especially at higher isothermal cure temperatures. The minimum vitriﬁcation
time is shorter in microwave cure than in thermal cure, especially for the
DGEBA/DDS system.

In the microwave processing study, three processing techniques, batch
processing, scale-up, and continuous processing, were investigated. The main results

of the processing study are summarized as follows.
In the batch processing study

* Unidirectional and crossply graphite ﬁber/epoxy composite laminates
consisting of 24, 72, and 200 plies were successfully processed in a 17.78 cm

tunable resonant cavity using 2.45 GHz microwave radiation.

* The heating rate and the temperature uniformity were strong functions of the
ﬁber orientation and the cavity length during microwave heating of fully cured
3.8 cm thick AS4/3501-6 composite. Both uniform planar temperature or
overall temperature can be achieved at proper ﬁber orientation and resonant
mode.

* Temperature uniformity during the processing and the ﬂexural properties of

231

the processed samples were shown to be strong functions of the resonant
heating modes.

The ﬂexural properties of the microwave processed unidirectional composites
were also affected by the ﬁber orientation of the composites with respect to the
electromagnetic coupling probe. The maximum ﬂexural properties of
unidirectional composites were observed when the sample was processed at an
orientation of 45 degrees to the coupling probe.

The ﬂexural properties of microwave processed crossply samples have been
found to be between those values measured for the microwave processed
unidirectional laminates cured with ﬁber orientations perpendicular and parallel
to the coaxial coupling probe.

The concept of a processing mode (PM) is deﬁned based on the ﬂexural
properties of the processed composites. Criteria deﬁning a processing mode
were obtained from a database of types of processing modes, radial ﬁeld
patterns during processing, and the flexural properties of processed
composites.

The ﬂexural strength of the unpressurized thermally cured unidirectional and
crossply composites were much lower than those of unpressurized microwave
cured samples. The ﬂexural modulus of the unpressurized thermally cured
unidirectional composites was comparable to that of unpressurized microwave
cured unidirectional samples with a ﬁber orientation of 0 degrees to the
coupling probe, but lower than that of microwave cured unidirectional samples

with other ﬁber orientations.

Comparing the ﬂexural properties of the microwave processed composites with

a processing time of 90 minutes and no pressure to those of autoclave

232

processed samples with a processing time of 300 minutes and a pressure of

100 psig

a) A comparable ﬂexural strength but lower ﬂexural modulus were
found in the microwave processed 24-ply unidirectional composites than
those of the autoclave processed samples.

b) Both ﬂexural strength and ﬂexural modulus of microwave processed
24-ply crossply samples were lower than those of autoclave processed

samples.

* The apparent interlaminar shear strengths (ISS) of microwave processed
composites increased with increasing extent of cure.

* Scanning electron microscope analysis for the fracture surface revealed that

a) Graphite ﬁber/epoxy bonding was higher in microwave processed
composites than in thermally processed samples.

b) Microwave processed composites failed in matrix failure mode for
samples of high extent of cure and in a combination mode of interfacial

and matrix failure mode for samples of low extent of cure.

c) The autoclave processed samples failed in a combination mode of
interfacial and matrix failure mode.

In the scale up study

* Fully cured DGEBA/DDS epoxy and Hercules AS4/3501-6 composite squares

were heated in a 17.78cm and a 45.72cm tunable cylindrical resonant cavities

233

with the same cavity geometry ratios, sample/cavity size ratios, and the input

power density.

* A microwave source operating at 2.45GHz was used for the 17.78cm cavity
and 915MHz was used for the 45.72cm cavity, in order to reproduce the same

empty cavity resonant mode for the same eavity geometry ratio.

* The scale-up of cavity length during microwave heating of low to medium loss
materials, such as an epoxy square, can be approximated by the scale-up factor

of empty cavity for a given resonant mode.

* The maximum temperature was linearly related to the maximum temperature
difference during microwave heating of graphite ﬁber/epoxy composite. This
relationship was the same for both the 2.45 GHz system and 915 MHz system.

In the continuous processing study

* A unique microwave applicator was invented in which continuous graphite
ﬁber reinforced prepreg tapes can be continuously processed using 2.4SGHz

microwave radiation without causing microwave leakage.

* The microwave power absorption efﬁciency was higher in the CH modes than
in the PS modes.

* Fully cured products were obtained in the controlled-hybrid modes while only
partially cured products were obtained in the pseudo-single modes under

experimental conditions.

234

In order to predict and control the temperature and extent of cure of the
composite during processing, a one dimensional processing model was developed for
both microwave and thermal processing. The main results were summarized as

follows.

* A ﬁve-parameter microwave power absorption model was derived to calculate
the microwave power dissipated inside the composite during microwave

processing.

* A FORTRAN program, combining the energy balance equation, the microwave
power absorption model, and the least squares optimization was coded to
generate the ﬁve parameters based on the temperature/time/position proﬁles
obtained from microwave heating of a fully cured composite.

* A set of ﬁve parameters was generated for a resonant mode at L,= 16.03 cm
and 0° ﬁber orientation in a 72-ply unidirectional AS4/3501-6 composite
loaded cavity. For this mode, P, increases linearly with input power to the
power of 0.3, AT and A,, increase linearly with the square root of the input

power, and 0,- and 0,, are not functions of the input power.

* The process model was coded numerically using Lee’s algorithm and Milne’s

method for processing simulation.

* The thermal simulation results show that the thickest AS4/3501-6 composite
that can be processed is 4.0 cm with the manufacture’s suggested cure cycle
and it is very important to keep a high effective heat transfer coefﬁcient at the

surface during thermal processing.

* The advantages of microwave processing over thermal processing were fully

demonstrated by the computer simulation.

CHAPTER 11
FUTURE WORK

This study systematieally demonstrated the feasibility and advantages of
microwave processing as an alternative in the processing of polymers and composites.
However, there are still some fundamental and application issues that need to be
further explored.

In the neat resin study, the further work that need to be done is:

* To detect the crosslinked network structural difference in the microwave and
thermally cured epoxy resins using wide-angle X-ray diffraction or solid-state
NMR. This study will give insights into the question of why microwave cured
epoxy samples have higher Tg.

* To determine the key functional groups in curing agents responsible for the
enhancement of reaction rate and Tg using a curing agent having an
intermediate dipole moment, such as diamino diphenyl methane. If an
intermediate behavior is observed in diamino diphenyl methane with
microwave radiation, the overall dipole moment of the curing agent is the
determining factor for microwave cure behavior. If the microwave cure
behavior is similar to the DDS system, the proximity of amines in mPDA have
inhibiting effects on microwave radiation. If the microwave cure behavior is
similar to mPDA, it means that the activity of the SO, group in DDS is
responsible for the signiﬁcant increase of reaction rates and T, during

microwave cure.

* To study the polymer rheology behavior during microwave processing.

* To study weight loss during microwave heating.

235

236
In the processing study, some application issues need to be further explored.

* Develop a pressure device to couple pressure into the processing.
* Study the microwave heating characteristics of complex shaped composites.

* Study the sealibility for the composite processing in the microwave
environment with the samples only scaled in two dimensions.

* Study the scalibility of complex shaped composites.

* Apply the microwave energy in various continuous processing techniques
using the microwave applicator invented in this study. These could include
extrusion, pultrusion, carbon ﬁber or wire production, ﬁlament winding, and

SO on.

In the modeling, the current one dimensional process model needs to be
extended into a three dimensional model and the rheology, mechanical property, and
void formation behaviors need to be coupled into the model.

50

250

Appendix I FORTRAN code for generating the TTI‘ diagram for
the DGEBA/DDS system

'l'l'I‘ Diagram Calculating Program for DGEBA/DDS System
variables declaration

REAL mwn,mwl,mwkllna,mwkle,mwk21na, mwk2e,mwxvit
REAL thn,thl,thkllna,thkle,thk21na,thk2e,thxvit

REAL n,l,tc(300),xgel,mwtggel,thtggel,p,f

REAL g,gg,ﬁ,delt,mwkl,mwk2,thk1,thk2

REAL tht,mwt,thtgel(300),thtvit(300),thx0,mwx0

REAL mwtgel(300),mwtvitl(300),mwtvit2(300),thx,mwx

OPEN(9, FILE=’tttdth.m’)

OPEN(ll, FILE=’tttdmw.m’)

constants

mwn=.15

mwl=.0

mwkllna=23.2l

mwk21na=34.8

mwkle=21000.0

mwk2e=30000.0

thn=.4

thl=.0

thk21na=25.01

thkllna=l6.07

thk2e=23270.0

thkle= 16040.0

xgel=0.57

mwtggel=339.7

thtggel=334.6

tg0=295.4

DO 501=l,200

mwtgel(i)=0.0

mwtvitl(i) =0.0

mwtvit2(i)=0.0

thtgel(i) =0.0

thtvit(i) =0.0

write(9,250)

write(11,350)

format(’ %For thermal cure of DGEBA/DDS’,//
l ’%th(i,j), th(l,j)=temp in K; th(2,j)=thtgel, in min’/
2 ’% th(3,j)=thtvit,in min., th(4,j)=thxvit’/// .
3 ’% T(K), thtgel(min) thtvit(min) thxvit ’/’th=[’)

237

'm Mg. a. -‘N- Agave tin—Ila
- 1

350

CO

UtJtht-d

238

format(’ %For microwave cure of DGEBA/DDS’ ,//

’%mw(i,j), mw(l,j)=temp in K; mw(2,j)=mwtgel, in min’/
’% mw(3,j)=mwtvit,in min. ,at mwx=mwxvit’/
’%mw(4,j)=mwtvit2, in min. where mwx=thxvit’/

