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Ann Arbor, MI 48106 The Pennsylvania State University The Graduate School Department of Engineering Science and Mechanics MICROWAVE PROCESSING OF CERAMICS: MODELING, CHARACTERIZATION AND APPLICATION A Thesis in Engineering Science and Mechanics by Xiang Dong Yu Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 1992 Date of Signature We approve the thesis of Xiang Dong Yu. _____________ Vasundara V. Varadan Distinguished Alumni Professor of Engineering Science and Mechanics and Electrical Engineering Thesis Advisor Chair of Committee Vij^y K. Varadan Distinguished Alumni Professor of Engineering Science and Mechanics and Electrical Engineering A- Sridhar Komameni Professor of Clay Mineralogy Deepak Ghodgaonkar Assistant Professor of Engineering Science and Mechanics Richard P. McNitt Professor of Engineering Mechanics Head of the Department of Engineering Science and Mechanics 10 ( W /*? 2- ABSTRACT This thesis presents the results of modeling, characterization and application of microwave energy to ceramic processing. Microwave processing of materials provides several advantages that in many cases help improve product quality, uniformity of grain structure, and yield. The ability of the microwave energy to penetrate and, hence, heat from within the product, helps reduce processing time, costs, and in some cases reduce the sintering temperature. There is also some evidence that microwave processing of materials actually provides improved microstructure and other properties. Microwave processing makes it possible to quickly remove binders, sinter ceramics without rupture or cracking, reduce internal stress, lower thermal gradients, and control the state of oxide. However, microwave heating of material and sintering of ceramics is a complex process that is not understood completely and there are many remaining problems that still need to be solved before this new technique can be transferred from the research laboratory to actual industrial use. The objectives of this thesis are: 1. To establish the basic equations for describing microwave processing of ceramics, namely sintering and heating. 2. To use the established equations for obtaining microwave deposition and temperature distribution considering dielectric properties variation with temperature and heat transfer boundary condition such as radiation and convection. 3. To use the Finite Difference Time Domain (FDTD) method to simulate a single mode cavity and related thermal insulation problems. 4. To establish a single mode high power microwave heating system for in situ microwave processing and characterization. 5. To establish a method of measuring the reflection coefficient using an impedance analyzer and calibrating the detectors on the impedance analyzer. 6. Use the microwave heating and sintering system as well as measurement technique to heat, sinter and characterize densified and green ceramic rod specimens. 7. Use the single mode high power microwave heating devise for ceramic processing application. The basic equations describing the interactions of ceramics with electromagnetic fields are derived first. Modeling the green ceramic during microwave sintering as a deformable dielectric, a continuum mechanics model is used to describe the interaction of electromagnetic waves with the deformable thermoelastic body. Combined with the thermal and mass diffusion equations into the consideration, such a description is unique, efficient and complete in modeling microwave processing macroscopically. A further application of this theory would enable the prediction of the stress fields generated in the ceramic body if temperature gradients occur. Using those equations, microwave heating of ceramics is modeled. The Finite Difference Time Domain (FDTD) method is also used to simulate heating of ceramics in a single mode cavity and analyze the insulation scheme. To further understand microwave sintering and heating of ceramics, an experimental study using a single mode high power microwave heating device was conducted. This device would make it possible to simultaneously heat and sinter ceramics and characterize the process with the same source. In this experimental work, a ceramic rod in the microwave cavity is modeled by an equivalent T network. The reflection caused by the ceramic rod, coupling aperture and variable short is measured by a modified reflectometer attached to the transmission line. The dielectric property of the ceramic rod is the function of the measured reflection coefficient. An inversion technique would allow for retrieving corresponding dielectric properties of the ceramics during heating or sintering. The ceramic rod samples are either a densified product obtained commercially or a green coupon obtained through an extrusion process. The single mode high power microwave cavity was also used in the ceramic processing. TABLE OF CONTENTS Page LIST OF FIGURES............................................................................................................. ix LIST OF TABLES...............................................................................................................xii ACKNOWLEDGEMENTS........................................................................ xiii Chapter 1 INTRODUCTION..............................................................................................1 1.1 Scope of the Thesis............................................................................................ 1 1.2 Statement of Problem..........................................................................................1 1.3 Ceramic Processing............................................................................................4 1.4 Microwave Processing of Ceramics...................................................................7 1.5 Thesis Organization..................................................................... 13 Chapter 2 BASIC EQUATIONS TO DESCRIBE MICROWAVE HEATING OF CER A M IC S.................................................................................................... 15 2.1 Introduction...................................................................................................... 15 2.2 Equations for Microwave Material Interaction................................................ 15 2.2.1 Maxwell’s Equations............................................................................... 15 2.2.2 Boundary Condition............................................................................... 18 2.3 Theory of Electromagnetic Field Interactions with Deformable Dielectrics... 19 2.4 Thermal Diffusion Equation............................................................................ 22 2.4.1 Mass Conservation Equation.................................................................. 23 2.4.2 Momentum Conservation Equation........................................................24 2.4.3 Moment of Momentum...........................................................................25 2.4.4 Energy Conservation Equation.............................................................. 27 2.4.5 Boundary Condition...............................................................................30 2.5 Mass Diffusion Equation.................................................................................32 2.6 Conclusion and Discussion.............................................................................. 35 Chapter 3 MODELING MICROWAVE PROCESSING OF CERAMICS..................... 37 3.1 Introduction......................................................................................................37 3.2 Theory.............................................................................................................. 38 3.3 Description of the Model.................................................................................. 40 3.4 Impedance Method...........................................................................................40 3.5 Computing Procedures..................................................................................... 44 3.6 Results and Discussion....................................................................................44 3.6.1 Microwave Energy Absorption by Ceramics......................................... 44 3.6.2 Simulating Microwave Heating of Ceramics......................................... 46 3.7 Conclusion....................................................................................................... 49 Chapter 4 MODELING SINGLE MODE CAVITY WITH FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)................................................................51 4.1 Introduction......................................................................................................51 4.2 Finite Difference Time Domain Method...........................................................53 4.2.1 Introduction.............................................................................................53 4.2.2 FDTD Method Formulation....................................................................55 4.2.3 Outer Radiation Boundary Condition (ORBC)..................................... 57 4.3 Structure of Model............................................................................................ 60 4.4 Computation Procedures.................................................................................. 63 4.5 Results and Discussion....................................................................................65 4.6 Conclusion....................................................................................................... 66 Chapter 5 EXPERIMENTAL SYSTEM.......................................................................... 71 5.1 Introduction......................................................................................................71 5.2 Magnetron.........................................................................................................71 5.3 Circulator.......................................................................................................... 73 5.4 Directional Coupler.......................................................................................... 75 5.5 Impedance Analyzer.........................................................................................75 5.6 The 4-Stub Tuner.............................................................................................77 5.7 Iris 5.8 Resonator Cavity.............................................................................................83 ................................................................................................................ 80 Chapter 6. IN SITU MICROWAVE HEATING, SINTERING AND CHARACTERIZATION........................................................................................... 88 6.1 Introduction..................................................................................................... 88 6.2 Historical Background.....................................................................................88 6.3 Mathematical Model for the Characterization.................................................. 90 6.4 Model Description........................................................................................... 95 6.5 Impedances of the Iris and the Variable Short.................................................97 6.6 Measurement of the Reflection Coefficient.....................................................98 6.7 Effect of Iris on the Reflection Coefficient Measurement............................ 103 6.8 Green Ceramic Rod Preparation.................................................................... 105 6.9 Experimental Set-up and Characterization Procedures..................................108 6.10 Results and Discussion.................................................................................. 109 6.11 Conclusions.................................................................................................... 119 Chapter 7 APPLICATION OF MICROWAVE ENERGY TO CERAMIC PROCESSING......................................................................122 7.1 Introduction.................................................................................................... 122 7.2 Insulation Considerations in Microwave Processing................................... 123 7.3 Temperature Measurement.............................................................................125 7.3.1 Introduction............................................................................................125 7.3.2 Pyrometer Measurement.......................................................................127 7.3.3 Thermocouple Measurement................................................................. 127 viii 7.3.3.1 Introduction............................................................................. 127 7.3.3.2 The Principle of the Thermocouple........................................ 128 7.3.3.3 Theory of Thermoelectricity.....................................................128 7.3.3.4 Thermocouple in the TE103 Cavity..................................................130 7.3.3.5 Summary........................................................................................... 133 7.4 Binder Burn-out of Tape-Casted Ceramics by Microwave Energy............... 133 7.4.1 Introduction........................................................................................... 133 7.4.2 Microwave Processing......................................................................... 134 7.4.3 Ceramic Tape Preparation..................................................................... 135 7.4.4 Characterization Method...................................................................... 135 7.4.5 Results and Discussion......................................................................... 135 7.4.6 Conclusion............................................................................................ 139 7.5 Microwave Sintering of A^Oj/c-ZrC^ Composites..................................... 140 7.5.1 Background........................................................................................... 140 7.5.2 Microwave Sintering............................................................................141 7.5.3 Sample Preparation................................................................... 142 7.5.4 Mechanical Properties.......................................................................... 142 7.5.5 Conclusion............................................................................................ 149 Chapter 8 CONCLUSIONS AND FUTURE W ORK..................................................152 REFERENCES ................................................................................................................ 155 APPENDIX EXPRESSION FOR q and Dj j ................................................................172 ix LIST OF FIGURES Figure Eagg 2.1 Differential Control Volume, dxdydz, for Species Diffusion Analysis................. 33 3.1 Ceramic Slab............................................................................................................ 41 3.2 Microwave Power Absorption for Slabs with Different Loss Factors.................. 45 3.3 Microwave Power Absorption for Slabs with Different DielectricProperties... .45 3.4 Procedures for Simulating Microwave Heating of Ceramics................................ 