’ % mw(5 ,j) =mwxvit’l/

’% T(K) mwtgel(min) mwtvitl mwtvit2 mwxvit’,/’mw=[’)
DO 100 j =l,230

tc(i)=273.2+25+j

mwxvit=(tc(i)/tg0-l.0)/(tc(i)*.82/tg0-.68)
thxvit=(tc(i)/tg0-l.0)/(tc(i)“.7/tg0—.56)

print *, tc(i),thxvit,mwxvit

mwkl =exp(mwkllna-mwkle/ 1 .987/tc(i))

mwk2 =exp(mwk21na-mwk2e/ l .987/tc(i))

thkl =exp(thkllna-thkle/1.987/tc(i))

thk2 =exp(thk21na-thk2e/ 1 .987/tc(i))

print *, ’thkl =’,thkl,’thk2=’,thk2,’mwkl =’,mwkl,’mwk2=’,mwk2

If (tc(i) .GE. 418.0) mwl=0.6
If (th) .GE. 438.0) thl=.8

IF (thxvit .GT. 1.0) then

thxvit=l.0

go to 300

end if

thx0=0.0

n=thn

L=thl

ﬁ=1.0

if (i .lt. 50) then

delt=10.0

else

delt=l.0

end if

DO 200 i=1,10000000

tht=(i-l)*delt

f=1.0—((1.0—n)*ﬁ+ﬁ**(n/2.0))/(2.0-n)

thx=((1.0-ﬁ)*(1-n)*(2.0-L)+2.0*(1.0-ﬁ**(n/2.0))*
(1 .0-L/n)-(2.0-n) *L'LOG(ﬁ))/2 .0/(2.0—n)

p=((2.0*(1-n)*ﬁ+n*ﬁ**(n/2.0))/(2.0-n)+L*f)*
(l .0—thx)*(thkl +thk2*f)

g=-((l.0-n)*(2.0-L)+n*(l-L/n)*ﬁ**(n/2.0-l.0)+
(2.0-n)*IJﬁ)/2.0/(2.0-n)

gg=p/g

ﬁ=ﬁ+gg*delt

 

200

25
300

95

1

239

IF ( (xgel-thx)"'(xgel-thx0) .LT. 0.0) THEN
thtgel(i)=tht
print ". ’thtgel(’.i.’)=’.thtselti)
end if
IF ( (thxvit-thx)‘(thxvit-thx0) .LT. 0.0) THEN
thtvit(i)=tht
print *. ’thtvit(’..i.’)=’. thtvit(i)
end if
IF ( thtgel(i) .NE. 0.0 .AND. thtvit(i) .NE. 0.0 ) GO TO 25
thx0=thx
CONTINUE

write (9,1000) tctj),thtgel(j),thtvit(i).thxvit

n=mwn

L=mwl

mwx0=0.0

ﬁ=1.0

if (mwxvit .gt. 1.0) go to 100

if (1 .le. 50) then

delt=10.0

else

delt=l.0

end if

if (i .GT. 85 ) then

delt=.1

end if

if (j .gt. 135) then

delt=.01

end if

if (i .gt. 180) then

delt=.001

end if

if (j . gt. 220) then

delt=.0001

end if

DO 400 i=1,100000000

mwt=(i-l)*delt

f = 1.0-((1 .0-n)*ﬁ+ﬁ*"‘(n/2.0))/(2.0—n)

mwx=((l.0-ﬁ)*(l-n)"'(2.0-L)+2.0*(l.0-ﬁ**(n/2.0))*
(l .0-L/n)-(2.0—n)*L*LOG(ﬁ))/2.0/(2.0—n)

p=((2.0*(1.0—n)*ﬁ+n*ﬁ**(n/2.0))/(2.0-n)+L*f)*

(1.0-mwx)*(mwkl +mwk2*f)

g=-((1.0—n)*(2.0-L)+n*(l.0-L/n)*ﬁ**(n/2.0-l.0)+

(2.0-n)*IJﬁ)/2.0/(2.0-n)

 

1

700

400
l l
1000
2000

100

3000

4000

240

gg=plg

ﬁ=ﬁ+gg*delt

print *, mwx

IF ( (xgel-mwx)*(xgel-mwx0) .LT. 0.0) THEN
mwtgel(j)=mwt

print ". ’mwtscl(’.j,’)=’. mwtgel(i)
end if

IF ( (mwxvit-mwx)‘(mwxvit-mwx0) .LT. 0.0) THEN
mwtvitl(j)=mwt

print *, ’mwtvitl(’,j,’)=’, mwtvitl(i),’at mwx=mwxvit’
end if

IF (thxvit .eq. 1) THEN

mwtvit2(j)=1000

go to 700

end if

IF ( (thxvit-mwx) I"(thxvit-mwa) .LT. 0.0) THEN
mwtvit2(j)=mwt

print *, ’mwtvit2(’,j,’)=’,mwtvit2(i),’at mwx=thxvit’
end if

IF ( mwtgel(i) .NE. 0.0 .AND. mwtvitl(i) .NE. 0.0
.AND. mwtvit2(i) .NE. 0.0) GO TO 11

mwx0=mwx

CONTINUE

print *, ’the result is wrong!’

print *, ’program is running’

write (11,2000) tcﬁ),mwtgel(i),mwtyitl(i),

mwtvit2(j),mwxvit
format(2x,f7.3,2x,e15.7,2x,e15.7,2x,f7.3)
format(2x,f7.3,2x,e15.7,2x,e15.7,2x,e15.7,2x,f7.3)

CONTINUE

write(9,3000)

write(11,4000)

format(’];’/’semilogx(th(:,2),th(:,1),th(:,3),th(:,l))’/
’hold’)

format(’];’/’semilogx(mw(:,2),mw(:,l),mw(:,3),mw(:,1))’/

’hold’l’semilogx(mw(:,4),mw(:,l))’)

close(ll)

close (9)

print *, ’end’

end

Appendix II FORTRAN code for calculating the parameters for
microwave power absorption model

Program ﬁtpararneters

implicit none

double precision pi,mn,tdx(72),sum,dx,dt,tdt(500), denm(500,72),er1,b(5), b1,
bet(72), er, ma(72,72), ttemp(500,72),expm(500,72),x(5,500,72),b(5),to,yin,a
integer cnt,m,pt,i,j,k,ml

common lblok8/ x

common lblkl/ yin,to,m

common lblk6/ expm,pt

common lblk7/ dx,dt,tdt

common lblkO/ ttemp

common Inwblk/ tdx,m1

open(unit=12,status= ’unknown’ ,ﬁle= ’exn4.m’)
open(unit=13,status= ’unknown’,ﬁle= ’exx4.m’)
call input(b)

pi=3.141592654

do i= 1,5

bet(i)=0

enddo

eall tempprof(b,ttemp)

call err(ml ,pt,expm,ttemp,er)
write(*,*),’Iteration:’,cnt,’ S=’,er
write(*,*),’Parameters’

do i= 1,5

write(".*).b(i)

enddo

erl =er

mn=er

cnt=0

do while((abs((erl-er)/erl) .GT. l.0e-3).OR.(cnt .LT. 1))
if (cnt .GT. 0) then

a=al

endif

do i= 1,5

call dpowc(b,i,denm)

do j = 1,pt

do k= l,ml

x(i,j ,k) = (denmﬁ ,k)-ttemp(i ,k))/ (0.00001 "b(i))
enddo

enddo

enddo

241

W—u'-_-—_'." - --_- _

242

do i= 1,5
eall rhsum(m1,pt,x,expm,ttemp,i,a)
ma(i,6) =a
do j = 1,5
eall lhsum(ml,pt,x,i,j,bl)
ma(i,j) =b1
c write(*.*).ma(iJ)
enddo
enddo
eall mat(ma,5,bet)
do i= 1,5
b(i) =ab8(b(i) +bet(i))
if ((i.EQ.4).OR.(i.EQ.5)) then
if(b(i).LT.0) then
116) =b(i)-2 ‘Pi"(-1 +int(b(i)/ (2 "pi)))
else
b(i) =b(i)-2 *Pi‘int(b(i)/ (2 *pi))
endif
endif
enddo
call tempprof(b,ttemp)
call err(m1,pt,expm,ttemp,erl)
cnt =cnt+ 1
write(*,*),’Iteration:’,cnt,’ S=’,erl
write(*, *), ’Parameters’
do i= 1,5
write(*, *).b(i)
enddo
enddo
write(*, I") , ’Parameters ’
do i= 1,5
write(*. *).b(i)
enddo
write(*, *), ’ Temperatures ’
write(*, *), ’Experimental’ , ’ Fitted’
write(l2,*),’exp=[’
. do i= 1,pt
write(*, *), ’Time: ’ ,tdt(i)
write(12160),
tdt(i),exr)m(i,1)TeXPm(i.2).expm(i,3).eXPm(i,4).expm(i,ml).ttemp(i.1).
ttemp(i,2),ttemp(i,3),ttemp(i,4),ttemp(i,ml)
do j = 1 ,ml
write(*,59),expm(i.j).ttemp(i.i)
59 format(F9.5,3X,F9.5)

 

60

243

enddo

format(F7.2,1X,F9.5,1X,F9.5,1X,F9.5,1X,F9.5,1X,F9.5,1X,F9.5,1X,F9.5,

1X,F9.5,1X,F9.5,1X,F9.5 )

61

62

63

65

wﬁtc(",*)

enddo

write(12.*).’l;’

write(l3,‘),’xsl =[’

do j =l,pt
write(l3.61).tdt(i)TX(1.j.1).X(1.i.2)TX(1J13)TX(l.i.4).X(l.ijl)
format(F7.2,lX,F12.5,1X,F12.5,1X,F12.5,1X,F12.5,1X,F12.5)
enddo

“1603.031?

write(l3.°).’xs2=[’

do j =l,pt
write(13.62).tdt(i)TX(2.j,1).X(2J,2).X(2.j.3).X(2.j.4).X(2,i,ml)
forrnat(F7.2,lX,F12.5,1X,F12.5,1X,F12.5,lX,Fl2.5,1X,F12.5)
enddo

write(l3.").’l;’

write(13.*).’xs3=[’

do j =l,pt
wﬁw03,63),tdt(i)TX(3,i.l)TX(3J,2).X(3.jT3).X(3.j,4)TX(3.ijl)
format(F7.2,lX,F12.5,1X,F12.5,1X,F12.5,lX,Fl2.5,lX,F12.5)
enddo

write(l3.*).’];’

write(l3.*).’xs4=[’

do j = 1,pt
write(l3.64).tdt(i)TX(4.i.1).x(4.i.2).x(4.j.3).X(4.j.4).X(4.j.ml)
forrnat(F7.2,lX,F12.5, 1X,Fl2.5,1X,F12.5,1X,F12.5,1X,F12.5)
enddo

write(l3.*).’l;’

write(13.").’xs5=[’

do j = 1,pt

writC(13.65),tdt(i)TX(5.j. 1).X(5J.2)TX(5J.3),X(5.i,4).X(5.i.ml)
format(F7.2,1X,F12.5,1X,Fl2.5,1X,F12.5,lX,F12.5,lX,F12.5)
enddo

write(13.").’l;’

end

subroutine rhsum(ml ,pt,x,expm,ttemp,n,a)

integer i,j,n,m,pt,ml

double precision a,ttemp(500,72),expm(500,72),x(5,500,72)
a=0

do i= 1,pt

do j = 1,m1

rm in: 112-«nut.

244

a=a+(expm(iJ)-ttemp(iJ))°X(n.iJ)
enddo

enddo

return

end

subroutine lhsum(ml ,pt,x,n,q,b)
integer ml,n,q,i,j,pt,m

double precision b,x(5,500,72)
b=0

do i= 1,pt

do j = l,ml
b=b+X(q,i.i)*X(n.i.i)

enddo

enddo

return

end

subroutine err(ml ,pt,expm,ttemp,er)
double precision er,ttemp(500,72),expm(500,72)
integer m1,i,j,pt

er==0

do i= 1,pt

do j = 1,m1
er=er+((expm(i,j)-ttemp(i,j))"2)
enddo

enddo

return

end

subroutine dpowc(b,q,dtem)
double precision dtem(500,72),c(5),b(5)
integer q,i

do i= 1,5

if (i .EQ. q) then

c(i) =b(q) +0.00001*b(q)
else

c(i) =b(i)

endif

enddo

call tempprof(c,dtem)
return

end

 

245

subroutine tempprof(b,ttemp)

implicit none

integer k,ml,m,i,j,tpt,pt

double precision
slpe,diff,hdll ,hdhl ,hdh2,cf,tdx(72),ptot,ft,hd2,expm(500,72), b(5),hd1,
rtemp(500.72).y5(72).DC(72).b(5).d1.wf.yav3.pl.p2.y2(72).y1(72).y3(72.72).
dt,dx,to,kavl ,kav2,kav3,kav,yavl ,yav2,t,yin,dmt,tdt(500),intl(72)

common lblkl/ yin,to,m

common lblk6/ expm,pt

common lblk7/ dx,dt,tdt

common lsblk8/ hdll ,hdhl ,hdh2,hd2

common lnwblk/ tdx,m1

common lblnk/ cf,intl(72)

t=0

do i= 1 ,m + l

if(tdt(l).EQ.0.0) then

do j = l,ml-l

if(((i- 1)*dx.LT.tdx(j + 1)).AND. ((i-l)*dx.GE.tdx(j))) then

81Pe= (intlﬁ + 1)-int1(i))/(th(i + 1)-th(i))

yl(i) = l.0/to*(intl(j) + s1pe"((i-l)"dx-tdx(i)))

endif

enddo

if(((i-l)*dx.GE.tdx(ml))) then

yl(i) =intl(ml)/to

endif

else

yl(i) = 1.00

endif

y5(i) =y1(i)

enddo

do i= l,m + 1

do j = l,m +2

y3(i,j) =0.00

enddo

enddo

tpt=l

call unimWP(b(1).b(2).b(3).b(4).b(5),m"dX.m,Pc,pt0t)

do while(tpt .LE. pt)
if (abs(t-tdt(tpt)).LT. 1.0e-12) then
k= 1
do while (k.LE.ml)
do i= 1 ,m + 1
if (abs((i-l)*dx-tdx(k)).LE.1.0e-6) then

246

rtemp(tpt,k) =to*y5 (i)
k=k+ 1
endif
enddo
enddo
tpt=tpt+1
endif

C

C Lees Method
if (t.GT.0) then
dmt=2*dt/(3"(dx"2.0))
yav3 =0.50*(y2(2) + y2(1))
call Th(yav3 *to, l ,kav3)
call Th(y2(l)"'to,l,kav)
call prod(yZ(l)*to,l,p2,wf,d1)
diff =to*abs(yin-y2(l))
if (diff.LE.27) then
hdl =hdll
endif
if ((diff.LE.43).AND.(diff.GT.27)) then
hdl =hdhl
endif
if (diff.GT.43) then
hdl =hdh2
endif
y3(1,1)=dmt*(2*dx*hdl +kav3 +kav) +p2
y3( l ,2) = -(kav3 +kav) *dmt

' ﬁ‘h dim —.“
W, .4 e . r . .='

y3(] ,m +2) =p2*y1(1)+dmt*((yZ(1)+y1(l))*(-2*dx*hdl-(kav+kav3))+
(kav+kav3)*(y2(2) +y1(2))+6"‘yin*hdl*dx) +2*dt"pc(l)/to+pl *yl(i)

do i=2,m

yavl =0.50"'(yZ(i) +y2(i + 1))

yav2 =0.50*(y2(i) +y2(i-1))

call Th(yavl*to,1,kavl)

call Th(yav2 *to, l ,kav2)

call prod(yZ(i)*to, 1 ,pl ,wf,dl)

y3(i,i-l) =-kav2"‘dmt

y3(i,i) =pl +dmt*(kavl +kav2)

y3(i,i+1)=—dmt*kav1

y3(i,m +2) =dmt*(kavl *(y2(i + 1) +y1(i + 1))-(kav1 +kav2)*(yZ(i) +y1(i)) +
kav2*(yZ(i-l) +y1(i-1))) +p1 *yl(i) +2*dt“pc(i)/to

enddo _

yav3 =0.50*(y2(m +1)+ y2(m))

247

eall Th(yav3‘to,1,kav3)

call Th(y2(m + 1)*to,1,kav)

call prod(y2(m+ l)*to,l,p2,wf,dl)

y3(m +1,m + l) =dmt*(2*dx"hd2 +kav+kav3) +p2
y3(m + l,m) = -dmt*(kav +kav3)

y3(m+1,m+2)=p2*yl(m+l)+dmt*((y2(m+l)+yl(m+1))*(-2"‘dx"hd2-
(kav+kav3)) + (kav+kav3)*(y2(m) +y1(m)) +6*yin*hd2*dx) +2*dt“pc(m+ l)/to
call tridag(y3,m+ l,y5)
do i=1,m+ l
ﬂm=ﬂ® l
ﬂ®=ﬁ® .
enddo 2
endif "1
C
C Crank Nicolson Method
if (t.EQ.0) then -‘
dmt=dt/(2*(dx**2)) 1“
yav3 =0.50*(yl(2)+yl(l))
call Th(yav3 *to, l ,kav3)
call Th(yl(l)*to, l ,kav)
call prod(yl(1)*to,l,p2,wf,dl)
diff =to*abs(yin-y 1 (1))
if (diff. LE.27) then
hdl =hdll
endif
if ((diff.LE.44).AND.(diff.GT.27)) then
hdl =hdhl
endif
if (diff. GT.44) then
hdl =hdh2
endif
y3(1, l)=dmt*(2*dx*hdl +kav+kav3) +p2
y3(l ,2) =—dmt*(kav3 +kav)
y3(1 ,m +2)=y1(1)*(p2-dmt*2*dx*hdl-(kav3 +kav)*dmt) +
dmt*(kav3 +kav)*y1(2) +4“yin *hdl ‘dx *dmt +pc( l)*dt/ to
do i=2,m
yavl =0.50*(yl(i)+yl(i+ 1))
yav2 =0.50*(yl(i) +y1(i-1))
call Th(yavl*to,1,kavl)
call Th(yav2 *to, l ,kav2)
call prod(yl(i)*to,l,pl,wf,dl)
y3(i,i-l) =-kav2*dmt
y3(i,i) =pl +dmt*(kavl +kav2)