47 3.5 Dynamic Temperature Profile 0.01 inside the Slab from the Incident Plane 3.6 Temperature Distribution over the Thickness of the Slab....................................... 48 3.7 Total Microwave Energy Absorbed by Slab vs. Temperature................................ 50 4.1 Yee Cell for Finite Difference Time Domain M ethod.............................................54 4.2 Single Mode Cavity for Material Processing...........................................................62 4.3 Structure of Ceramic Rod and Surrounding Microwave Susceptor...................... 62 4.4 Truncated Gauss Pulse............................................................................................64 4.5 Frequency Response of an Empty Cavity...............................................................67 4.6 Convergence Analysis of the FDTD Method......................................................... 67 4.7 Field Distribution in the Cavity with a Ceramic Rod.............................................. 69 4.8 Microwave Energy Absorbed by a Ceramic Rod in the Cavity............................. 69 4.9 Microwave Energy Absorption by Both Ceramic Rod and 48 Microwave Susceptor at Room Temperature.......................................................... 70 4.10 Microwave Energy Absorption by Both Ceramic Rod and Microwave Susceptor at lOOO'C...........................................................................70 5.1 Single Mode High Power Microwave Processing and Characterization System................................................................72 5.2 Multicavity Oscillator Magnetron.............................................................................74 5.3 A 3-Port Circulator...................................................................................................76 X Directional Coupler................................................................................. ..76 Impedance Analyzer................................................................................ ..78 A 4-Stub Tuner........................................................................................ ..79 Characteristics of One Stub..................................................................... ..79 Three Coupling Iris for Material Processing Cavity.............................. ..81 Electric and Magnetic Polarization in the Iris......................................... ..82 Field Distribution in TE103 Cavity........................................................ ..84 Equivalent Circuit of a Ceramic Rod in the Transmission Line............. ..92 Equivalent Circuit of a Ceramic Rod in the Cavity................................. ..96 A Vector Network Analyzer..................................................................... ..99 Measurement of the Impedance of the Iris.............................................. 100 Measurement of the Impedance of the Variable Short............................ 100 Three Detector Reflectometer................................................................... 102 Input Microwave Power vs. Output DC Voltage from Detector............. 104 Thermal History of Microwave Heating and Characterization.............. 111 Position of the Variable Short vs. Time................................................. 111 Reflection Coefficient vs. Time.............................................................. 112 Measured Dielectric Properties of Coors AD-998.................................. 113 Microwave Sintering of Alumina Ceramic Rod History..................... . 115 Density vs. Sintering Tim e..................................................................... 116 Variation of the Ceramic Rod Diameter with Sintering Tim e................. 117 Variation of Dielectric Properties of the Green Ceramics During Microwave Sintering.................................................................. 118 Insulation Scheme for Microwave Processing....................................... 126 Measuring Temperature with Thermocouple in the Cavity..................... 131 Ceramic Tape Manufacturing Flow Chart............................................... 136 Temperature as a Function of Time for Microwave Binder Bum O ut.. 137 Temperature as a Function of Time for Conventional Binder Bum Out. 137 Dielectric Properties of Ceramic Tapes at Different Temperatures........ 138 xi 7.7 Conventional Sintering Schedule...........................................................................143 7.8 Microwave Sintering Schedule...............................................................................143 7.9 Diametral Compression Test...................................................................................148 7.10 Diametral Compression Strength...........................................................................148 7.11 Micrograph of Conventional Sintered Alumina-4% Zirconia................................150 7.12 Micrograph of Microwave Sintered Alumina-4% Zirconia...................................150 7.13 Micrograph of Conventional Sintered Alumina-10% Zirconia............................. 151 7.14 Micrograph of Microwave Sintered Alumina-10% Zirconia................................151 xii LIST OF TABLES Table. Page 4.1 Resonant Frequencies for Different Cavity Structures........................................... 68 6.1 Characteristics of Non-Propagating Mode in the Waveguide............................... 107 6.2 Real Part of the Dielectric Constant of Teflon and Quartz................................ 120 7.1 Comparison of Results from Microwave and Conventional Sintering................144 7.2 Vickers Hardness of the Sintered Ceramics..........................................................147 ACKNOWLEDGEMENTS The author would like to take this opportunity to sincerely thank Dr. Vasundara V. Varadan, thesis advisor, and Dr. Vijay K. Varadan for providing the most important source of support and strength throughout the course of this work as well as during graduate study. Special thanks also go to the author’s doctoral committee of Drs. Sridhar Komameni and Deepak Ghodgaonkar for their valuable instruction and suggestions throughout this study. The author also would like to thank the colleagues in the Research Center for the Engineering of Electronic and Acoustic Materials for their help during his stay at the center. Finally, the author thanks his parents, BinQi Yu and ShuFang Chen, and his wife, Jenny, for their patience and encouragement. 1 Chapter 1 INTRODUCTION 1.1 Scope of the Thesis This thesis attempts to address three areas which need to be researched in detail before microwave processing can be widely used in the ceramic industries. They are modeling, characterization and application. In the modeling part, the equations that describe microwave processing, modeling microwave heating of ceramics and application of finite time domain method to simulate a single mode microwave cavity are given. In the characterization part, an in situ microwave heating and characterization system was established, densified as well as green ceramics are simultaneously either heated or sintered and characterized. In the application part, processing considerations for insulation design and temperature measurements with thermocouple in the microwave field are presented. Also, applications of microwaves processing to binder burn-out and sintering of AI2 Q 3/Z1O 2 arc studied. 1.2 Statement of Problem Microwave processing is a typical interdisciplinary area for researching. Knowledge of both microwave fields and components and ceramic processing techniques is necessary for success. Although both microwave technology and ceramic processing are mature fields by themselves, combining them to form a new technology is not an easy task. Today, almost all the work done in the area of microwave processing of ceramics is at a small scale at various laboratories. The experimental results have shown that microwave processing has many advantages over conventional methods. These advantages include lowered sintering temperature, reduced activation energy, accelerated diffusion rate, fine 2 microstructure, etc. The application of such a technique to a large scale production line requires more detailed research in the area of modeling, characterization and engineering design. Microwave processing of ceramics is a complicated process. It is necessary, however, to theoretically understand such a physical process to guide industrial design and application. Another reason is that experimental research work consumes time and could be very costly in practice. The interactions between microwave fields and ceramics cause an energy transformation from microwaves to heat Such a thermal energy would rapidly raise the temperature of the ceramic materials. At high temperatures, various diffusion processes will occur and ceramics are densified. For the green ceramics, which are homogeneous mixture of ceramic powders and pores, a complete description of microwave interactions lends itself to the theory of wave propagation in a random medium, a topic well understood only by the specialists. To overcome the difficulty in describing this complicated process, an effective medium approach would be the first choice to be considered. With such a choice, ceramic materials, which are to be processed in a microwave field, can be seen as a continuum medium with properties equivalent to that of the mixtures. The densification process can be treated as the interactions of microwave fields with deformable dielectrics. Coupled with appropriate thermal transfer and mass diffusion equations, a complete description of the microwave process can be arrived. For densified ceramics, a regular approach can be used for modeling which only requires consideration of dielectric property changes with temperatures. For a complicated structure such as the case involving insulation, ceramic samples and microwave susceptor, numerical methods have to be employed to obtain fields distribution and more importantly microwave energy absorbed by the ceramic sample. In developing microwave processing of ceramics, problems such as basic scientific studies on microwave-materials interactions and loss mechanisms need to be solved. There is also a critical need for a broad data base on dielectric properties of materials at high temperature over different frequencies. Recently, dielectric properties of ceramics has been studied via a so-called free space method. However, the free space 3 method suffers from long characterization times that are needed for the large planar sample to reach thermal equilibrium. Heating at high temperature for such a period of time may very well alter the microstructure of the material. More importantly, because the material being characterized was heated in the conventional furnace rather than microwaves, the microwave-material interaction mechanisms were not revealed. Contrary to the free space characterization method which requires heating of large planar specimens, the in situ microwave heating and characterization method uses microwaves to heat a thin ceramic rod and the same field is also used to detect the effect of the dielectric property change with temperature. Such a technique can be used to continuously measure dielectric permittivity as a function of temperature and offers a unique approach for understanding the interactions between microwaves and ceramics. It is applicable to both densified as well as green ceramic specimen. The use of microwave energy for processing of ceramics is still veiy much in its infancy. Preliminary investigation has shown that it is applicable to drying, slip casting, calcination, sintering, joining, plasma assisted sintering and chemical vapor deposition. Its potential is considerable since a number of very distinctive advantages have been claimed. Since ceramic processing usually requires high temperature, insulation is necessary to avoid any temperature gradients which may cause fracture of the ceramic specimen. For microwave processing, insulating materials have to be microwave transparent as well as possess good insulation properties. For the ceramic materials which have very low loss at microwave frequency, initial heating has to be provided for smooth operation. Those arrangements should not block penetration of microwaves into the ceramic sample. An important issue in microwave processing of ceramics is the proper measurement of temperature. Because of microwave interference, temperature measurements in the electromagnetic field with a thermocouple have been difficult in most cases, which prevents proper assessment of the effect of microwaves on ceramic processing. Since microwave processing is a very fast densification process, its mechanical and microstructural properties have to be evaluated so that a complete processing technique can be developed to obtain products with the desired quality. 4 1.3 Ceramic Processing Ceramics are defined as inorganic and nonmetallic materials. One important characteristic of ceramics is that it is basic to the operation of many industries. For example, refractories are basic components for the metallurgical industry. Abrasives are essential to the machine tool and automobile industry. Glass products are essential to the automobile as well as to the architectural, electronic and electrical industries. Cements are essential to the architectural and building industry. Various special electrical and magnetic ceramics are essential to the development of computers and many other electronic devices. Recently developed structured ceramics and their composites are promising materials for engine application operated at high temperature. As a matter of fact, almost every industrial production line, office and home depend on ceramic materials. A major characteristic of ceramics familiar to everyone is that they are brittle and fracture with little or no deformation. This behavior is in contrast to metals, which yield and deform. As a result, ceramics cannot be formed into shapes by the normal deformation process used for metals or plastics. There are two basic processes used in the ceramic industry for shaping ceramics. One is to use fine ceramic particles mixed with a liquid or binder or lubricant or pore spaces, a combination that has Theological properties which permit shaping. Then by heat treatment the fine particles are agglomerated into a cohesive useful product. The essentials of this procedure is first to find or prepare fine particles, shape them, and then stick them back together by heating. The second basic process is to melt the material to form a liquid and then shape it during cooling and solidification; this is most widely practiced in forming glass. Since the emphasis of this research is in the polycrystal ceramics, a little more review is given on the first method. Ceramics, depending on their chemical composition which determine their properties, are used in electronic devices, structure materials, chemical processing components, refractory structures, construction materials, and domestic products. Generally, ceramic products are obtained through raw powder material processing, forming 5 and sintering. Although recent development in raw material processing and forming has tremendously increased the engineering control of the raw material property and forming technology, the final stage of ceramic processing, i.e., sintering of ceramics, is still done using the old technique, where green ceramic products are placed in the furnace and the temperature is slowly increased to one-half to two-thirds of the melting point. Such a process is slow and often needs a long time for a complete cycle which results in waste of time and cost of energy. In processing of polycrystal ceramics, mixed-oxide