 

us
y3(i,i+l)=-dmt“‘kavl

y3(i,m+2)=dmt*(kav1*(yl(i+1))-(kav1+kav2)*(yl(i))+kav2*(yl(i-l)))+
dt‘pc(1)/to +pl *yl(i)

enddo

yav3 =0.50*(yl(m+ 1) +y1(m))

eall Th(yav3 *to, l ,kav3)

call Th(y1(m+1)*to,1,kav)

eall prod(yl(m+l)"'to,l,p2,wf,dl)

y3(m+ l,m+ 1) =dmt*(2*dx*hd2+kav+kav3)+p2

y3(m + l,m) = -dmt*(kav3 +kav)

y3(m +1, m +2)- -p2*y1(m+ l) +dmt*((yl(m +1))*(-2*dx*hd2-(kav+kav3))+
(kav +kav3)*(yl(m)) +4* ym*hd2*dx) +dt‘pc(m + 1)/to
call tridag(y3, m+ l,y5)
do i= 1 ,m + 1
y2(i) =y5(i)
enddo
endif
t=t+dt
enddo
return
end

subroutine Th(a,x1,kl)

double precision a,xl,kl,kf,km

common lblk2/ kf

km =(0. l6l +xl*(0.00147*a—0.4l7))/100.0
kl =(0.02514*km)/(.02514+km)

return

end

subroutine prod(y,x2,retl,ret2,ret3)
double precision yl,xl,y,x2,r2,rho,sp,retl,ret2,ret3
y1=y

x1=x2

call den(yl,xl,rho)

call Cp(yl,xl,sp,r2)

retl =rho*sp

ret2=r2

ret3=rho

return

end

249

subroutine den(a,xl ,ret)
double precision a,ret,rf,rm,x1
common lblk5/ rf

ret=rf

return

end

subroutine Cp(a,xl,ret1,ret2)
double precision intl(72),a,xl ,rf,tem,wf,vf,cf,retl ,ret2
common /blk4/ vf

common lblk5/ rf

common lblnk/ cf,intl(72)
call den(a,xl,tem)
wf=tem*vf/rf

retl =cf

ret2=wf

return

end

SUBROUTINE

UNIMWP(Pl ,AT,AB,ANGLET,ANGLEB,L,N,PTOTAL,psum)

000 00 00800

C

"U

P1,P2,PTOT AL - POWER ABSORPTION RATE DUE TO SIDE WAVES,

AND BOTTOM WAVES, AND ALL TEM WAVES, IN W/M‘3
AB,AT - MAGNITUDES OF BOTTOM AND TOP INCIDENT TEM
WAVES IN V/M
L - COMPOSITE TI-IICKNESSJN M.
AN GLEB,ANGLET - POLARIZATION ANGLE OF BOTTOM AND TOP
TEM WAVES,

0 TO 2*PI.

N - THE NUMBER OF OUTPUT DATA DESIRED.
implicit none

INTEGER N ,I

double precision P1, AT, AB, ANGLET,ANGLEB,EPSO,MUO,FREQ,L
double precision PI,P2(72),P'I‘OTAL(72),PSUM, DELTH,PI2,EXY(100)
DOUBLE COMPLEX J,EX(72),EY(72)

DOUBLE COMPLEX EPSPC, EPSQC,EXT,EYT,EXB,EYB
COMPLEX * l6 EPSPU, EPSQU

DOUBLE COMPLEX KP, KQ

DOUBLE COMPLEX EXAVG(100),EYAVG(100)
EPSPC=(1.0,-25000)

DATA EPSQU /(33.0,-53.3)/

 

225

250
EPSQC=(14.5,-75.8)

OPEN INPUT AND OUTPUT DATA FILES
CONSTANTS

PI=3.14159

J=(0.0, 1.0)

Wﬁtd‘.‘)11

FREQ=2.45"'1.0E+9

EPSO=1.0/(36*PI)"'1.0E-9

MUO =4'PI'1 .0E-7
EPSPC=EPSPC'EPSO
EPSQC=EPSQC*EPSO

INCIDENT TOP AND BOTTOM TEM WAVES
EXT=AT*COS(ANGLET)
EYT=AT*SIN(ANGLET)
EXB=AB*COS(ANGLEB)
EYB=AB*SIN(ANGLEB)
KP=2*PI*FREQ*SQRT((MUO*EPSPC))
KQ=2*PI*FREQ*SQRT((MUO*EPSQC))

DELTH =0.01 *L/N

DO 200 I=1,N+ 1

EX(I) =EXB*EXP(J*KP*(I-N-l)*DELTI-I) +
EXT*EXP(-J *KP*(i-1)*DELTH)

EY(I) =EYB*EXP(J*KQ*(I-N- l)*DELTH) +
EYT*EXP(-J *KQ*(i-1)"‘DELTH)

EXY(I) = sqrt(abs(EX(I))"2 +abs(EY(I))**2)

P2(I)=-FREQ*PI*(imag(EPSPC)"ABS(EX(I))"2+
imag(EPSQC)*ABS(EY(I))**2)
PTOTAL(I)=P1+1.0e-6*P2(I)

PSUM=0.0

DO 225 I=1,N

EXAVG(I) =(EX(I) +EX(I + 1))/2

EYAVG(I) =(EY(I) +EY(I+ 1))/2

P12=-FREQ*PI*(IMAG(EPSPC)*ABS(EXAVG(I))"2+
IMAG(EPSQC)*ABS(EYAVG(I))"2)

PSUM=PSUM+(P1+P12)*9.0*.0254*.0254*DELTH

return
END

251

SUBROUTINE TRIDAG(c,n,b)

DOUBLE PRECISION c(72,72),bet(72),g(72),b(72)
INTEGER n,i,j

bet(l)=c(l,l)

s(1)=C(1,n+1)/bet(1)

do i=2,n

bet(i) =c(i,i)-c(i,i-1)*c(i-1,i)/bet(i-1)

80) = (66.11 + 1)-C(i.i-1)"s(i-l))/bet(i)

enddo

b(n)=s(n)

j=n

do 1111111410151)
PG)=1gO)-c(j.j+l)*b(i+l)/bet(i)
1:]-

enddo

return

END

SUBROUTINE MAT(C,N,B)
DOUBLE PRECISION arnx,C(72,72),A(72,72),B(72),TEMP,MAX,RAT
INTEGER r,xm,mx(72),N,I,J,K,M,tr
K=l

DO I=1,N

DO I = 1,N +1

A(IJ)=C(1J)

ENDDO

ENDDO

do i=1,n

mx(i)=i

enddo

DO WHILE (K.LE.N)
MAX=A(K,K)

tr=0

do i= 1 ,n

if (abs(a(i,k)).GT.abs(max)) then
max=a(i,k)

tr=l

r=i

endif

enddo

if (tr.EQ.l) then

do j = l ,n + 1

temp=a(k.j)

a(k.j)=a(r.i)

252

a(rJ)=temp
enddo
endif
arnx=abs(max)
tr=0
do i=l,n
if (amx.LT.abs(a(k,i))) then
arnx=a(k.i)
r=i
tr= 1
endif
enddo
if (tr.EQ.l) then
xm=mx(r)
mX(r)=mx(k)
mx(k)=xm
do i=l,n
temp=a(i,k)
a(i,k)=a(i,r)
a(i,r)=temp
enddo
endif
max=a(k.k)
DO 25 J = 1,N +1
A(KJ)=A(KJ)/MAX
CONTINUE
DO 301 = 1,N
IF (1 .NE. K) THEN
RAT=A(I,K)
DO 27 J = 1,N-+1
AOTD=A(I.J)-RAT*A(KJ)

CONTINUE
ENDIF
CONTINUE
K=K+l
ENDDO
DO 401 = 1,N
B(mx(i))=A(I,N+1)
CONTINUE
RETURN

END

253

subroutine input(b)

implicit none

integer ch1,ch, m, i, ml

double precision

intl(72), hdh2, cf ,spx, fx, tb, sdx ,,tdx(72) expm(500, 72), hdll ,hdhl h,c12,vf,rf,kf,dt,dx,l,,to

t1n,yin, tf, td ,tdt(SOO), b(5)

integer j,pt

common Iblkl/ yin,to,m

common Iblk2/ kf

common lblk4/ vf

common lblk5/ rf

common lblk6/ expm,pt

common lblk7/ dx,dt,tdt

common /sb1k8/ hdll ,hdhl ,hdh2,hd2

common Inwblk/ tdx,ml

common lblnk/ cf,intl(72)

read(*,"'),ch,chl,rf,vf,kf,cf,hdll,hdhl,hdh2,hd2,l,tin,to,dt,m

do i= 1,5

md(*,*).b(i)

enddo

if (chl.EQ.O) then

read(*,*),sdx,fx,ml

spx=(fx-sdx)/ml

do i=1,m1

tdx(i)=sdx+(i-1)*spx

. enddo

else

read(*.*).m1

do i=l,ml

md(*.*).th(i)

enddo

endif

if (ch.EQ.O) then

fwd(*.*).tb,td,pt

pt = (tf-tb)/td + 1

do i= 1,pt

tdt(i) = (i-1)*td

enddo

else

read(*.*).pt

do i= 1,pt

r@Il€1("'."‘).tdt(i)

do j = 1 ,ml

readC“,"‘).¢=J(pm(i,j)

254

if((i.EQ.1).AND.(tdt(i).EQ.O)) then
intl(i) =expm(i,j)

endif

enddo

enddo

endif

yin = tin! to

dx =1/m

return

end

V—*-"_'_— -T -.T

Appendix III FORTRAN code for processing model

PROGRAM MWCURE

implicit none

integer st,timcnt,i,j,cntr

double precision
endt,diff,hdll,hdhl,ptot,pc(72),b(5),vf,cf,emt(10),em(lO,2),a.3,e3,
hdl,hd2,rf,kf,dl,wf,xav1,xav2,xav3,xt(72),rl(72),yav3,pl,p2,y2(72),yl(72),y3(72,72
),x(72),xl(72),x2(72),x3(72),dt,dx,l,td,tf,to,tin,al,a2,el,e2,kavl,kav2,kav3,kav,yavl,
yav2,t,dh,tdt,
Vimdmt.hdh2.templ(5).rte(72).Y5(72).err0r.y4(72).sudt.dtl.r2(72).r3(72).r4(72),xt1(72
). f1(72).t2(72),t3(72).f4(72),y0(72).error1

integer m,o

common lblkl/ a1,32,a3,e1,e2,e3,to

common lblk2/ kf

common /b1k3/ xl,x2,x3

common lblk5/ rf

common lblok/ st

common Iblokll em,emt

common lblnk/ cf

common lnblk/ vf

common /tb1kl/ m

common ltblk2/ endt

common lblock/ rte

common /ovlk/ y0,yl,y2

common lfblk/ f1,f2,f3,f4

rced(*,*),rf,cf,lcf,vf,hdll,hdhl,hdh2,hd2,l,dh,st,to,endt
write(*,*),rf,cf,kf,vf,hdll,hdhl,hdh2,hd2,l,dh,st,to,endt
md(*.*),cmt(l)
em(l,1)=0
do i=2,st+1
md(*.*),em(i,l)
write("‘,*),em(i-l,l),’ < =t< =’,em(i,l)
write(*,*),’Final Temperature: ’
read(*.*).emt(i)
em(i,2) = (emt(i)-emt(i-1))/(em(i, 1)—em(i-l , 1))
write("‘,*),’Slope=’,em(i,2)
enddo
read(*,*),al,a2,a3,el,e2,e3,dt,m,tf,td
C write(*,*),al,a2,a3,el,e2,e3,dt,m,tf,td
do i=1,5
md(*.*).b(i)

255

10

000

256

enddo

dx=llm

do i=1,5

md(*. *). tcmplﬁ)

enddo

open(unit=10,status= ’unknown’,ﬁle= ’n4.m’)
open(unit= ll,status= ’unknown’,ﬁle=’n5.m’)
open(unit= l2,status= ’unknown’,ﬁle= ’n6.m’)
Wﬁte(*.*).dx

all unimWP(b(l).b(2).b(3).b(4).b(5).l.m.rte,Pt0t)
do i= 1,72

PC(i)=0

enddo

dh=dhlto

do 10 i = l,m-+1

y0(i)=l.00

yl(i)=l.00

y2(i)= 1.000

x(i) =0.0000

x1(i) =0.0000

x2(i) =0.0000

x3(i)=0.0000

continue

t=0

tdt=0

timcnt=l

yin=emt(l)/to

do i=1,m+1

do j = l,m+2

y3(i,j) =0.00

enddo

enddo

Wﬁte(12,*).’tt=[’

mw(ll.*).’xt=[’

Start Loop In Time

do while (t.LE.tf)
if (t.GT.0) then
call POWStO’ZmC.)
else

call powstolmc)
endif

 

 

257
if (abs(t-tdt).LT.1.0e-6) then

C write(".")

C write(‘,11),t

11 format(5X,’Time :’,F12.4)

C write(*, 12)

C write(*,*),’Cure Temp’,yin*to

12 format (5X,’Distance’,8X,’Cure’,9X,’Temperature’,4X,’Power’,4X,’Rate’)
C write( 10, 13),int(100*t/60)

13 format (’txd’,15,’=[’)

write(l 1,48),t/60.00,x(int(templ(l)/dx)+ 1),x(int(templ(2)/dx)),x(int(templ(3)/dx)),
x(int(templ(4)/dx)),x(int(templ(5)/dx))
14 format(F12.3,’ ’,FlO.6,’ ’,F10.6,’ ’,F10.6)

if (t.GT.0) then

write(l2,48),t/60.00,to*y2(int(templ(l)/dx) + l)-273. 15 ,to*y2(int(templ(2)/dx))
-273.15,to*y2(int(templ(3)/dx))-273. 15 ,to*y2(int(templ(4)/dx))-273. 15,to*y2(int(temp1
(5)/dx))-273. 15,pc(l)

else

write(l2,48),t/60.00,to*y1(int(templ(1)/dx)+ l)-273. 15 ,to‘yl(int(templ(2)/dx))

-273. 15 ,to*y1(int(templ(3)/dx))-273. 15 ,to'y1(int(templ(4)/dx))-273. 15,to"'yl(int(templ
(5)/dx))-273. 15,pc(1)

endif
48 format(F12.3,lX,F8.3,lX,F8.3,1X,F8.3,lX,F8.3,lx,F8.3,lx,F8.3)

do i= 1 ,m + 1

if (t.GT.0) then
C write(*.15).(i-l)*dX.X(i).t0*y2(i).PC(i).r2(i)
C write(lO,20),to*y2(i),x(i)

else
C write(*.15).(i-1)*dx,X(i),t0*y1(i),PC(i).r1(i)
C write(lO,20),to*y1(i)-273. 15,x(i)

endif
15 format (5X,F8.5,5X,F8.5,5X,F12.5,5X,F8.5,5X,F8.5)
20 format (F10.6,1X,F12.6)

enddo
C write(10.*).’l;’

tdt=tdt+td

endif

if (t.G'I‘.O) then
cntr=0
sudt=0
dtl =dt
errorl = 100
do while (sudt.LT.dt)

258

do i= l,m + l
mu =y2(i)
xt(i) =x(i)
call mtt’v()'2(i).x(i).r2(i))
enddo
dmt=2*dt1/(3*(dx**2.0))
0 =0
error=0
do while (((o.EQ.0).OR.(error.GT.1.0e-4)).AND.(o.LE.50))
call powstosmc) T
do i= l,m + l
y4(i) =)'5(i)
xtl (i) =xt(i)
call rat¢>(y5(i),xt(i).r3 (1))
r4(i) = l .0/3.0*(rl(i) +r2(i)+r3(i))
enddo
yav3 =0.50*(yZ(2) +y2(1))
xav3 =0.50"'(x(2) +x(l))
call Th(yav3 *to,xav3,kav3)
call Th(y2(1)*to,x( 1),ltav)
call Pmd02(l)"‘t0.X(l).PZ.Wf.dl)
diff = to*abs(yin-y2( 1))
if (diff.LE.27) then
hdl =hdll
endif
if ((diff.LE.43).AND.(diff.GT.27)) then
hdl =hdh1
endif
if (diff. GT.44) then
hdl =hdh2
endif
y3(l , l) =(dmt‘(2‘dx*hdl +kav3 +kav) +p2)
y3(l ,2) = -y5(2) *(kav3 +kav) *dmt
y3(l ,m +2) =p2*yl(1) +dmt*((y2(l)+y1(1))*(-2*dx*hd1-(kav+kav3)) +
(kav +kav3)*(yZ(2) +y1(2)) +6*yin*hdl *dx)-2*dt*r4(l)*dh*(l-wt)*dl +2*dt"pc( 1)/to
y5(1)=(y3(l,m+2)*y3(1 ,2))/y3(1. 1)

do i =2,m

yavl =0.50*(yZ(i) +y2(i + 1))
yav2 =0.50*(y2(i) + y2(i- 1))
xavl =0.50*(x(i) +x(i+ 1))
xav2 =0.50*(x(i) +x(i-l))
call Th(yavl *to,xavl ,kavl)
call Th(yav2 *to,xav2,kav2)

 

if“: .4 ".._‘ 5.11. -‘ "-18.1“: . . .

259

“11 Pmd02(i)*t0.X(i).P1.wf.dl)
C write(*,*),kavl ,kav2,pl ,wf,dl ,rl

y3(i,i- 1) =-kav2*dmt"‘y5(i- l)

y3(i,i) =pl +dmt*(kavl +kav2)

y3(i,i+ l) =-dmt*kavl *y5(i + l)

y3(i,m +2) =dmt*(kavl "(yZ(i + l) +y l (i + 1))-(kav1 +kav2)‘(y2(i) +y1(i)) +
kav2 *(yZ(i- 1) +y1(i- 1)))-2*dt"'r4(i)"dh‘( 1 ~wf)*dl +p1*y 1 (i) +2‘dt‘pc(i)/to

ysa) =(yam+2)-y3(i,i-1)-y3(i.i+ 1))/y3(i,i)

enddo

yav3 =0.50*(y2(m+ 1) +y2(m))

xav3 =0.50*(x(m + 1) +x(m))

call Th(yav3 *to,xav3,kav3)

call Th(yZ(m +l)*to,x(m+1),kav)

call prod(yZ(m+1)*to,x(m+1),p2,wf,dl)
C write(*,*),kav3,kav,p2,wf,dl ,r2

y3(m+ l,m+ l) =dmt“(2"dx*hd2 +kav+kav3) +p2

y3(m + 1 ,m) =-dmt*(kav3 + kav)*y5(m)

y3(m+ l,m+2)=p2*y1(m+ 1)+dmt*((yZ(m+l)+yl(m+1))*
(-2 *dx*hd2-(kav +kav3)) + (kav +kav3)‘(yZ(m) +y1(m)) +6*yin*hd2*dx)—2 'dt‘r4(m +
l)*dh* (1-wt)*dl +2*dt"pc(m+ l)/to

y5(m + 1) =(y3(m+ l,m+2)-y3(m+ l,m))ly3(m+ l,m+ l)

error=0

call cure(timcnt,y5,xt1,m+ l,dt1,xt)

do i= 1 ,m + 1

if (error.LE.abs((y$(i)-y4(i))/y5(i))) then

enor=ab8((yS(i)-y4(i))/y5(i))

endif

if (xt(i).GT.1.0e-7) then

if (error.LE.abs((xt(i)-xt1(i))/xt(i))) then

error=abs((xt(i)-xt1(i))/xt(i))

endif

endif

enddo

o=o+1

enddo

if (o.GT.lOO) then
if(errorl .LE.error) then
goto 21

else

dtl =dt1/2

errorl =error
cntr=cntr+l

21

260

endif

else

sudt = sudt+dtl
do i= l,m + 1
xl(i) =x2(i)

x2(i) =x3(i)

x3(i) =x(i)

x(i) =xt(i)

y0(i) = yl(i)
ﬂ®=ﬂ®
y2(i) =y5(i)

r1 (i) =r2(i)

r2(i) =r3(i)

if (timcnt.GT. 3) then
f 1 (i) =--- f2(i)

12(i)= f3(i)