industrial chemical are commonly produced by calcining a mixture of carbonates, hydroxides, sulfates, nitrates, acetates, oxalates, alkoxides, and so on. In general, the reaction produces an oxide and a volatile reaction product (e.g., C 0 2, S i0 2, H 2 0,...). During calcination process, the reaction may be controlled ( 1 ) by the reaction rate at the reaction surface, (2 ) by gas diffusion or permeation through the oxide product layer, or (3) by heat transfer. In general, the calcination reaction is heterogeneous. Hence, the reaction occurs at a sharply defined reaction interface. Drying is the removal of organic binder or liquid from a porous material by means of its transport and evaporation into a surrounding area. It is an important operation prior to firing in processing bulk raw materials. Drying cost is a significant factor in the selling price of industrial minerals. In drying ceramic ware, the initial drying rate is independent of the water content and depends solely on the temperature, humidity, and rate of movement of the air over the surface of the ware. The rate of drying is equal to the rate from a free water surface. If an enlarged cross section of the ware is observed, it appears that there is a continuous film. At a water content such that the particles just come in contact, this water film disappears from the surface and the rate of drying suddenly decreases. The lower rate of drying is due to the resistance to the flow of liquid to the surface or may be caused by vaporization in the interior and diffusion of the vapor out to the surface. Measurements of shrinkage during drying process indicate that the major part of shrinkage occurs during the constant rate period. The shrinkage is essentially completed during the constant drying period. The water films decrease in thickness until at the critical point, at which the rate of drying and also the 6 rate of shrinkage sharply change, the particles have just come in contact. This is the end of the shrinkage and the beginning of a lower rate of drying. Sintering is the term used to describe the consolidation of the product during firing where the temperature in the product exceeds one half to two thirds of the melting temperature, which is sufficient to cause significant atomic diffusion for solid state sintering or viscous flow when a liquid phase is present or produced by a chemical reaction. Consolidation implies that within the product, particles have joined together into an aggregate that has strength. Sintering is often interpreted to imply that shrinkage and densification have occurred. The driving force for sintering is the reduction in the total free energy AGT of the system AGt = AGv + AGb +AG s (1.1) where AGy, AGB and AGS represent the changes in free energy associated with the volume, boundaries, and the surface of the grains, respectively. The major driving force in conventional sintering is AGS, but the other terms may be significant in some stages for some material systems. The mechanism for transport during sintering are surface diffusion, evaporation condensation, boundary diffusion, lattice diffusion, viscous flow and plastic flow. Surface diffusion is a general transport mechanism that can produce surface smoothing, particle joining, and pore rounding, but it does not produce volume shrinkage. In materials where the vapor pressure is relatively high, sublimation and vapor transport to the surface of lower vapor pressure also produce these effects. Diffusion along the grain boundaries and diffusion through the lattice of the grains produce both neck growth and volume shrinkage. The mechanism of bulk viscous flow and plastic deformation may be effective when a wetting liquid is present and mechanical pressure is applied, respectively. Microstructure change during sintering could be classified into initial, intermediate and final stages. In the initial stages, particles become smooth, grain boundaries form and 7 neck grow and interconnected. Open pores will be rounding, active and segregated dopant start defuse, the porosity decreases about 12%. Shrinkage of open pores intersecting grain boundaries, significant mean porosity decreases and slow grain growth are the main phenomena in the intermediate stage. During final stage, closed pores containing kiln gas when density is less than 92%. Closed pores intersect grain boundaries, others shrink to a limited size or disappear. Pores larger than grains shrink relatively slowly, grains of much larger size appear rapidly. Pores within large grains shrink relatively slowly. 1.4 Microwave Processing o f Ceramics Most of the early work on using microwaves to process materials dates back to the early 1960’s and has led to some commercial applications in foundry and investment casting industry such as work done by Schoroeder and Hackett [1971], Valentine [1973, 1977], Stengel [1974]. The application of microwaves to the ceramic processing started in late 1960’s, when Tinga and Voss [1968] and Tinga [1969] published papers on theories of using microwaves to heat and sinter ceramics. Berteaud and Badot [1976] were able to use microwaves to heat refractory materials. Interests grew in the early 1980’s. Schubring [1983] successfully sintered an alumina spark-plug insulator. Johnson and Brodwin [1984] constructed a single mode cavity to sinter ceramics and subsequently used it for characterizing a densified ceramic rod. Roy et al. [1985] discovered that microwaves could be used to melt gels for ceramic powder processing. The research work on the subject of microwave processing of ceramics has been booming from the 1980s to the present. To accommodate these developments and for better communications among researchers, the Material Research Society and the American Ceramic Society organized four symposia in 1988, 1990, 1991 and 1992 which resulted in four proceedings. Those symposia dealt extensively with theoretical modeling, dielectric property characterization, equipment design and laboratory experiments on the microwave processing of ceramics and their composites in the area of calcining, drying, sintering, etc. Microwave processing has many advantages over the conventional method. Tian 8 and Johnson [1988] have demonstrated that microwave processing ceramics can produce ultra fine microstructure. The ceramic material they used is AI2 Q 3 . Jenny and Kimrey [1988] have sintered AI2 O 3 + 0.1% MgO in a vacuum using 28GHz microwave power instead of the most common 2.45GHz, the results suggested that more power can be absorbed by the ceramic compact because of the high value of the loss factor at 28GHz. They also reported that microwave heating lowered the sintering temperature and lowered the activation energy and increased the diffusion rate during microwave processing. To explain the observed phenomena, Meek et al. [1991] suggested that the electric field and power density are greatly intensified in the neck region between the grains; such a high power density could induce extreme high temperature in at neck region. This high temperature at the grain boundary could slightly melt ceramics there, which would change solid state sintering to liquid phase sintering; therefore, sintering rate is enhanced. Further work by Katz et al. [1991] indicated that relaxation type loss mechanism may be operable in the microwave frequencies during microwave heating of ceramics. That loss mechanism increases the correlation factor for diffusion. Later, Booske et al. [1992] proposed an ionic motion model to account for non-thermal effects during microwave heating of crystalline solid. Theoretical prediction of the microwave absorption during microwave sintering was also done by Varadan et al. [1988] by using rigorous multiple scattering theory. Similar advantages can be obtained for other ceramics with microwave processing. Krage [1981] sintered fenite with microwaves and found that the properties obtained from the microwave energy are comparable to the conventional. Desgardin et al. [1986] did microwave sintering of BaTiOs based ceramics. Aliouat et al. [1990] have used microwaves to sinter spinel-type oxides such as LiFe3 Og. By comparing mechanical properties of between microwave and conventionally processed alumina, Patterson et al. [1991] found that microwave processed alumina has a higher toughness and lower hardness. Noncrystalline T i0 2 was also sintered with microwaves by Eastman et al. [1991]. Mcmahon et al. [1990] tried a variety of ceramic materials with the microwave sintering technique. The microwave sintering technique was also used to process ceramic composites 9 such as sintering of partially stabilized zirconia by Wilson and Kunz [1988], 50% dense alumina compact plus 10% vol. SiC whisker by Meek et al. [1987c], A12 0^ plus 10% vol. SiC whisker in the 2.45 GHz cavity by Katz et al. [1988a] and titanium diboride by Katz et al. [1989] and Holcombe and Dykes [1991], zirconia-toughned alumina composite by Kimrey et al. [1990] and Patil et al. [1991], Yttria-2wt.% Zirconia by Holcombe et al. [1988]. Macdowell [1984] used microwaves to heat glass-ceramic composite and found that sodium nepheline was useful in converting microwave energy to heat. By using a gas pressurized cavity at about IMPa, Tian and Johnson [1988] sintered A l2O3-30% wt. TiC composite and obtained 95% density. Those composites are generally difficult to densify by conventional sintering methods. By applying microwave energy to internally combustible material, Ahmad et al. [1991] fabricated A12 Oj /TiC composites from Compacts of T i0 2 + Al + C. Microwave energy used for preparing fine ceramic powder has been demonstrated by the work of Komameni et al. [1988] and Kladnig and Horn [1990]. Singh et al. [1991] used microwave plasma to synthesize several non-oxide ceramic powders. Johnson [1991] has demonstrated that microwave generated plasma can be used to sinter ceramics. He also gave some considerations on the possible mechanisms of enhanced diffusion. Microwave heating is also applicable to melting and making fine ceramic powders from ceramic gel as shown by Roy et al. [1985], Komameni and Roy [1986] and Surapanani et al. [1991]. Sintering of non-oxide ceramic materials with microwave energy was studied by Katz et al. [1988b] on Br2 C, Tiegs et al. [1991,1991], Ferber. et al. [1991] and Kiggans et al. [1991] on SiN, Kumar et al. [1991] on SiC. Holcombe and Dykes [1990,1991] have developed a casket scheme for ultra high temperature microwave sintering of general non oxides. Effect of particle size on microwave processing of alumina was studied by Arindam et al. [1990]; it was found that microwave heating culminates in accelerated densification 10 with a better uniformity and homogeneity of microstructural vis-a-vis conventional fast firing, Ahmad et al. [1988] used microwaves to calcine, sinter and anneal Y I^C ujG y.* super conducting pellets. The microwave processed pellets have more refined microstructure, low porosity, improved oxygen content and higher super conducting transaction temperature over conventional processed samples. Aliouat et al. [1990] and Hyoun et al. [1991] also used microwave energy sintered Y I ^ C ^ C ^ . Microwave processing is also applicable to glass processing. Hassler and Johansen [1988] have used microwaves to heat fused quartz in the optical fibers fabrication process. Kao and Mackenzie [1991] used magnetite to act as a microwave susceptor during microwave sintering of soda-lime glass, the finished product has a function of microwave shielding and absorbing effects. Pope [1991] constructed a microwave furnace to sinter sol-gel derived silica glass. Walkiewicz [1988] used microwaves to selectively heat minerals. Hamlyn and Bowden [1992] have applied microwaves to process earthware ceramics. Wright et al. [1989] used microwaves to process ilmenite and titania-doped hematite. In Standish and Womer [1990] study, microwave energy was also used in the reduction of metal oxides with carbon. In fact, microwave heating technique is applicable to all phases of ceramic processing, i.e., drying, calcining, binder burnout and sintering, as demonstrated by Harrison et al. [1988]. Similar work was also done by Selmi et al. [1992] on Barium Strontium Titanate. In Selmi’s work, it was observed that microwave calcination enhanced the solid state reaction and achieved required calcination much faster than the conventional method because of the high microwave power absorption ability of the carbonate. Application of microwaves to ceramic processing in a large scale was studied by Katz and Blake [1991]. A variety of material systems have been investigated, together with a number of different approaches. The microwave applicator used during microwave processing could be both multimode such as work done by Harrison et al. [1988], Ahmad et al. [1988], Katz et al. [1988a], Krage [1981] and single mode such as work done by Tian and Johnson. [1988]. Patil et al. [1991] used circular waveguide cavity to sintering alumina rod, the resonant mode in the circular cavity can be changed to accommodate samples of the different shape. To process ceramics composites, Tian et al. [1988] used a closed cavity with gas pressure to aid microwave sintering. Jow et al.[1987], Kimrey and Janney [1988] have discussed the design principle for high power microwave cavities. One parameter which is important in designing and applying microwave energy to ceramic processing is the dielectric property of ceramic materials. Early work was done by \b n Hippie [1954] and his co-workers at the MIT Laboratory for insulation research. Subsequent work was continued at the same institution by Westphal [1975,1977,1980]. Their monumental work has established the database for microwave dielectric properties. To satisfy recent interests in the area of microwave processing of ceramics, dielectric properties of the materials at broader band of frequencies and higher temperatures have to be obtained to provide bases for designing and applying microwave heating for industrial applications. Recently, Fuller et al. [1984], Ho [1988] and Varadan et al. [1991] have established methods to characterize ceramic dielectric properties at high temperature using either free space or cavity method. While most of the research work in microwave processing of ceramics is done in experimental form, only few of them are on the modeling of this process. It is necessary, however, to theoretically understand such a physical process to guide industry design and practical application. Another reason is that experimental research work consumes time and could be very costly in practice. A relatively new approach which is being taken by some researchers is the construction of models which attempts to address the use of microwaves in the processing of materials from the theoretical point of view. With regard to microwave processing of ceramics, these models currently fall into two main approaches; an atomistic approach where the interactions between microwaves and materials are considered in terms of the effect on parameters such as atomic diffusion as done by Kenkre et al. [1990] and Kenkre [1991], Bykov et al. [1991], Meek [1987a], Meek et al. [1988,1991], Katz et al. [1991] and Gupta and Evens [1991] and a ‘micro’ approach where the material is 12 considered a homogeneous continuum and power deposited is set against heat losses as done by Iskander[1990], Smyth [1990], Iskander et al. [1991], Chatterjee and Misra [1991], Eugene and Snider [1991], Kriegsmann [1991], Ultimately, these two approaches will be combined. The interactions between microwaves and materials are strongly materials-property dependent and most of the ceramics are low loss materials at room temperature. Three approaches have involved the use of coupling aids, microwave susceptor and high microwave frequencies. The concept of trying to rind a second phase which will aid coupling of the green body with the incident microwaves appears to be fundamentally sound. The additives must aid coupling and the densirication process but not be detrimental to the properties of the sintered body, for example, by leaving residual glassy grain boundary phase. Roy et al. [1985], Meek et al. [ 1987b], Komameni and Roy [1986] have suggested and used some additives for microwave processing. The use of a microwave susceptor to reduce the severity of the inverse temperature gradient with microwave heating alone attract more and more interests in the designing of microwave processing. Research work by Humphrey [1980], Krage [1981], Wilson and Kunz [1988], Harrison et al. [1988] and Janney et al. [1992] have shown microwave susceptor provides a degree of conventional radiant heating so that a uniform heating can be achieved and fast heating rate can be realized. An inherent disadvantage is that microwave susceptor reduces the efficiency of the process by absorbing some of the incident energy. To obtain consistent microwave sintering results, Aliouat et al. [1990] have published results on using a control algorithm for microwave sintering in resonant system. The use of high frequency microwaves ( typically in the 20-40GHz range) has resulted in the ability to dissipate high level power within ceramic materials. Results presented by Jenny and Kimrey [1988] showed that the material has a higher microwave absorbing ability for it has a high loss tangent and more uniform heating rate due to shorter wavelength. However, the depth of penetration is reduced at high frequencies. This could lead to reduced uniformity of heating at high temperature and for large sample. Ultimately, the deciding factor for microwave processing will depend upon 13 economics. Studies have been performed in this area by Jolly [1972], Das and Curlee [1987], Schmidt [1986], Patterson [1975], Sanio and Schmidt [1988] and Some conflicting results have been obtained. One of the key reasons for the uncertainty is that a number of initial assumptions have to be made at present because of absence of suitable data. In Das’s study, it is concluded that microwave sintering of ceramics will not conserve energy, when the conversion of fuels to electricity and the conversion of electricity to microwave energy are considered. According Paterson’s[1975] study, microwave sintering results in an energy saving of as much as 90% over conventional electric furnace techniques. 1.5 Thesis Organization This thesis has eight chapters. In the first chapter, the scope of the thesis and the problems addressed in this thesis are mentioned. The historical backgrounds regarding conventional and microwave processing of ceramics are also reviewed. Chapter 2 presents the equations describing microwave processing of ceramics. Ceramics undergoing microwave processing are treated as deformable dielectrics. Heat transfer and mass diffusion equations are also included in the description. To understand microwave processing of ceramics, chapter 3 is devoted to simulate microwave irradiation of a ceramic slab. To account for the temperature dependence of the dielectric properties of the material, the impedance method is used to determine the electromagnetic energy absorbed by the ceramics; direct time integration is used to treat the nonlinearity of the problem. In chapter 4, the method of finite difference time domain method is used to simulate the structure of a single mode cavity. In the simulation, a complicated structure of insulation material, microwave susceptor and ceramic sample are considered. Chapter 5 details the high power microwave heating system which is used for characterization as well as processing of ceramics. The system consists of a magnetron, 14 circulator, directional coupler, impedance analyzer, 4-stub tuner, iris, variable short and microwave resonator cavity. Chapter 6 describes the method of in situ microwave heating and characterization of ceramics. The ceramic rod in the microwave resonant cavity is represented by an equivalent T-circuit. The variation of the dielectric properties of the ceramic rod is sensed by measuring the reflection coefficient through a reflectometer modified from an impedance analyzer. Both densified and green ceramic rods are used for characterization. Chapter 7 shows the application of microwaves to ceramic processing. Qualitative analysis of insulation scheme, temperature measurement with pyrometer and thermocouple are given. The advantages of the microwave processing method over the conventional one are discussed. In this chapter, microwave-assisted binder burn-out, microwave sintering of zirconia toughened alumina and resulted structural and microstructural properties are presented. Chapter 8 gives the conclusion and the work that needs to be done in the future. 15 Chapter 2 BASIC EQUATIONS TO DESCRIBE MICROWAVE HEATING OF CERAMICS 2.1 Introduction As stated in the previous section, sintering is a complicated process where green ceramics densify through various diffusion paths at high temperature. Micro or nano scale description of the sintering is still an active research subject in the material research community. For the problem to be solved here where the knowledge of temperature and microwave power absorption are important, the densification process can be described as the interactions between electromagnetic fields and deformable dielectric body. In this section, the basic equations which could properly describe microwave processing of ceramics are derived. By including mass diffusion equation into the consideration, a complete description of microwave processing of ceramics can be done. 2.2 Equations for Microwave Material Interaction y 2.2.1 M axwell’s Equations It is known that ceramics are dielectric materials independent of whether are in the green or densified form. The equations describing interactions between microwaves with ceramics are commonly known as Maxwell’s equations. It was the genius of J.C.Maxwell in 1865, who corrected the inconsistencies of Ampere’s Law and thus enabled to study the propagation of electromagnetic waves. Although there is no difference in using integral or differential form of Maxwell’s equations in general, the integral form is preferred here to account for the densification process of the green ceramics during microwave processing of ceramics. The integral form of Maxwell’s equations are q dV ( Coulomb's Law) h - l dS = 0 (Absence of Free Magnetic Poles) 1 B j E dl = - | H dl = j J (2.1) B dS ( Faraday's Law) dS + D dS ( Ampere's Law) (2.2) (2.3) (2.4) Using the international system of units, E and H are electric and magnetic field intensity and have the units of Volt/meter (V/m) and Ampere/meter (A/m). D and B are electric field flux and magnetic field flux and have the units of Coulomb/met2 (C/m2) and Weber/meter2 (Wb/m2). J and p are electric current density and electric charge density. The units for J and p are Ampere/meter3 (A/m3) and Coulombs/meter3 (C/m3), respectively. Considering that the densification process occurs during microwave sintering of ceramics, the derivative with respect to the time in the above listed equations should be taken as a material derivative. Therefore, the differential form of Maxwell’s equations to describe the field variation during microwave processing of ceramics is as follows 17 V x H =J + ^ at + (V-D)v (2 .8) with local velocity v coupled in the Maxwell’s equation, it is necessary to consider the thermal diffusion and mass diffusion equations in order to solve this problem. In practice, an iterative method has to be used to solve those complicated and coupled equations. At each iterative step, it is assumed that microwave would not see the movement of the body but would see the different boundary with different electromagnetic properties in different iterative steps. In so doing, the regular differential form of Maxwell’s equations are recovered. For a complete description of the behavior of the medium under influence of the fields, the constitutive relations which were obtained from experiments need to be used D = £r£oE B = p rp 0 H (2.9) .1 = a E where £q and |i 0 are the permittivity and permeability of free space. They are 109/36ti (F/m) and 4rcxl0 ' 7 (H/m), respectively. ^ and |ir are the relative dielectric constant and relative permeability, a is the conductivity of the material and has a unit of mho/meter (Cl /m). After some mathematical manipulations, the vector Helmholtz equations used to describe electromagnetic wave propagation in the medium can be obtained as follows V2 E + k 2 E = — - iw J (2.10) V2 H + k2 H = - Vx J (2.11) £ For the case considered here, there would be no source charge or the electric current in the region of considerations. 18 2.2.2 B oundary C ondition At the surface which separates two regions with different physical properties, the electric and magnetic fields will satisfy the boundary conditions which can be derived from Maxwell’s equations. (1) At the boundary where both 1 and 2 are dielectric media, the boundary conditions are n * ( D i -D 2 ) = qs n ■( B i -B 2 ) = 0 (2 . 12) n x ( Ei -E 2 ) = 0 nx(Hi-H2 )=Js where qs and J s are the surface charge and surface current (2) At the boundary where 1 is the perfect conductor and 2 is the dielectric medium, the boundary conditions are nD 2 =qs n • ( B i -B 2 ) = 0 (2.13) n x ( E i -E2 ) = 0 n X H2 = J s In using microwaves to process ceramics, the amount of microwave energy absorbed by the materials is important to know. The time instantaneous Poynting vector is the flux of the electromagnetic power through the medium, which is defined as P = E x H* (2.14) The time average Poynting vector is thus found to be p(t)d(cot) the net time average power flux enter a close surface S is then (2.15) By using Maxwell’s equations | (E x H*) • dS = j ( o j [ nolir H H * . e 0 e' E • E*]dV (2.17) + | |e oeoe' e EE E ¥* dV + | | J • E dV In the above equation, the imaginary part is the energy stored in the electric and magnetic field, while the real part is the energy which will be transformed to the heat. So the power absorbed by the dielectric medium per unit volume is Pabsorb = ^ ( Coe" (o E- E* + J • E ) (2.18) = i-EoCefftl) ( E E * ) 2.3 Theory of Electrom agnetic Field Interactions w ith D eform able Dielectrics To model microwave processing of ceramics, theory of electromagnetic fields interacting with a deformable dielectric need to be reviewed for better understanding. In this section, a basic theory of electromagnetic fields interacting with deformable dielectrics especially with ceramics is presented. According to the response to the electromagnetic field, ceramics can be classified as ferroelectric, diamagnetic, ferromagnetic and ferromagnetic materials, etc. In modeling microwave processing of ceramics, all the interacting mechanisms of microwave with 20 ceramics have to be studied. A complete description of those mechanisms is obviously beyond the scope of this thesis. For simplicity, the structure of material which will interact with electromagnetic fields can be envisioned as a body containing the following classes of charges and current distributions. 1. A free charge q per unit volume and a free current J, which consists of either charges and or currents from an exterior source or conduction current and space charges which belong to the material. 2. Bound point charges 8 qa located at xa and moving with velocities v a . The bound charges are distributed in such a way that over any finite material volume the average charge vanishes: (2.19) a 3. Microscopic current loops 8 jp, which cause the magnetization of the body. Over the closed Loops lp it follows that (2 . 20) when the material with above mentioned electric structure is exposed to an electric field E and a magnetic field B, it will experience a force f(em), a torque l(em), and a rate of energy change w(em) per unit volume given by f(em)dV = ( qE + JxB)dV + £ {8qaE(x«) + 6qavax B(xa)} a (2 .21) l(em)dV =[qx xE + xx ( Jx B )] dV + 2 a (&qaxx E(xa) + 5qaXaX[vaxB(xa)]| 21 +2 I Xp X [8 jpX B( xp) ]dl (2.22) ph W(em)dv = J-Edv + X 8 qa v„-E( xa) + X I % E ( x p ) d l P h (2.23) The summations inthe abovethree equations are over the bound charges and microscopic current loops that arecontained in the material volume dV. If x is used to indicate the position of the centroid of dV and and 1;^ are defined such that Xoc= x + S<a) (2.24) xp= x + S(p) (2.25) then by employing a formal power series expansion for the values of the fields at x E(*«).E(*)+E,i(x)4) (2-26) B(xp) = B(x) +Bfi(x) Sip, (2.27) and by defining the polarization P and the magnetization U as Pdv = X 8 qa^(a) (2.28) a Udv = £ ( 8 j i) X ^ )dl P (2.29) the force f(em), the moment l(em), and the rate of energy production w(em) as given by eq. 2.21,2.22 and 2.23,can be written in the first approximation (by neglecting second-order terms in £(a), etc.) as f<em) i = qE i + e ijkJjBk + E i,kPk + ®ijk (Pj + PjVl.l) Bk + ejjkVjBkiiPi + UkBki l(em) = XX f(em) + P x £ + U XB (2.31) 22 W(em) = V- f(em) + p £ • fr - U B + J E (2.32) C = E + vxB (2.33) where C, is defined as ■fr is the polarization density per unit mass pfr = P (2.34) and the dot above a quantity indicates material differentiation f ■ $ > *Pj-v (2.35) in the derivation of above equation, the following relation has been employed : ( P + P d iv v ) = X ^(a)5q« (2.36) a which follows from the fact that the summation is over all particles. If the above forms for f(em), l(em) , and w(em) are accepted, the balance laws of a continuum interacting with electromagnetic fields can be obtained as demonstrated in the following sections. In the case where free space charges, microscopic current is neglected and the dielectric is not deformable, the work done by the electromagnetic field on the dielectric is the same as obtained before which is the last term of eq. 2.32. 2.4 Thermal Diffusion Equation To predict the behavior of ceramics during microwave processing, the temperature distribution has to be known. Basic equations such as mass, momentum, energy conservation and related boundary conditions are found to be useful. Combined with the Maxwell’s equations as well as the mass diffusion equation, the microwave sintering process can be completely described. The goal of this section is to use a general set of basic principles and establish 23 thermal diffusion equations to model thermal behavior of ceramics in the microwave fields. 