13(i) = f4(i)

endif

enddo
timcnt=timcnt+ 1
endif
write(*,*),o, ’ ’ ,error, ’ ’ ,dtl , ’ ’,sudt
enddo

endif

if (t.EQ.0) then

call cure(timcnt,yl ,x,m+ l,dt,xt)
do i= l,m+ l '
xl(i) =x2(i)

x2(i) =x3(i)

x3(i) =x(i)

x(i) =xt(i)
mummwmmunm)

enddo

call powst(yl .pc)
dmt=dt/(2*(dx**2.0))

yav3 =0.50*(yl(2)+y1(1))

xav3 =0.50*(x(2) +x(1))

call Th(yav3*to,xav3,kav3)

call Th(yl(l)*to,x(l),kav)

call Pr0d(yl(1)*t0,X(l),p2.wf.dl)
diff = to*abs(yin-yl(l))

if (diff. LE.27) then

hdl =hdll

261

endif

if ((diff.LE.43).AND.(diff.GT.27)) then

hdl =hdh1

endif

if (diff.GT.44) then

hdl =hdh2

endif

y3(1 , 1) =dmt"‘(+2"dx*hdl +kav+kav3)+p2

y3(1 ,2) = -dmt*(kav3 +kav)

y3(1,m+2)=y1(l)*(p2-dmt*2*dx*hdl-(kav3+kav)*dmt)+
dmt*(kav3 +kav)*yl(2) +4*yin*hdl‘dx’dmt-dl*(1-wt)*dt“rl(1)*dh+dt"'pc(l)/to

do i=2,m

yavl =0.50*(yl(i) +y1(i+ 1))
yav2 =0.50*(yl(i) +y1(i-1))
xavl =0.50*(x(i) +x(i+ 1))
xav2 =0.50*(x(i) + x(i-1))

call Th(yavl*to,xavl,lcavl)
call Th(yav2 1"to,xav2,kav2)
can prod(yl(i)*to.x(i).p1.wf.d1)
y3(i,i-l) =-kav2"‘dmt

y3(i,i) =pl +dmt*(kav1 +kav2)
y3(i,i+ l) =-dmt*kavl

y3(i,m+2) =dmt*(kavl *(yl(i+ 1))-(kav1 +kav2)*(yl(i)) +kav2“(yl(i-l)))-
dl*( 1-wt)*dt*rl(i)*dh +pl*y1(i)+dt"pc(i)/to
enddo

yav3 =0.50"‘(yl(m+ l) +y1(m))

xav3 =0.50*(x(m +1)+x(m))

call Th(yav3 *to,xav3,kav3)

call Th(yl(m+1)*to,x(m+l),kav)

call prod(yl(m+ l)*to,x(m+ l),p2,wf,d1)
y3(m+1,m+1)=dmt*(2*dx*hd2+kav+kav3)+p2
y3(m + 1 ,m) = -dmt*(kav3 +kav)

y3(m + l,m +2) =p2*yl(m + 1) +dmt*((yl(m + l))*(-2*dx*hd2-(kav+kav3)) +
(kav +kav3)*(yl(m)) +4*yin*hd2*dx)-dl *(l-wt)*dt*rl(m +1)*dh +dt‘pc(m + 1)/to
call mat(y3,m+1,y2)
timcnt=timcnt+ 1
endif
t=t+dt
call tinp(t,dt,yin*to,tin)
yin =tin/to

262

enddo
write(12.*).’];’
write(ll,*).’];’
close(lO)
close(l 1)
close(12)
end

subroutine Th(a,x,k1)

double precision a,x,k1,kf,km

common lblk2/ kf

km =(0.161+x*(0.00147*a-0.417))/100
kl =(0.02514*km)/(.02514+km)

return

end

subroutine prod(y,x,ret1,ret2,ret3)
double precision yl,xl,y,x,r2,rho,sp,retl,ret2,ret3
y1=y

x1 =x

call den(y1,x1,rho)

call Cp(yl,xl,sp,r2)

ret1=rho*sp

ma=ﬂ

ret3=rho

return

end

subroutine den(a,x,ret)
double precision a,ret,rf,x
common lblk5/ rf

ret=rf

return

end

subroutine Cp(a,x,retl ,ret2)

double precision rf,tem,vf,cf,a,x,retl,ret2
common lblnk/ of

common lblk5/ rf

common lnblk/ vf

call den(a,x,tem)

ret2 =tem*vf/rf

ret1=cf

return

263

end

subroutine tinp(ti,dt,yp,re)

double precision yp,ti,dt,re,em(10,2),emt(10)
integer fnd,i,st

common lblok/ st

common Iblokll em,emt

i=2

fnd=0

do while((i.LE.st+l) .AND. (fnd.EQ.0))

if ((ti.GE.em(i—l,l)) .AND. (ti.LE.em(i,l))) then
re=yp+dt*em(i,2)

fnd = 1

endif

i=i+ l

enddo

if (ti.GT.em(st+l,l)) then
re=yp+dt*em(st+ 1,2)

endif

return

end

subroutine powst(yd,px)
implicit none

double precision a1,a2,a3,el,e2,e3,tdx,endt,to,yd(72),px(72),ptot,mxa,rte(72)
integer i,m

common lblkll a1,a2,a3,el,e2,e3,to
common ltblkl/ m
common /tblk2/ endt
common lblock/ rte
mxa=yd(l)*to

do i=2,m+l

if (mxa.LE.to*yd(i)) then
mxa=to*yd(i)

endif

enddo

write(*.*), mxa

if (mxa.GE.endt) then

do i= l,m + l

px(i)=0

enddo

endif

if (mxa .LT. endt) then

264

do i= l,m + 1
PX(i)=rt6(i)
enddo
endif
C wﬁte(*,').px(l).px(5).px(m+l)
return
end

subroutine cure6,t,x,n,h,xi)

integer n,i,j

double Precision tcmpZJl(72).f2(72).f3(72).f4('72).00n.t(72),X(72).xi(72),h.
x1(72).x2(72).x3(72).tm1(72).th(72).tm3(72),nX(72).orr

common fblk3/x1,x2,x3

common lovlk/ tml,tm2,tm3

common lfblk/ fl,f2,i3,f4

con = l .0/sqrt(2.0)

if (i.LE.3) then

do i= l,n

(all mw(t(i).X(i).fl(i))

call rate(t(i),x(i)+0.50‘h'f1(i),f2(i))

call rate(t(i),x(i)+(con-0.50)*h*f1 (i) +(1-con)*h*f2(i),f3(i))

call rate(t(i),x(i)-con*h*f2(i) + (1 +con)*h*f3(i),f4(i))

Xi(i)=x(i)+h/6.0*(fl(i)+2*(l-con)*f2(i)+2"'(l +con)*13(i)+f4(i))

if ((rea1(xi(i))-l.000).GE. 1.0e-7) then

xi(i) = 1.0000

endif

enddo

endif

if (j.GT.3) then

do i= l,n

if (i.EQ.4) then

call rate(tml(i),x1(i),fl (1))

call rate(tmz(i),x2(i),f2(i))

call rate(tm3(i),x3(i),f3(i))

endif

call rate(t(i).x(i),f4(i))
xi(i)=x1(i)+4.0*h/3.o*(2*f4(i)-13(i)+2*f2(i))
orr= 1000

do while(orr.GT.l.0e-2)

nx(i)=xi(i)

call rate(t(i),nx(i),tcmp2)
xi(i)=x3(i)+h/3.0*(temp2+4*f4(i)+l3(l))

265

orr=abs((xi(i)-nx(i))/xi(i))

enddo

if ((real(xi(i))-l .000).GE. 1.0e-10) then
xi(i) = 1.0000

endif

enddo

endif

return

end

subroutine rate(y,x,ra)

double precision al,a2,a3,el,e2,e3,to,y,x,ra
common lblkl/ al,a2,a3,el,e2,e3,to
double precision k2,k3,k4,R
R=8.314

if ((reeﬂ(x)-l .000).LT. 1.0e-8) then
k2=a1*exp(-el/(R*y*to))

k3 =a2*exp(-e2/(R*y*to))
k4=a3*exp(-e3/(R*y“'to))

if (x.LE.O.30) then
ra=((k2+k3*x)"'(1-x)*(0.47-x))
else

ra=(k4*(l-X))

endif

endif

if ((real(x)-1.000).GE. 1.0e-10) then
ra=0.0000

endif

return

end

10.

ll.

12.

13.

14.

15.

16.

REFERENCES
N. Vlahalcis and D. Van Dorn, 25th joint Propulsion Conference, Monterey,
CA, July, (1989).
B. Tittmann, MRS Bulletin, April, p. 21, (1988).

J. Wei, J. Jow, J.D. DeLong, and M.C. Hawley, SAMPE Journal, 21 (1),33
(1991).

W.I. Lee and 6.8. Springer, J. Composite Materials, 18, p. 387 (1984).

J. Wei, Y. Chang, B. Thomas, and M.C. Hawley, Proceeding of ICCM/VIII,
1, 10—L (1991).

Agrawal, R. and L. T. Drazl, J. Adhesion, (1989).
J. Jow, Ph.D dissertation, Chem. Eng. Depart., MSU, (1988).
W.I. Lee and 6.8. Springer, J. Composite Materials, 18, p. 357 (1984).

J. Wei, J. Jow, J.D. DeLong, and M.C. Hawley, 2nd Topical Conf. on
Emerging tech. in mat., AIChE annual meeting, Nov. 1989, SF, CA.

J. Wei, J.D. DeLong, M. DeMeuse, and M.C. Hawley, ”Comparison of
Microwave and Thermal Cure of Epoxy Resins" , Polym. Eng. & Sci. (in
press) (1992).