2.4.1 Mass Conservation Equation A green ceramic sample, which is formed by a static press, is a homogeneous mixture of ceramic particles and pores. If the pores are considered massless, the total mass of the ceramic green sample and the densified product are the same, then the mass conservation equation of the classical mechanics can be invoked. Consider an arbitrary volume V fixed in space, bounded by surface S, if continuous medium of density p(t,x,y,z) fills the volume at time t, the total mass in V is (2.37) In the sintering process, the density depends on the position and time. The rate of mass increase in the volume is (2.38) If the mass of air pores and the evaporation of the ceramics at high temperature are neglected, the total mass will not be created or destroyed inside volume V, this must also be equal to the rate of inflow of mass through the surface. Since the integral vanishes for arbitrary choice of the volume V, it follows that the integrand must vanish at each point of a region in which no mass is created or destroyed. The resulting equation, a consequence of the conservation of mass, is known as the continuity equation. ^ +V(pv)=0 (2.39) 24 2.4.2 Momentum Conservation Equation The momentum conservation equations are actually the realization of Newton’s second law of motion. For a differential control volume in the green ceramic body, this states that the sum of all forces acting on the control volume must equal to the net rate at which momentum leaves the control volume. At an instant of time t, the linear momentum of all particle contained in a domain V is 111 = jv P V' dV (2.40) if the body is subjected to surface tractions T;, surface pressure p and body force per unit volume fj, the resultant force is F i= ((T i-p )d V + ' ( fi + f(em)i)dv I (2.41) £ n i = Fi (2.42) Newton's law states that Hence, according to eq 2.40 and 2.41, it is found that pui dV = | cn- P) dS + I ( f(em) i +fi) dV (2.43) where p is the mass density, pv; the total momentum density, T; the total stress vector acting on the body, fj the applied body force density per unit mass due to exterior sources ( assumed independent of the electromagnetic fields). According to Cauchy's formula, the surface traction may be expressed in terms of the stress field <rij, so that T^OjjVj, where Vj is the unit vector along the outer normal to the boundary surface S of the domain V. a,j is the stress field. On substituting Gj jVj for T, and 25 transforming the surface integral into a volume integral by gauss’ theorem, eq. 2.43 becomes I. T r ^ (pv‘vJ)]dv = / J § L- £ +f<“ H dv (2 -44> since above equation must hold for an arbitrary domain V, the integral on the two side must be equal. Thus dpvj d i r 4| : ,pViVj)=§ i - | i +f''”>i+f (2.45) The left-hand side of eq. 2.45 is equal to T?l 3vj dvj +P at +VjaXj (2.46) The quantity in the first parentheses vanishes according to the equation of continuity, while the second is the acceleration dv/dt. Hence the celebrated Eulerian equation of motion of a continuum is obtained. P d t ” dxj 3xj (em)l + t (2.47) 2.4.3 Moment of Momentum Newton’s law also states that the rate of change of moment of momentum is equal to the total applied torque about the origin. The moment of momentum of a body occupying region V of space with boundary S at an instant of time is Hi = j ( a + eijkXjpDk) dV (2.48) where Qj is the spin density. If the body is subjected to a surface traction Tj, surface pressure p, body force per unit volume fj surface torque A,; and applied body couple lj of non-electromagnetic origin, the resultant moment about the origin is 26 = j^ (®ijkxj[fk (®ijkxj[fk + ^(em) id — f(em) id l(em),i + + ^| ) ddV V + ejjk[XjTk e]jk - XjVfc ] + A.[ dS (2.49) Euler’s law states that, for any region V, ^ « i = Li (2.50) so the integral form of Euler’s law is & j ( oi + eijkxjpuk) dV = J ^ (2.51) (®ijkxj[fk "**■ + l(em)i ++ lil i ) dV + I 6^ijklxj jjk[XjTk - XjVk ] + A<i dS ■f(em) k] + dV + For the problem considered here, it is assumed that there is no spin density, the surface torque and the applied body torque are identically zero: Ci = 0 M O h=0 (2.52) Introducing Cauchy's formula into the the first term at the right side of equation 2.51 and transforming the result into a volume integral by Gauss’s theorem. It can be shown that ^ I eijk xj pvk dV = (2.53) 'v { eijk (xj^lk),l" ®ijk(xjp),k + eijk xj ( f(em) k +fk) + l(em)i }dV I Evaluating the material derivative and using eq. 2.53, following equation can be found eijk xj|(p v k ) + ^ - ( e ijk xj pvkvi) (2.54) 1 = eijk (xjCTlk),l“ eijk(xjp),k +eijk xj ( f(em) k +fk) + l(em)i The second term in the above equation can be written as eijk pVjVk +eijk xj pvkvi) = 0 + eijk xj pvkvi) (2.55) 27 hence, eq. 2.56 becomes The sum in the square bracket vanishes by the equation of motion hence, eq. 2.56 reduces to (2.57) The effect of the electromagnetic fields causes the asymmetry of the stress tensor. 2.4.4 Energy Conservation Equation The law of conservation of energy is the first law of thermodynamics. Its expression for a continuum can be derived as soon as all forms of energy and work are listed. There are three forms of energy: the kinetic energy K, the internal energy E and the gravitational energy G. The kinetic energy contained in a regular domain V at time t is (2.58) The internal energy is written in the form (2.59) The gravitational energy depends on the distribution of mass and may be written as (2.60) Where E is the internal energy per unit mass. The first law of the thermodynamics states that the energy of a system can be changed by absorption of heat Q and by work done on the system. Expressing this in term of rates, it is found that 28 ^ (K + E + G) = 0 + W (2.61) The heat input into the body must be imparted through the boundar. Let dS be a surface element in the body with unit outer normal Vj. The heat transferred to the body is assumed to be representable as hjVjdS. If the medium is moving, it is assumed that the surface element dS be composed of the same particles. The rate of heat input is, therefore (2.62) != - | hiV*dS The work done on the dielectrics by the body force per unit volume fj and f(em)i in V, surface traction T and surface pressure p on S and the energy inputted by the electromagnetic field w(em) and exterior source q can be expressed as W = | { ( fi +f(em)i ) Vj + W(em) + q }dV + J ( T; - pV; ) Vi dS (2.63) Using eq. 2.61, it is obtained that A dt ^pVjVi + p£ ] dV N (2.64) =Jf(fjVi + f(em) jvj + w(em) + ( OijvOj - f~ +<i)dV Using the formula to compute the material derivative, it is easy to obtain the following result after some calculations 2 dt 2 dt + ^rP 2 pd3l + K dt v+ dt dt +^ a ~ + p £ div v + dt ^2 5 5 ) +d)p div v K = fiVi + f(em) iVi + W(em) + ( OijvOj - +q The above equation can be simplified greatly if the equations of continuity and motion are used. Here, f ; is the total body force per unit mass. The difference between fj and Fj is the gravitational force, by definition 29 (2.66) since dt at (2.67) ldxi and for a gravitational field that is independent of time, where (2.68) at Combining eq. 2.39 and eq. 2.47, eq. 2.65 becomes + p t* = FiVi+f(™>i V i + w < e m ) + ( ^ ' ^ ? +<i (2.69) But Ipvi^XL = lp S lv i T 1 dt ¥ dt (2.70) and if heat transfer obeys Fourier’s law, . . aT hi=-k^ r (2.71) where k is the thermal conductivity and T is the absolute temperature. Let E = cT (2.72) where c is the specific heat, then eq. 2.69 becomes p ( l F +Vi div (cT ■* =^ (k^ ) + FiVi + f(em) iVi + W(em) + (aijV i)|j ” I k f * + ** (2.73) 30 2.4.5 B oundary Condition The heat conduction equation is the second-order in space and requires known conditions on either T or its normal derivative at every boundary point. At the boundary, heat will transfer between ceramics body and environment due to temperature difference. According to the physical mechanisms that underlie the heat transfer modes, different boundary conditions have to be constructed to account for the heat transfer at the boundary. 1. Known temperature: In the conventional heating of ceramics, the furnace is set to a certain temperature. The temperature at the boundary is therefore a constant. Hence (2.74) Tb = T0 2. Insulated Walls: If ceramics are completely insulated by the material which is microwave transparent, the heat transfer at the boundary in the normal direction is zero. Therefore (2.75) 3. Conduction: Conduction may be viewed as the transfer of energy from the more energetic particles to the less energetic ones of a substance due to interactions between particles. If the heat flux at the boundary is known, the heat transferred is ■k — lb • n = q0 (2.76) dn 4. Convection: Convection is defined as the conveying of heat through a liquid or gas by motion of its parts. The convection heat transfer is comprised of two mechanisms. In addition to energy transfer due to random molecular motion, there is also energy being transferred by bulk, or macroscopic motion of the fluid. This fluid motion is associated with the fact that, at any instant, large number of molecules are moving collectively or as aggregates. Such a motion, in the presence of the temperature gradient, will give rise to heat transfer. Because the molecules in the aggregate retain their random motion, the total heat transfer is then due to a superposition of energy transport by the random motion of the molecules and by the bulk motion of the fluid. Regardless of the particular nature of the convection heat transfer mode, the appropriate rate is of the form 9T - k — lb - n = h«, ( Tb - T „ ) dn (2.77) where, the convective heat flux is proportional to the difference between the surface and fluid temperature, Tb and T, respectively 5.Radiation Thermal radiation is the energy emitted by matter that is at a finite temperature. Radiation is an electromagnetic phenomenon and which travels easily through a vacuum at the speed of light. The energy of the radiation field is transported by electromagnetic waves. While the transfer of energy by conduction or convection requires the presence of a material medium, radiation does not. According to the Stefan-Boltzmann law, the maximum flux at which radiation may be emitted from a surface is given by dT - k — lb n = o R Ti> dn (2.78) where Tb is the absolute temperature of the surface and o R is the Stefan-Boltzman constant. Such a surface is called an ideal radiator or blackbody. The heat flux emitted by a real surface is less than that of ideal radiator and is given by dT - k — lb n= o r F r T{ dn (2.79) where FR is a radiative property of the surface and is called emissivity. This property indicates how efficient the surface emits compared with an ideal radiator. In a lot of situations as well as in the microwave heating of ceramics, the ceramic body which has a small surface is completely surrounded by a much larger surface. The net rate of radiation heat exchange between the surface and its surrounding can be expressed as 32 Mewing all the boundary conditions, it is obvious that convection and radiation heat transfer are the two main heat loss mechanisms in the microwave processing of ceramics. Since the temperature for processing of ceramics is often very high, radiation radiation heat loss becomes a dominant heat transfer mechanism because of the fourth-power relationship. Since there is almost no fluid flow inside microwave furnace or single mode cavity, radiation is still a significant heat transfer mechanism even it is close to room temperature. For the reason mentioned above, the radiation boundary condition is applied in modeling microwave processing of ceramics. 2.5 Mass Diffusion Equation In microwave sintering of ceramics, diffusion will occur in the homogeneous mixture of ceramic powders and pores. The pores will be diffused out during solid state reaction. The diffusivity of the mixture depends on the temperature, pressure, external electromagnetic field and material to be sintered. Such a phenomenon can be veiy well described by the mass diffusion equation. The mass diffusion equation is analogous to the heat equation. Consider a homogeneous medium that is a binary mixture of ceramics powder and pores and stationary. That is, the mass average velocity is everywhere zero and mass occurs only by diffusion. The resulting equation can be solved for the species concentration. Applying this equation to microwave sintering process, the entire densification process can be described. Allowing for concentration gradients in each of the x,y and z coordinate directions, it is wise to define a differential control volume dxdydz, as shown in figure 2 . 1 , within the medium and consider the process that influences the distribution of ceramics. With concentration gradients, diffusion must result in the transport of ceramics through the control surfaces. The conservation equation can be written as Mc,in +>*C,g-Mc.out = d! £1='toc,st (2.80) According to the Fick’s law of diffusion, the species transport rates at opposite 33 n" A,Z+dz A,y+dy _lt ,, x+ dx A, x II n A, z Figure 2.1 Differential Control Volume, dxdydz, for Species Diffusion Analysis 34 surface is related by d nr Ydydz nC,x+dxdydz=nC,xdydz +-1 — T dx (2.8X.a) d nr vdxdz nC y+dydxdz= nC)ydxdz +-±— ^ ------ - (2 .8 1 . b) n ir d n r .dxdy nc,z+dzdxdy=nc,zdxdy +- L—r — <2-8 1-c) dz In addition, there may be volumetric chemical reactions occurring through the medium. The rate at which the ceramic is generated within the control volume due to such reactions may be expressed as Mc=ricdxdydz (2.82) Where M "c is the rate of increase of the mass of ceramics per unit volume of the mixture. Finally, these processes may change the mass of ceramics stored within the control volume, and the rate of change is Me st = —^-^dxdydz dt (2.83) The net mass that flows in must equal to the rate of storage of ceramics plus their generation within the volume. Using eq. 2.80, it can be seen that . ! * £ . ^ a + lic = i?PC dx dy dz dt (2.84) using the definition nC = PC V c The conservation equation becomes (2.85) 35 dt (2.86) + “ (PCUc) +~{pCvc ) + ~(pCW c) = nc dx dy dz The species velocity can be expressed by the bulk velocity, so that L2 - .2 - .2 dy2 dz2 , 8 p c +i P c +l £ c ' dt dx dz dy dx2 (2.87) For a stationary medium, the mass average velocity V is zero. Hence the final mass diffusion equation is as follows when mass generation is neglected dpC = Dcp d2pC , d2pc | ^ P C dx2 dt dy2 ' (2.88) dz2 where D™, up is the diffusion coefficient. Boundary Condition 1. Known concentration condition 2 Me, st = -^ d x d y d z dt (2.89) -C D cp^ L |x=o = Jc s dx (2.90) . Constant Species Flux when Jc,s= 0 , this condition is called impermeable surface condition. 3. Initial Condition pc( 0 ,x,y,z)=po (2.91) 2.6 Conclusion and Discussion In this chapter, a set of equations which could be used to completely describe microwave processing of ceramics are derived. They include Maxwell’s equations, thermal and mass diffusion equations. It is believed that such a description is complete. However, it is difficult to solve them without making many assumptions, for there are many parameters that need to be experimentally obtained. Theses parameters are diffusion coefficient and elastic coefficients. Equations 2.5, 2.6, 2.7, 2.8, 2.73 and 2.88 are the core of the descriptions. If the diffusion coefficient and elastic constants which relate the deformation and stress are known, those equations can be solved with the finite element method as shown by Lewis and Schrefler [1987]. For a basic understanding of the microwave processing of ceramics, a simplified problem is considered in the next section. There, a ceramic slab radiated by a TE plane wave is modeled to simulate microwave heating of ceramics. 37 C hapter 3 M ODELING MICROWAVE PROCESSING O F CERAM ICS 3.1 Introduction The use of microwave energy is a new and exciting approach in ceramic processing. Since microwave heating is a volumetric process, it could provide uniform heating so that the temperature gradient which is observed in conventional rapid heating method can be avoided. Rapid and uniform heating are important in sintering of ceramics. On the contrary, non-uniform heating is often observed during this thesis research with microwave sintering. Therefore, it is of practical interest to simulate the phenomenon of microwave heating for better control and more efficient use. In spite of the significance of the problem, there is no comprehensive analysis available which would describe the behavior of ceramic materials exposed to electromagnetic radiation. Research by Iskander [1988] and Watters et al. [1988] has revealed some of the mechanisms of microwave heating of ceramics. However, the simulation of microwave heating of ceramics with a temperature dependent dielectric property is still lacking. In this chapter, a method of simulating microwave heating of ceramics with temperature dependent dielectric properties is developed. The impedance method is used to find the microwave energy absorbed by ceramics. A non-linear finite element method is developed to determine the dynamic temperature profile in the ceramics during microwave heating. Using this method, the thermal runaway phenomenon in microwave heating of ceramics is successfully simulated. With detailed analysis of the microwave energy absorption pattern in the ceramics, the effects of dielectric properties on microwave energy absorption by ceramics are discussed. The causes of non-uniform heating by using microwave energy alone are also investigated. In doing so, a better understanding of microwave heating of ceramics is realized. 38 3.2 Theory In order to simulate microwave heating of ceramics, it is necessary to find the electric and magnetic fields strength inside ceramics and absorbed microwave energy. Electric and magnetic Helds are linked by Maxwell's equations, a group of linear differential equations. Assuming an e 'i£0t harmonic time dependence, Maxwell's equations can be expressed as follows, V (eoerE) = pe (3.1) (3.2) V x E = j n o n rH (3.3) V xH = c E - j t o e e rE (3.4) where e0 and |X0 are the permittivity and permeability in the vacuum, e,. and (Xj. are the relative permittivity and permeability of the material, o is the conductivity of the material. E and H are the electric and magnetic field strength, respectively. The propagation of the energy in the electromagnetic fields can be deduced from this equation system and leads to Poynting's theorem (3.5) which states that the mean energy, P, flowing into a surface, S, depends on the amplitude, distribution and prevailing phase of the electric and magnetic field. By using Gauss' law, equation 3.5 can be converted into the volume integral which can then be resolved into three single integrals 39 P = jo) I (ioP^H • H*)dv - jcoj eoe^E’E*)dv (3.6) + col £()£r(E 'E*)dv V The first two integrals take account of the magnetic and electric fields respectively while the third represents the energy dissipation in the dielectric in a general form. Therefore, the energy that is converted into heat by the alternating field is (3.7) It increases with the frequency, the square of the electric field strength and the imaginary part of the dielectric constant. Once the profile of e/' as a function of temperature and the electric field strength in the homogeneous body are known, it may be possible to study the thermal runaway conditions through the source-incorporated heat-diffusion equation. The diffusion of thermal energy in a homogeneous bounded volume V is determined by the partial differential equation pCp Kh V2T = P (3.8) where p, Cp and Kh are the mass density, the specific heat, and the thermal conductivity of the material, respectively. P is the microwave energy density absorbed by the material. At the boundary of the volume V, the boundary condition K h n VT = h(T-T0) + F ^ T 4 - Tq) (3.9) must be satisfied. In which h is the heat convection coefficient, Fr and or are the emissivity of the material and Stefan-Boltzmann constant. T 0 is the ambient temperature. The initial condition is T(r,0) = Ti This heat diffusion equation is analogous to the forced Fisher equation (3.10) 40 Tt = Txx + G(T) (3.11) which is known to have chaotic behavior for specific initial and boundary conditions as investigated by Fisher [1937] and Rothe [1981]. For the densified ceramics to be considered here, the diffusion equation is not needed. 3.3 Description of the Model For the simplicity, a ceramic slab with Finite thickness under plane wave radiation is considered here, as depicted in figure 3.1. The incident electric field is a monochromatic plane wave propagating in the z-direction and is polarized along the y-axis. To account for material non-linearity during microwave heating, the slab is further divided into layers so that finite element method can be used accurately. It is assumed that each element will have the same material properties during microwave heating process at each iterative step. Since the ceramic slab is assumed to be very large, the problem becomes one-dimensional. In the following discussion, layers with smaller thicknesses will be considered as different media since they may have different material properties such as dielectric constant and loss factor which are functions of temperature during the microwave heating process. To simulate microwave heating of ceramics, the microwave energy absorbed by the ceramic slab must first be calculated. Therefore, the electric and magnetic fields in the ceramic slab have to be determined. The electromagnetic field inside a dielectric body of arbitrary shape is difficult to be determined. For the model considered here, an impedance method which is given by Varadan and Varadan [1988] is applied and proven to be effective to account for material non-linearity. 3.4 Impedance Method The system of figure 3.1 is a ceramic slab discretized into n regions for the consideration of dielectric properties change with temperature. The ceramic slab is radiated 41 k-1 Z2 Zk-1 Zk k+1 n-1 Zk+1 Zn-1 Incident Wave Reflected Wave To sources Slab Figure 3.1 Ceramic Slab 42 by a normally incident, time harmonic wave in region 1. There are forward- and backward-travelling Helds for each region. The total electric field for each region can be written as E|c(z) = E£e-Ykz + Ejje+Tfcz = Efc-Tkzj i + p e +2 TfkzJ = Eje-7kz(l+ r(z)) (3.12) where / ( z) is defined as the reflection coefficient at any location in the the region and it is the complex ratio of the reflected wave to the incident wave Hz) =3f€+2Tkz (3.12) The corresponding total magnetic field is Hr(z) = Hje-Vkz + Hfce+Ykz = E^ Z| l - =^ ^ (l- T[z)) (3.14) A total field impedance Z(z) is defined at any location z by the ratio of the electric field to the total magnetic field kt ) 1 - Tkfz) A reverse expression can be obtained for T(z) in terms of Z(z) by solving for T(z) Ik(z) ' T1k Zk(z) + Tik (3.16) The reflection coefficient T(z) at location z can be obtained in terms of reflection coefficient H z') at location z' in the same region, i.e. H z) =F(z')e2Y(z-z’) (3.17) At any interface separating two regions, T(z) and Z(z) satisfy the following two conditions. 1. The total field impedance Z(z) is continuous across the interface; that is, at an interface separating kth and k+lth layer Zk{zk) = Zk+i(0) (3.18) 2. The reflection coefficient is discontinuous across the interface. In practice, the reflection coefficient f n(0) in the region n and the incident field E+j in the region 1 are usually known. The total field impedance for the nth region is (3.19) 1 • UO) Since Z„.i(zn. 1) = Zn(0) (3.20) the reflection coefficient can be obtained at Z=zn . 1 rr n-l(z)-= . i ~ \ — Zn-l(zn-l)~ fln-1 — 7 -— r— — ^ (3.21) Zn-l(zn-l)'*'^ln-l At Zn.j = 0 the reflection coefficient is r n-i(0) =r(zn.i)e-2 Vn-i(zn-i) (3.22.a) the iteration procedure from eq.3.19 to eq.3.22 can be used n-1 time, so that the reflection coefficient in each region and at any position can be found. At the region 1 , the incident field is known so that the reflected field is E i= E |ri(0) (3.22.b) Using the tangential electric field continuous condition, it is easy to find that E i(0)=E2(0) (3.23) E+=e { [ 1 1 M ] \ i+ r 2(o)/ (3.24) E^E ^O ) (3.25) and again, using iteration procedure from eq.3.22(a) to eq.3.25e n-1 time, the field distribution for each region can be obtained. For the lossy dielectric material, the energy absorption in each region is 44 3.5 Computing Procedures When the microwave energy absorbed by the ceramic slab is known, the heat diffusion equation can be used to calculate the temperature variation with time and position. Since the dielectric constant and loss factor are functions of temperature, the microwave energy absorbed by the ceramic slab is also a function of temperature. Hence, the heat diffusion equation becomes non-linear. A non-linear finite element method is therefore needed to find the dynamic temperature distribution profile. At the surface of the ceramic slab, radiation link elements are used to account for radiation heat loss. Conduction loss is neglected since the radiation loss is the prime heat loss at high temperature. The detailed implementation of the non-linear analysis is as follows. The time step to do the non-linear analysis is designated first. The temperature at the end of time step is then estimated. The material properties at the middle of the temperature increment are used for each media. The microwave energy absorbed by each media with different dielectric properties is calculated according to the technique described above. The temperature distribution is then obtained by using power absorption data. The computed results will be compared with the prior estimated temperature. Such an iteration procedure will continue until the difference between the estimated and calculated temperature reaches a prescribed value. 3.6 Results and Discussion 3.6.1 Microwave Energy Absorption by Ceramics By using the technique stated above, the effects of dielectric constant and loss factor on the power absorption by ceramics are considered. Figure 3.2 gives the comparison of power absorption by ceramic slabs with the same dielectric constant and different loss factors. The increase in loss factor will dramatically increase the ability of the Normalized Power Absorption Density 45 e = ( 9.0, 0.01 ) o.o. OX) 3.1 0.2 0.3 0.4 0 .5 0.6 Distance From Incident Plane ( Incident W avelength ) Normalized Power Absorption Density Figure 3.2. Microwave Power Absorption for Slabs with Different Loss Factors. — e = ( 10.0,0.1 ) e = ( 9.0, 0.01 ) I I I I | IT I I | I I I I | 1 I 1 I | I I I I | I I I I .0 0.1 0.2 0.3 0.4 0.5 Distance From Incident Plane ( Incident Wavelength ) Figure 3.3. Microwave Power Absorption for Slabs with Different Dielectric Properties. 0.6 46 ceramic slab to absorb microwave energy. Figure 3.3 shows that with the increase of both dielectric constant and loss factor, ceramic absorbs more microwave power and the distribution of the absorbed microwave energy becomes more uniform. 3.6.2 Simulating Microwave Heating of Ceramics The developed non-linear finite element method is used in this case to find the dynamic temperature profile of a ceramic slab under plane wave radiation considering changes in both dielectric constant and loss factor with temperature. The analysis procedure is displayed in figure 3.4. The data of dielectric constant and loss factor change with temperature are taken from Fukushima et al.[1987]. The incident microwave power flux is taken to be 30kw/m2. The mass density, specific heat and thermal conductivity are taken to be 4g/cm3, 1.125 j/g°C and 10w/cm2, respectively. The thickness of the plate is taken to be 5.08 cm. The finite element analysis routine on ANSYS is implemented in the calculation. The calculation is done on a VAX-11/780 computer. The CPU time is 79 seconds. Figure 3.5 gives the temperature variation with time at 0.01m inside the slab from the edge of ceramic slab at the microwave incident side. The temperature increases slowly at the beginning and rapidly after 600°C. Figure 3.6 displays the temperature profile over the thickness of the slab. An appreciable temperature gradient is observed. This temperature gradient is caused by nonsymmetric microwave radiation of the ceramic slab and radiation heat loss at the boundary which subsequently results in a non-uniform power absorption by the ceramic slab. If symmetric radiation is realized, i.e., microwave radiation is from both sides of the slab, the uneven heating will result only from boundary radiation heat loss and the center of slab will have the highest temperature. Hence, the radiation heat loss at the boundary is the major cause of the non-uniform heating with microwaves observed in the early research where the ceramic sample is melted at the center while the boundary is still intact. In order to prevent this effect, good insulation must be used at the boundary. Also, in practice, the 47 Input geometry data & :reate finite element N . mesh Input initial temperature and physical parameters Obtain microwave energy absorbed by each v element Prescribe AT & estimate A t' obtain complex dielectric constant at T+AT/2 Calculate T + AT' N adjust At =AT/AT'*At' recalculate temperature distribution T + AT y Calculate electromagnetic field strength inside the slab Obtain complex dielectric constant according to calculated temperature distribution STOP Figure 3.4. Procedures for Simulating Microwave Heating of Ceramics. 48 TEMPERATURE ( °C ) 2000 1000- 0. 0 0.1 0.2 0.4 0.3 TIME ( hour) Figure 3.5. Dynamic Temperature Profile 0.01m inside the Slab from the Incident Plane. 2000 18.480 (min.) 1600 17.701 (min.) 1200 16.312 (min.) 800 13.301 (min.) U o 2 a 2 8. E Q) H 12.000 (min.) 400 7.988 (min.) 0.761 (min.) 0 0 1.3 2.5 3.8 5.1 Distance From Incident Plane (cm) Figure 3.6. Temperature Distribution over the Thickness of the Slab. 49 ceramic sample needs to be rotated continuously to prevent any uneven radiation. Figure 3.7 shows the variation of total microwave energy absorbed by the ceramic slab against the temperature variation at 0 .0 1 m inside the slab from the edge of the ceramic slab at incident side. It indicates that as the ceramic becomes hot, its energy absorption ability is increased. Therefore, thermal runaway is simulated in microwave heating of the ceramics. 3.7 Conclusion A method of modeling microwave heating of ceramics is developed here. The results show that increasing the dielectric constant could increase microwave power absorption uniformly while increasing the loss factor could increase the material's ability to absorb microwave energy. It is found that non-uniform heating observed in the laboratory can be caused by boundary radiation loss and non-uniform radiation by the microwave source. Through this research, it is observed that the dielectric property of a material at elevated temperature has a very important role in designing microwave processing. In microwave sintering of ceramics, the green sample changes its microstructure during microwave heating. Therefore, the dielectric properties change not only with temperature but also with microstructure. Hence, the characterization of the dielectric property of ceramics during microwave sintering is very important. In doing so, it would be able to control the sintering process and understand thoroughly the mechanism of microwave sintering. Therefore, it is necessary to develop a method to dynamically characterize the dielectric properties of ceramics during processing. Chapter 6 will introduce a system for in situ microwave heating, sintering and characterization. It is also useful to model microwave heating for the ceramic samples of complicated shapes. For the case where green ceramic is surrounded by insulation material and placed in a microwave applicator, the impedance method to find electromagnetic fields distribution is no longer applicable. The numerical approach is desirable for predicting field distribution in the ceramic specimen as well as in insulation material, so that uniform temperature distribution can be achieved during microwave processing of ceramics. 50 1000 O £ 2 0 0 0 Temperature ( ° C ) Figure 3.7. Total Microwave Energy Absorbed by Slab vs. Temperature. 