J. Wei, B. Thomas, and M. C. Hawley, ANTEC’92, Detroit, May, (1992).

J. Wei, M. DeMeuse, and M. C. Hawley, ACS meeting, SF, CA, April,
(1992).

J. Wei, B. Thomas, and M. C. Hawley, 37th Intern. SAMPE, Anaheim, CA,
March, (1992).

U. Hottong, J. Wei, R. Dhulipala, and M.C. Hawley, Ceramic Transactions,
21, 587 (1991).

M.C. Hawley and J. Wei, MRS Symp. Proc. , 182, 413 (1990).

J. Wei, J.D. DeLong, and M.C. Hawley, Proceeding of 5th Amer. Soc.
Comp., 239 (1990).

266

17.

18.
19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.
32.

33.

34.

267

Larry Fellows and Martin C. Hawley, 37th Intern. SAMPE, Anaheim, CA,
March, (1992).

J. Mijovic and J. Wijaya, Macromolecules, 23(15), 3677 (1990).
J. Mijovic, A. Fishbain, and J. Wijaya, Macromolecules, 25(25), 986 (1992).

A. Gourdenne, Proceedings, International Conference on reactive Proceeding
of Polymers, p. 23, (1982).

J. Jow, J. DeLong, and M. C. Hawley, SAMPE Quart., 29(2), 46 (1989).

D. A. Lewis, J. C. Hedrick, T. C. Ward and J. E. McGrath, Polymer
Preprint, 28(2), 330, (1987).

J. low, M. C. Hawley, M. Finzel, J. Asmussen, and R. Fritz, AIChE Conf.
Emerg. Tech. Mat., Aug. (1987).

J. Jow, M. C. Hawley, and M. Finzel, Rev. Sci. Instr., 69(1), 96 (1989).

JD. DeLong, J. Jow, and M.C. Hawley, AIChE 2nd Conf. Emerg. Tech.
Mat., Nov, (1989).

Lewis,D.,T.Ward,J.Summers,and T.Ward,Polymer Prep., 22, 1 (1988).

G.L. Vogel, J. Jow, J.D. DeLong, and M.C. Hawley, Proc. Am. Soc. Comp.
4th Tech. Conf. Comp. Mat., Oct., (1989).

L. K. Wilson and J. P. Salerno, Technical Report, AVRADCOM TR-78-46,
(1978).

Normans Strand, Modern Plastics, 54, 64 (1980).

A. Gourdenne et a1. , 3rd International Conference on Composite Materials,
Paris, 2, 1514 (1980).

A. Gourdenne and Le Q. Van, Polymer Preprints, 22, 125 (1981).
E. Karmazsin and P. Satre, Thermochemics Acta, 23, 305 (1985).

J. Jow, M. C. Hawley, M. Finzel and T. Kern, Polym. Eng. Sci., 28, 1450
(1988).

Jullien, H., and H. Valot, Polymer, 26, 506 (1985).

35.

36.

37.

38.

39.

40.

41.

42.

43.

45.

46.

47.

48.

49.

268

J. Wei, Fabrication of experimental setup and procedure for microwave curing
of AS4/3501-6 composite laminate, Research Complex, MSU, (1989).

J. Wei, V. Adegbite, Y. Chang, and M.C. Hawley, 6th annual ASM/ESD
Advanced Comp. Conf./Exp., Session of Emerg. Struct. Comp. tech. 1, Oct.,
(1990).

Chen, Y. F., and C. Y. C. Lee, Polym. Mat. Sci. Eng. (Am. Chem. Soc.) 60
(1989), 680-684.

. J. Mijovic, and J. Wijaya, Polymer Composites, 1,1, 184 (1990).

E.E.W. Kassner, ”Apparatus For the Generation of Short Electromagnetic
Energy” U.S. Patent 2 109 843, Oct. 5, (1937).

M. Rothstein, ”Dielectric Heating“ in Encyclopedia of polymer science, vol
5. , New York: Interscience Publisher, (1966).

R.W. Bruce, Mat. Res. Soc. Symp. Proc., 125, 3,(l988).

R.F. Harrington, ”Time-Harmonic Electromagnetic Fields” , McGraw-I-Iill Book
Company,1nc., (1987).

CC. Ku and R. Liepins, ”Electrical Properties of Polymers: chemical
principles” , Macmillan Publishing Company, (1987).

P. Debye, Polar Molecules, Chemical Catalog Co. , New York, Chap. V. ,
(1929).

T.W. Gault and LB. Wilson, ”Radio Frequency Rcdrying of Veneer - A Case
Study”, J. Microwave Power, p.179, (1987).

Irving J. Chabinsky, Mat. Res. Soc. Symp., 125, 17, (1988).

G. Lightsey, et. a]. , J . Microwave Power and Electromagnetic Energy, 23(1),
11 (1988).

Jean-Marie Thiebaut, C. Akyel, and G. Roussy, IEEE Trans. Instrum. Meas.,
31(1), 114 (1988).

G. Lightsey, C. George, and L.D. Russell, J. Microwave Power, p.86,
(1986). .

‘6

50.
51.
52.
53.
54.
55.
56.
57.

58.

59.

61.
62.

63.

65.

67.

68.

69.

269
R.D. Smith, EPRI Report EM-3645, (1984).

Stuart 0. Nelson, J. Microwave Power, p.65, (1985).

J .P. Vetsuypens and W. Van Loock, J. Microwave Power, p.110, (1986).
N.S. Strand, Modern Plastics, p.64, (1980).

DA. Lewis, MRS Symp. Proc., 269, Ll.2 (1992).

W.H. Sutton, MRS Symp. Proc., 262, L1.l (1992).

L. Dauerman, MRS Syrup. Proc., 182, L3.3 (1990).

J. Suzuki, et. al, J. Microwave Power, 25(3), 168 (1990).

SJ. Oda, Inter. Microwave Power Inst., 26th Symp. Proc., Buffalo, NY,
(1991).

SJ. Oda, MRS Symp. Proc., 262, L101, (1992).

R.J. Giguere, et. al, J. Tetrahedron Lett., 21, 4945 (1986).

R.N. Gedye, F.E. Smith, and KC. Westaway, Can. J. Chem., 66, 17 (1987).
R.J. Giguere et. al, J. Tetrahedron Lett., 28, 6553 (1987).

R.J. Giguere, MRS Symp. Proc., 262, L11.1, (1992).

RC. Collin, “Field Theory of Guided Waves”, 2nd ed., IEEE Press, (1990).
Haw-Hwa Lin, Ph.D. dissertation, Michigan State University, (1989).

E. Karmazsin, P. Satre, and J. F. Rochas, Thermochimiea Acta, 23, 305
(1985).

J .Brandrup, E.H.Immergut,polymer handbook,3rd edition John
Wiley&sons,inc., (1989).

Turi, Edith, Thermal Characterization of Polymeric Materials, Academic Press
Inc. (1981).

BA. Rozenberg, Kinetics, Thermodynamics and Mechanism of Epoxy
Oligomers with Amines. Advances in Polymer Science, Ed. K. Dusek, Part

70.

71.

72.

73.
74.

75 .
76.

77.

78.

79.

80.

81.

82.

83.

85.

86.

87.

270
II, 110, 113, (1985).

Acitelly,M.A.,Prime,R.B.,and Sacher,E.,Polymer,pp 335-342, 1971.

Hagnauer,G.L., Pearce, P.J., LaLiberte,B.R., and Roylance, M.E.,
Thermorheology of Thermosetting Polymers, pp 25-47, 1983.

Dutta, A. and Ryan, M.E., Journal of Applied Polymer Science, 24, 635
(1979).

Mijovic, J.,Journal of Applied Polymer Science, 31, 1177 (1986).

Moroni, A., Mijovic,J., Pearce, E.M., and Fauna, C.C.,Journa1 of Applied
Polymer Science, 32, 3761 (1986).

Sichina, W.J. ,Dupont Company Applications Brief, No. TA-93.

J .K. Gillham, ”Developments in Polymer characterization -3", J.V. Dawkins,
p. 159, Applied Science, London (1982).

Sourour, S. and Kamal, M.R., Thermochimic Acta, 15, 41 (1976).
L. Schechter, J. Wynstra, and R.P. Kurkjy, Ind. Eng. Chem. 48, 94 (1956).

LA. O’Neil and CF. Cole, J. Appl. Chem. (London) 6, 356 (1956).
H. Dannenberg, Soc. Plast. Eng. Trans, 3, 78 (1963).

K. Horie, H. I-Iiura, M. Sawada, I.‘ Mita, and H. Kambe, J. Polym. Sci., A—l,
8, 1337 (1970).

K. Dusek, M. Ilavsky, and S. Lunak, J. Polym. Sci. Polym. Symp., 53, 29
(1976).

QC. Riccardi, H.E. Adabbo,.and R.J.]. Williams, J. Appl. Polym. Sci., 22,
2481 (1984).

Mark Finzel, Ph.D. Dissertation, Michigan State University, 1991.

CC. Riccardi and R.J.J. Williams, ”Crosslinked Epoxies",B. Sedlacek, and J.
Kahovec Eds., Walter de Gruyter & Co. NY, (1987).

Mark Finzel, Lab notes, Depart. Chem.Eng., MSU. (1989).

W.I. Lee, A.C. Loos and 6.8. Springer, J. Comp. Mater., 16, 510 (1982).

88.

89.

91.

92.

93.

94.

95.

96.
97.
98.

100.

101.

102.
103.

104.

105.

271

I. Havlicek and K. Dusek,”Crosslinked Epoxies",B. Sedlacek, and J. Kahovec
Eds., Walter de Gruyter & Co. NY, (1987).

E. Rabinowitch, Trans. Faraday Soc., 33, 1225 (1937).
M. Gordon, and W. Simpson, Polymer, 2(4), 383 (1961).

M.L. Williams, R.F. Iandel, J.D. Ferry, J. Am. Chem. Soc., 21, 3701
(1955).