51 Chapter 4 MODELING SINGLE MODE CAVITY WITH FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) 4.1 Introduction Theoretically, microwave heating is a volumetric process so that uniform and rapid heating is possible. However, it is shown in chapter 3 that nonuniform heating can occur in practice because of nonuniform field distribution and heat loss at the boundary. It is therefore necessary to provide suitable insulation to avoid any temperature gradient Also, initial heating is necessary for ceramics which are low loss material at room temperature. A popular method of obtaining both initial heating and good insulation is to place some microwave susceptor around ceramic specimen. To understand such a complicated structure, a numerical technique has to be employed to find fields strength and energy absorption by the ceramic specimen as well as microwave susceptor. In this chapter, FDTD is introduced and used to analyze a single mode cavity for ceramic processing. In doing so, the design of insulation scheme will be provided and full advantages of microwave heating can be utilized. Microwave processing of materials provides several advantages that in many cases help improve the products quality, uniformity of grain structure, and yield. The ability of the microwave energy to penetrate and hence, heat from within the product, helps reduce processing time, costs, and in some cases of ceramic processing, reduce the sintering temperature. There is also evidence that microwave processing of materials actually provides improved microstructure and other reported advantages of microwave processing of ceramics include removal of water, binders and gases without rupture or cracking, reduction of internal stress, lowering of thermal gradients, and possibility of controlling the state of oxidation. There are several research and development activities in the area of microwave 52 processing of materials. Those activities show that much progress has been made. Some experiments and studies that support some of the anticipated advantages of microwave processing have been successfully completed. However, these studies suggest that microwave heating is a complicated process and that there are many remaining problems that still need to be solved before this new processing technique can be fully understood and/or optimized. For example, some basic interaction mechanisms that may have resulted in the apparent reduction in activation energy of the reported lowering in sintering temperature need to be studied in more depth. The role of frequency and the sample insulation combination on sintering results also need to be better understood. From the engineering and technology point of view, other types of problems such as the design of optimum processing parameters for improved quality, yield, and economics also need to be studied. To this end, it is believed that computational techniques and numerical methods can play an important role in the future research and development activities in the area of microwave processing of materials. Computational techniques may be used to calculate and predict microwave power deposition patterns in materials. These calculations may be made for a wide variety of heating technique and more cost effective computer modeling and sample-insulation combinations. Computer modeling and computational techniques can also be used to calculate temperature distribution and heating patterns in processed samples and may help identify critical parameters in controlling heating process. Single mode microwave resonant cavity has a simple structure and a known field distribution, It has received a lot of attention from those who are interested in using it as a microwave applicator. However, the field distribution, when a dielectric body is inserted into the cavity, is still unknown especially when microwave susceptor is used for preheating. By not knowing such a field distribution and power absorption by the dielectric materials, thermal runaway and nonuniform heating could occur. In this chapter, efforts are made to simulate a single mode cavity to be used for material processing by using the finite difference time domain method. By analyzing different insulation schemes, a better understanding of interactions between microwaves and ceramics is achieved. In the initial 53 simulation, the resonating frequencies of the cavity with different structures are obtained to analyze the effect of the cavity structure on the resonance frequency and to validate the applicability of the FDTD method to the current problem. During subsequent calculation, the cavity was subjected to a continuous incident wave. The field distribution in the cavity and energy absorption by the inserted dielectrics are obtained to analyze the current microwave processing insulation scheme. 4.2 Finite Difference Time Domain Method 4.2.1 Introduction The finite-difference time-domain method is a technique that has recently generated a great deal of interests and has been widely applied to 3D dielectric objects. The solution procedure is based on expressing the time-domain Maxwell’s curl equation in terms of their finite-difference representations. Such a representation has been made elegantly in terms of Yee’s [1966] cell as shown in figure 4.1. In general, FDTD method is useful and attractive in the area of microwave processing, because ( 1 ) in the microwave processing, the computational domain is physically enclosed by the conducting walls of the cavity. Hence, most of the computational difficulties and the inaccuracies are eliminated and replaced by “solid,” well defined boundary conditions on perfectly conducting walls. (2) The FDTD method involves the development of the stead-state solution of the field quantities iteratively and as a function of time. Hence, for the microwave processing of materials research where the electric characteristics of the materials are constantly changing with temperature, the varying properties of materials may be included in the FDTD calculations. (3) The FDTD method may be used to calculate the number of modes with and without the sample, the field distribution in each case, and may even provide some guidelines to the optimum size of the sample that may be heated uniformly, without cracks, and in the shortest period of time. (4) The FDTD solution has the unique advantage of dealing accurately and routinely with dielectric interfaces and inhomogeneities in the properties of materials to be microwave Figure 4.1 Yee Cell for Finite Difference Time Domain Method 55 processed. The technique simply utilizes different finite-difference equations that explicitly involve the properties of materials at the dielectric interfaces. Hence, the FDTD solution procedure may be used to model various insulation-sample combinations that provide optimum sintering results using microwave heating. In recent publications, Sheen et al. [1990] and Moore and Ling [1990] have shown that FDTD method can be used to model microstripline structure. Alinikula and Kunz [1991] and Kunz et al. [1991] have also used FDTD in predicting aperture coupling for a shield wire and wave guide problems. Those computations show that FDTD can be used for analyzing aperture coupled cavity problems. 4.2.2 FDTD Method Formulation For Maxwell’s equation, given in chapter 2, when all three constitutive parameters, e, jx, o, are present the curl equations for isotropic material with no electric sources can be written as, dH = --MV xE) dt |i (4.1) dE = _ £ E + I ( V x H) dt e e (4.2) By using separate field formalism, the total field can be expressed as £_£total = gincident +£scattered (4 3 ) jj_ |jto ta l = [^incident + |jscattered (4 4) The incident fields satisfy the free space Maxwell’s equations and can be specified analytically throughout the problem space while the scattered fields are found computationally and only the scattered fields need to be absorbed at the problem space faces. In the media of the scatter, the total fields satisfy 56 VxEt0,al = ^ (4.5) dt VxHtolal = e -E^ tal- + oE t0tal dt (4.6) while the incident fields traversing the media satisfy free space conditions VxEinc = - n o ^ . (4.7) VxH“c = eo^=r— ■ dt (4.8) rewriting the total field behavior as VxfEinc+Escal) = -p d(H,nc+HScat) (4.9) V x jH ^ + H 8031) = e ^ Em-C^ ESCa-) + a (Einc+EScat) (4.10) ot by subtracting the incident fields above to obtain the equations governing the scattered fields in the media VxEscat = - p ^ f ^ ^ dt , v d H inc (^"^°)“ a T VxH scat = E ^ ^ - + oE inc ( e - e o j ^ - + o E inc dt L' outside the scatter in free space the scattered fields are (4.11) (4.12) muscat VxEscat = -|i 0 —3 :— dt (4.13) VxHKat = E o ^ ^ (4.14) the above equations can now be rearranged so that the time derivative of the fields are expressed as a function of the remaining terms for ease in generating the appropriate difference equations. dfjscat _ - (|i-|io)dHinc UVxFScat) dt " |i dt V ] ^ SCat_ O g scat. f fg in c . ( E - e o ^ + i ( V x H scat) dt E E (4.15) (4.16) 57 Substituting derivatives with corresponding time and position differences and using Yee’s cell as reference, the difference form for the Ex and Hx can be written as | H § q j,K )n -Hg(I,JtK -l)n J Az At 'e +cAt H |(I,J,K ) n + 1 =H|(I,J,K)n - j ^ ) ( H lx(IJ,K )n+1 -H jl(I,J,K)n) (4.18) n \^ ( I J ,K ) n+2 -E |(I,J-l,K ) n + 2 , / nEf(I,J,K )n+2 -E ^(I,J,K -l ) n + 2 Ay vM1/ Az Both E and H are evaluated over successive cycles where E is evaluated first at n+1/2 and H at n+1. This interleaves E and H temporally and results in a temporal center difference or “leapfrog” approach when coupled with the spatial central difference. The above notation shows E at N=n+l/2 updated from its prior value at n-1/2 and the curl of H at n. Similarly H is evaluated at N=n+1 from its earlier value at n and the curl of E at n+1/2. 4.2.3 Outer Radiation Boundary Condition (ORBC) Since FDTD technique uses a finite problem space, there must be components on the surface of the problem space. These components, unlike the remaining interior components, are not completely surrounded by their neighbors. As a result, when calculating the update of the surface components according to the formulation given in previous section, there is not enough information to correctly calculate the updated value. An outer radiation boundary condition must be employed to approximate the missing field components. If the approximation were perfect, then the outermost component would be updated as if the scattered field passed through the surface component's location were 58 unaffected by the truncation of the problem space. Any real ORBC is an approximation. It will approach the ideal and will generate some reflected radiation, typically small, from the outmost or outer boundary components. There have been a number of ORBC developed prior to the now widely used Mur’s radiation condition. Taflove and Brodwin [1975] used the conducting layer approach in his early work with FDTD technique. Gilbert [1976] have examined the legendre polynomial analytic continuation approach. Mur [1981] derived an outer radiation boundary as well as more generalized treatment starting with the wave equation. Following Mur’s treatment, the wave equation for a single field component W may be written as (4.19) where c0 is the velocity of propagation. Assuming that the FDTD problem space is located at x>0 , there is a boundary at x= 0 that a scattered wave will reach and be reflected unless an ORBC is imposed. The ORBC is found from the above wave equation. The scattered wave has velocity components V x, Vy, V z such that V x 2 + V y 2 + V z 2 = C02. It is also possible to define inverse velocity components Sx, Sy, Sz where C — 1 Ox ~ c _ i Oy — 1 c , Oz — 1 (4.20) (4.21) The scattered wave can be approximated by a plane wave where W = Re (\|/(t + Sxx +Syy +Szz)) (4.22) and, by expressing Sx as (4.23) 59 the wave equation becomes W= Re (4.24) when (4.25) i.e. Sx k 0 the wave is traveling in the -x direction at some unspecified angle i.e. Sy and Sz are not specified. This wave satisfies, as can be seen by direct substitution, the first order boundary condition at the boundary x = 0 (4.26) The W satisfying the above equation is therefore a wave traveling in the -x direction and is outgoing and may therefore be characterized as absorbed. Since Sy and Sx are not specified, a solution is not determined for the above first order boundary condition. Approximations of the first and second order will allow W to be found at the boundary x=0. The first order approximation is that ( l -(C g S y )2 - (C q S z )2 j>2 - 1 + 0 (C Q S y J2 + 0 ( 0 0 8 2 ) ? a 1 (4.27) so that the first order boundary condition is approximated to first order by (4.28) The second approximation is 60 ( l - ( C g S y ) 2 - (C g S z )2}i - o ((( ( C o S y )2 1- (C g S y )2 -K C q S z)2 ) + ~ 2^ (c0^yF "KcqSz)2) - K c q S z J 2 ) )^ ) (4.29) so that the second order boundary condition is approximated as — - 1- i( (CQSyf2-H coS^ Wlx=0= 0 dx 1 (4.30) dt By taking the time derivative of the above equation and multiplying by l/c 0, the second approximation becomes The approximate boundary condition for FDTD application can be obtained through the appropriate discretization. 4.3 Structure of Model The cavity to be modeled, as shown in figure 4.2, is a rectangular cavity which is used in this thesis research for processing of ceramics such as sintering, joining and simultaneous microwave heating, sintering and characterization of ceramics. The resonant cavity is primarily resonated in TE103 mode at 2.45GHz. The cross section of the cavity has a dimension of WR430. The length of the cavity can be adjusted so that the cavity will always be in resonance at 2.45 GHz with the insertion of different dielectric materials in practice. A section of the waveguide is connected ahead of the cavity which acts as a transmission line for sending the incident wave. The computing mesh is constructed as follows. The cubic cell used for modeling has a dimension of 3.413mmx 3.413mm x3.413mm. The thickness of the cavity wall is 61 molded by four nodes. The cavity is assumed to be made of copper. The Mur [1981] absorbing boundary condition is used at the incident side of the buffer to limit the calculation space. The position of the variable short can be readily adjusted by a parameter specified in the program. In doing so, the cavity can be tuned to the resonant position. The iris used in processing of materials could be rectangular or circular. In this calculation, rectangular iris is used for the easy implementation. Circular iris has to be approximated by many rectangular. Three particular structures are considered in this analysis. The first one is an iris coupled empty cavity. Its resonance characteristics is analyzed. In doing so, the applicability of the FDTD method to the current problem can be validated and its accuracy can be estimated. The effect of the iris on the resonance of the cavity can also be analyzed. In the second case, a ceramic rectangular rod is inserted into the cavity alone which is simulating the case where simultaneous microwave heating, sintering and characterization are researched. The ceramic rod can be either densified or green. In the third case, the ceramic rod is surrounded by four SiC rods as shown in figure 4.3. These SiC rods are used as the microwave susceptor in microwave sintering of ceramics for initial heating purpose because of their high microwave absorbing ability. Such an arrangement has been experimentally proved to be essential in achieving uniform temperature distribution during microwave processing of materials. At room temperature, most of the ceramic materials have very low dielectric loss. Their ability to absorb microwave energy is very little. When dielectric material which has low loss is inserted into the cavity, it can cause large reflection and low heating rate. Such a large reflected microwave energy can be detrimental to other microwave components such as the magnetron, etc. Also, uniform temperature distribution can not be realized with dielectric material alone because of heat loss at the boundary. Holcombe and Dykes [1991] and Janney et al. [1992] have done research work to construct the right insulation scheme for providing initial heating and maintaining uniform temperature distribution for the materials to be processed. Theoretical investigation is still lacking for analyzing those structures in the guiding of practical design. 