K.C. Rusch, J. Macromol. Sci-phys., B2(2), 179(1968).

G. Wisanrakldt and J.K. Gillham, Journal of Coatings Technology, @083),
35 (1990).

C.W. Macosko and D.R.Miller, Macromolecules, 2, 199 (1976).

Flory, P.J. , Principles of Polymer Chemistry, Cornell unv. press. ,
Ithacas,New York, 195 3.

DR. Miller and C.W. Macosko, Macromolecules, 2(2), 206, (1976).
R.F.Boyer,Rubber Chem. Technol., 63, 1303 (1963).
Wisanrakkit, G.; Gillham,].K. Polym. Mater. Sci. Eng., 52, 969 (1988).

Bair, H.E., Polym. Prepr. (Am.Chem.Soc., Div.Polym.Chem.), 26(1), 10
(1985).

W.H. Jo and KJ. Ko, Polym. Eng. Sci., 31(4), 239, Feb. (1991).

I. Havlick and K Dusek, in Crosslinked Epoxy, B. Sedlacek and J. Kahovec
Eds. p417, De. Gruyter, Berlin, 1987.

DiBenedetto, A.T., J. Polym. Sci., Part B: Polym. Phys., 26(9), 1949 (1987).

LP. Pascault and R.J.J. Williams, J. Polym. Sci., Part B: Polym. Phys., 28,
85 (1990).

E. Mertzel and J. Koening, Adv. Polym. Sci., Part B: Polym. Phys., 28, 85
(1990).

D.W. Schiering, J.E. Katon, L.T. Drza], and V.B. Gupta, J. Appl. Polym.

106.

107.

108.

109.

110.

111.

112.
113.
114.
115.
116.

117.

118.

119.

120.

121.

272
Sci., 3, 2367 (1987).

A. Moroni, J. Mijovic, E.M. Pearce, and QC. Foun, J. Appl. Polym. Sci.,
32, 3761 (1986).

J.B. Enns, and J.K. Gillham, J. of Appl. Polym. Sci., 28, 2567 (1983).

P.G. Babayevsky, and J .K. Gillham, J. of Appl. Polym. Sci., 11, 2067
(1973).

J.K. Lee, and K.D. Pae, J. of Polym. Sci., part C: Polymer letters, 28, 323
(1990).

J. GaJy, J.P. Pascault, and M.F. Grenier-Loustalot, in Crosslinked Epoxy, B.
Sedlacek and J. Kahovec Eds., De. Gruyter, Berlin, 1987.

Bidstrup, S.A., Sheppard, N.F. Jr., and Senturia, S.D., Technical Report 8,
Ofﬁce of Naval Research, 1986.

M. DeMeuse, unpublished results.

N.F. Sheppard, Jr. and Senturia, Polym. Eng. & Sci., 26(5), 354 (1986).
Nielsen, L.E.,L_Mag;6m_QL_S_QL-Revs. macromol. Chem., 63(1), 62, 1969.
SM. Singer, J. Jow, J. DeLong and M.C. Hawley, W 1989.
S.Y. Lee and GS. Springer, W vol. 22, 15, Jan 1988.

Asmussen. J., H. H. Lin, B. Manring and R. Fritz, Rev. Sci. Instrum. 58, p.
1477, (1987).

J. Jow, M.C. Hawley, M.C. Finzel, J. Asmussen, H.H. Lin, and B. Manring,
IEEE, Microwave Theory and Techniques, 35(12), 1435 (1987).

J. Jow, M.C. Hawley, M.C. Finzel, and J. Asmussen, Rev. Sci. Instrum., 60,
1, p. 96, (1989).

W. 1. Lee, A. C. Loos and G. S. Springer, J. Composite Material, 16, 510,
(1982).

Threshold limit values and biological exposure indices for 1988-1989,
American Conference of Governmental Industrial Hygienists, Cincinnati, Ohio,

273
p103, (1988).

122. AL. VanKoughnett and J.G. Dunn, J. Microwave Power, 8 (1), 101 (1973).
123. J. Ippen, Rubber Chem. Technol., 44 (l), 294 (191).

124. C.S. Wang, Rubber Chem. Technol., 51 (l), 134 (1984).

125. Ben Manring, Ph.D. thesis, Michigan State University, (1992).

126. W.I. Lee and GS. Springer, J. Comp. Mater., 11, 135 (1983).

127. W.M. Sun, unpublished results, MSU, 1990.

128. V. Volpe, J. Comp. Mater., 14, 189 (1980).

129. M.A. Morgan, D.L. Fisher, and EA. Milne, IEEE Transaction on Antennas
and Propagation, Ap;33(2), 191 (1987).

130. O. Hashimoto and Y. Shimizu, IEEE transactions on Microwave Theory and
Techniques, mu 1), 1202 (1986).

131. J .V. Beck and K.J. Arnold, ”Parameter Estimation Engineering and Science”,
p410, John Wiley & Sons, Inc. (1977).

132. J. Wei, A. Sundaram, and M.C. Hawley, MRS Symp. Proc., 262, Lll.8,
(1992).

133. M.R. Dusi, et. al., J. Appli. Polym. Sci., 30, 1847 (1985).

134. M.L. Williams, R.F. Landel, and JD. Ferry, J. Am. Chem. Soc., 12, 3701
(1955).

135. Y.A. Tajima, and D.G. Crozier, Polym. Engr. Sci., 28, 491 (1986).

136. T.H. Hou and J .M. Bai, ”Chemoviscosity Modeling for Thermosetting Resin
systems - III”, NASA-CR-181718, NASA Langley Research Center (1988).

137. J .C. Halpin, A. Apicella, and L. Nicolais, Polymer Processing and Properties,
G. Astarita and L. Nicolais, eds, Plenum Press, New York, p143, (1984).

138. EN. Da C. Andrade, Phil. Mag., 11, 699 (1934).

139.

140.

141.

142.

143.
144.

145.

146.

147.

148.
149.
150.
151.

152.

153.
154.

155.

156.

274

F. Bueche, ”physical Properties of Polymers", p.91, Interscience Publishers,
New York (1962).

MB. Roller, Polym. Engr. Sci., 13(6), 406 (1975).

M.R. Dusi, GA. May, and J.C. Seferis, Organic Coatings and Applied
Polymer Science Proceedings, 41, 635 (1982).

J.W. Lane, J.C. Seferis, and M.A. Bachmann, Polym. Eng. & Sci., 26(5),
346 (1986).

J.M. Kenny, SAMPE J. 21(2), 39 (1991).
D.R. Day, International SAMPE Symp., 31, 1095 (1986).

D.R. Day, TJ. Lewis, ILL. Lee and S.D. Senturia, J. Adhesion, 18, 73
(1985).

D. Kranbuehl, S. Delos, M. Hoff and L. Weller, lntemational SAMPE
Symp., 31, 1087 (1986).

PR. Ciriscioli and GS. Springer, International SAMPE Sypm., 31, 312
(1989).

Loos A.C., G.S. Springer, J. Comp. Mater. 11, 135 (1983).

Dave, R. J.L. Kardos and M.P. Dudukovic, Polym. Comp., 8, 29 (1987).
Gutowski, T.G., T. Morigaki, and Z. Cai, J. Comp. mater. 21, 172 (1987).
Lindt, J .T., SAMPE Quarterly, 3,14 (1982)

A. Pourstartip, G. Riahi, and G. Smith, Proceedings of ICCM/8, 10-G-3,
Honolulu, (1991).

R. Dave, J. Comp. Mater., 24, 22 (1990).
Springer J. Comp. Mater., 16, 400 (1982)

R. Dave, J.L. Kardos, and M.F. DuDuKovic, Proc. of Amer. Soc. for
Comp., p137, (1986).

AR. Mallow, F.R. Muncaster and RC. Campbell, Proc. of Amer. Soc. for
Comp., p171, (1986).

157.

158.

159.

160.

161.

162.
163.

164.

165.

166.

167.

168.

275

J-B. Donnet and R.C. Bansal, When, Marcel Dekker, Inc., New
York, p151, (1984).

K.Y. Yoon, T.W. Kim, W.I. Lee, and E]. Jun, Proc. of 6th ICCM & 2nd
ECCM, vol. 1, p1.81, (1987).

A. Noordam et al. in ”crosslinked Epoxies", B. Sedlacek and J. Kahovec eds.,
W. de Gruyter, Berlin, p373, (1987).

R.J.J. Williams et al., Polym. Eng. & Sci., 39(18), 1140 (1990).

I. Dobas et al., in ”Crosslinked Epoxies", B. Sedlacek and J. Kahovec eds.,
p81, W. de Gruyter, Berlin, (1987).

T. Igarashi, et. a1. , Polymer, 29, 301 (1979).
A.K. Srivastava and LR. White, J. Appl. Polym. Sci., 22, 2155 (1984).

M.H. Loh and J.V. Beck, ASME winter annual meeting, Atlanta, GA,
December, (1991).

T.A. Bogetti and J.W. Gillespie, Jr., J. Comp. Mat., 23, 239 (1991).

G. D. Smith, Numerical Solutions of Partial Differential Equations: Finite
Difference Methods, 2nd ed. , Oxford University Press, Oxford (1978).

B. Camahan, H.A. Luther, and 1.0'. Wilkes, ”Applied Numerical Methods”,
Robert E. Krieger Publishing Company, Inc. ( 1990).

J .M. Kenny, A. Trivisano, and LA. Berglund, SAMPE J. 21(2),39 (1991).

UN

nrcnran STATE
lll1|"WMlIIHIIWWI
3129300

 

IV. LIBRQ
l H HIIHNHI‘INHIIIH
2

89928