62 Iris Ceramic Rod Variable Short Figure 4.2. Single Mode Cavity for Material Processing Ceramic Rod Microwave Susceptor Figure 4.3. Structure of Ceramic Rod and Surrounding Microwave Susceptor 63 4.4 Computation Procedures In the initial calculation, the response of the cavity in the frequency domain is needed to determine the frequency at which the cavity is resonating. To do so, a truncated electric Held pulse which polarized in the Y direction and has a Gauss form is used to illuminate the cavity. Mathematically, the pulse has the form iL _ y(t) = e | (Beto*DTj2 | (4.32) Figure 4.4 illustrates three pulses for different values of Beta. DT is the time step for the iteration. The response of the cavity is recorded on time domain. Such a time domain response is then transformed to the frequency domain by using the Fast Fourier Transformation method. In the subsequent simulation, the continuous incident electromagnetic waves have to be used for the calculation. Since the frequency for microwave heating is designated to be 2.45 GHz. The mode of wave propagation in WR430 waveguide can only be TE10, while other modes will be evanescent. With such an argument, the precise incident wave distribution in the waveguide ahead of the cavity can be specified. For the case where a TE10 mode of waves propagating in Z-direction and polarizing in Y-direction, the electric field has the form of Ey = Eyo sin (®*-) cos (cot - pg z) (4.33) where a is the width of the waveguide, pg is the phase constant in the waveguide. The field distribution inside the cavity will reach harmonic state after a sufficient number of iteration steps. Hence, the field distribution will have the form Exye"^03t. To find the spatial field distribution, the following method is used. Since 1.00-1- 0.75— Beta= 125 Beta= 100 0 .5 0 — Beta= 75 0 . 0 . 00- - -0 .2 5 t— i—i— i—I— i—i— i—i— |—i— i—i— i—|— r —i— i— r - 1 .0 - 6.5 0 .0 0.5 Time ( nano-second) Figure 4.4 Truncated Gauss Pulse 1'.0 Assume the iteration time step is At, then let N =^4At (4.36) N E (x ,y )= fflJ Ey( nAt ) (4.37) n=0 The microwave energy absorbed by the lossy dielectric material will be P Absorption = 2 ^ (4.38) 4.5 R esults an d Discussion Figure 4.5 shows the frequency response of a closed cavity when it is illuminated by a microwave pulse. Each maxima gives a resonant mode. Figure 4.6 gives the frequency response of a closed cavity which is designed to be resonant at 2.45GHz in TE103 mode after different number of iteration steps. The results in figure 4.6 illustrate that more accurate results could be realized with increased number of iteration steps. Comparing theoretical and FDTD results, it is found that the relative error is about 0.1% when 8192 iteration steps are used. Table 4.1 tells the resonant frequencies for different cavity structures. The number of iteration steps used here is 8192. Lower resonant frequency is obtained when dielectric rod is inserted into the cavity. Such a result is consistent with the phenomena observed when the cavity is examined by connecting it to a vector network analyzer. This result would also be useful for calibrating a very popular dielectric measurement technique, that is, the cavity perturbation method for measuring dielectric properties of various materials. More importantly, the knowledge of the frequency response would enable us to adjust the position of the short so that the cavity is always in 66 the resonant state at 2.45GHz. Figure 4.7 shows the electric field distribution inside the cavity when a ceramic rectangular rod is inserted into the cavity. It is obvious that the field distribution inside the cavity has the same shape as that of the empty cavity. The field distribution in the dielectric region is essentially suppressed due to the ceramic material which has a higher dielectric constant. At the region close to the iris, the field distribution is disturbed by the presence of the iris as compared to the cavity with no iris. Figure 4.8 gives the microwave energy absorbed by the dielectric ceramic rod. Since the ceramic rod has low loss, the energy absorbed by the ceramic rod is smaller despite it is located at the maximum electric field region. Such a power absorption knowledge can be used to obtain the temperature distribution in the ceramic rod. Figures 4.9 and 4.10 show the microwave energy absorbed by the ceramic and SiC rods at two temperatures. The result for the energy absorption at room temperature shows the SiC rod absorbing much more microwave power than the ceramic rod. Hence at the beginning of microwave processing of ceramics, the ceramic rod was mainly heated by the radiation from the SiC rods. At 1000*C, the ceramic rod absorbs almost the same amount of microwave energy as the SiC rod and is heated mainly by transforming absorbed microwave energy. The microwave energy absorbed by SiC rod is decreased because SiC rod has large conductivity at high temperature which prevents microwave penetration. The heat generated by SiC rods will be continuously radiating to the alumina rod for the insulation purpose rather than heating. In so doing, uniform and volumetric heating process can be realized. 4.6 C onclusion It is shown here that the finite difference time domain method is applicable for analyzing aperture coupled single mode cavity for microwave processing of ceramics. It not only indicates the working condition of the cavity but also gives the specific electromagnetic field distribution and energy absorption by material to be processed. 67 Ey 750_: “ (v/m) 500 40 2.0 Frequency ( GHz) Figure 4.5. Frequency Response of an Empty Cavity 2.50 Resonant Frequency 2 .45 - 2 .40 ■ 9 Calculation Theoiy 2.35 - 2.30 0 5000 10000 15000 20000 Num ber of Iteration Steps Figure 4.6. Convergence Analysis of the FDTD Method 68 Table 4.1 Resonant Frequencies for Different Cavity Structures "v^^sM O D E TE101 TE102 TE103 THEORY RESULT FOR PERFECT CAVITY 1.529 1.926 2.448 CLOSED COPPER CAVITY 1.532 1.922 2.447 1.542 1.932 2.435 1.319 1.877 2.249 1.319 1.877 2.136 COPPER CAVITY WITH IRIS CLOSED CAVITY & DIELECTRIC ROD COPPER CAVITY & IRIS & DIELECTRIC Figure 4.7. Field Distribution in the Cavity with a Ceramic Rod Figure 4.8 Microwave Energy Absorbed by a Ceramic Rod in the Cavity 70 - 25000 P(w) Figure 4.9 Microwave Energy Absorption by Both Ceramic Rod and Microwave Susceptor at Room Temperature Figure 4.10 Microwave Energy Absorption by Both Ceramic Rod and Microwave Susceptor at 1000°C 71 Chapter 5 EXPERIMENTAL SYSTEM 5.1 Introduction The main equipment used for this thesis research is the single mode high power microwave heating system as shown in figure 5.1. It consists of a 120v-high voltage power supply, a magnetron, a circulator, a directional coupler for power measurements, an impedance analyzer, a 4-stub tuner, a section of three quarter waveguide, a variable short, a pyrometer and attached computer data acquisition machine. Other equipments used are a 8510 network analyzer and its related free space set-up and waveguide measurement system, a scanning electron microscopy, a X-ray diffractometer, a multi-mode microwave oven and a conventional electric furnace. Since the single mode high power microwave heating system has a direct effect on the characterization and processing, a brief explanation of the function of each component is given in this chapter. 5.2 Magnetron The continuous wave magnetron is the microwave tube most commonly used in the microwave processing system. The first magnetron was developed prior to World War n in the late 1930s and early 1940s. A concentrated effort was made to advance the technology during World War II because of the need for higher frequency operation, primarily for radar systems. Compared with klystron, a magnetron has limited electronic tuning and modulation capabilities. Because of its low cost and high efficiency on the order of 80 percent, it has being increasingly used in industry heating, diathermy equipment, microwave ovens, etc. A schematic diagram of a multicavity magnetron oscillator is shown in figure VAX CLUSTER HP 9000 PC I HP 3497A DATA ACQUISITION SYSTEM POWER SOURCE CONTROL UNIT DUMMY LOAD TEMPERATURE LINEAR VOLTA<3E DISPLAYCEMEN1 TRANSI)UCER POWER METER PYROMETER i T T -55 dB JMAGNETROI' HEAD 0TO 3K W 2.45 GHZ CIRCULATO -55 dB J WAVEGU1DI COUPLER ' HIGH POWEF PHASE IMPEDANCE ANALYZER HIGH VSWR 4 -STUB TUNER IjADJUSTABLE ^W A V E G U ID E | j SHORT VARIAB IRIS CAVITY APPLICATOR HIGH VOLTAGE SUPPLY Figure 5.1 Single Mode High Power Microwave Processing and Characterization System CERAMIC SAMPLE -j to 73 5.2.(a). As may be seen from the figure, the magnetron consists of a multicavity anode block, a coaxial cathode, means of coupling the generated microwave power to the outside and a permanent magnet to provide a magnetic field along the axial length of the cathode and at right angle to the dc electric field in the radial direction. When the anode is at ground potential, a negative high voltage is applied to the cathode. Electrons are emitted from the heated cathode and, without the presence of a magnetic field, the electrons would travel radially to the anode. When a magnetic field is applied in a direction parallel to the tube's axis, the electrons follow a curved path to the anode and form a rotating electron cloud as shown in figure 5.2.(b). If the magnetic field is strong enough, the electrons are prevented from reaching the anode and the magnetron current is cut off. Anode current flow excites oscillations in the resonant cavities which, in turn, influence the shape of the rotating electron cloud. Some electrons are slowed down and others are speeded up, depending on the direction of the electric field on the circuit as the electrons pass through it. The electrons with increased velocity return to the cathode and release secondary electrons as they strike the cathode. The slowed down electrons, which have lost most of their energy to the resonant circuits along the way, eventually end up impinging on the anode at low velocities. The rotating electron cloud assumes a spoke shape and rotates at a constant velocity, giving up large amounts of energy to the microwave field. The microwave energy is coupled by means of a probe from one of the resonant cavities into an output coupling or antenna where it is launched into a transmission line, usually waveguide or coaxial line. 5.3 Circulator Operation of a magnetron at higher than specified VSWR may result in unstable performance and possible damage to the tube. A Circulator is used when necessary between the generator and the load to reduce the power reflected back to the generator from a mismatched load and thus reduce the VSWR at the magnetron output. The circulator is a three or four port device which has the unique ability to couple 74 Magnetic field Anode Microwave Output Cathode Figure 5.2 Multicavity Oscillator Magnetron 75 energy between adjacent ports on one direction only and isolate between non-adjacent ports. Referring to figure 5.3, energy entering port 1 exits only at port 2 and energy entering port 2 exits only at port 3. For the three port device, energy entering port 3 exits only at port 1. Thus, the energy circulates around the device in only one direction. Port 1 is the input, port 2 is the output, and port 3 is terminated with a matched dummy load which absorbs energy reflected from output port 2. Very little energy is coupled from port 3 to the input port 1. The circulator, like the isolator, relies on the non-reciprocal properties of ferrite for its operation. 5.4 Directional Coupler The directional coupler is a calibrated power sampler which has the ability to distinguish between two directions of power flow. The directional coupler consists of a main line and an auxiliary line that are separated from each other except in the coupling region where some of the energy in the main line is coupled in one direction into the auxiliary line. In figure 5.4, energy entering terminal 1 of a coaxial directional coupler exits at terminal 2 except for a small amount of energy coupled into the auxiliary line and appearing at terminal 3. Energy applied to terminal 2 appears at terminal 1 and almost no energy at terminal 3. Dual directional coupler for simultaneous sampling of power in both directions consists of two directional couplers back to back in one package. Coupling is defined as the ratio of main input power to auxiliary line output power. Directivity is defined as the ratio of auxiliary line power as a result of incident main power to the auxiliary line power as a result of reflected main power. Values of coupling are normally anywhere from 3dB to 50 dB. Directivity is usually in the order of 30 or 40 dB. 5.5 Impedance Analyzer The function of the phase/impedance analyzer is to provide additional degree of freedom in matching difficult load situation where the load impedance can change with 76 Dummy Load Mismatched Termination Power Source Figure 5.3 A 3-Port Circulator Pi Auxiliary Line Main Line Figure 5.4 Directional Coupler 77 power level, power source frequency. A multistub is often used in high power impedance matching where reflected power is utilized to determine stub insertions. Such an approach could cause performance difficulty such as in case where there is more than one tuner solution and the solution selected is the one with most sensitivity to load impedance fluctuations or the load impedance might be such that the reflected power has to be increased before the optimum tuner solution can be found. Therefore, using only reflected power as the tuning indicator can be difficult The impedance analyzer, which equipped with phase sensitive detectors that are inserted in the waveguide, is located between the power source and the tuner and is connected to a suitable oscilloscope where the load impedance is displayed in a polar format identical to the Smith chart. With this display, the action of the tuner can be precisely interpreted and the optimum tuner position can be obtained. Figure 5.5 shows the construction of the impedance analyzer. 5.6 The 4-Stub l\in er In operating the single mode microwave heating device, it is important to match an arbitrary load impedance to the transmission line. In doing so, maximum power transfer can be obtained and the standing wave which could appeared in the transmission line will be eliminated. Although impedance matching could be realized by functions of the variable iris and the variable short, ability to provide extra matching tuning is always necessary when fixed iris is used or difficult matching load is encountered. The 4-stub tuner is utilized in this set-up to tune single mode cavity during microwave processing of ceramics. The construction of the 4-stub tuner is illustrated in figure 5.6. It consists of four variable depth screws mounted on a fixed carriage free to move longitudinally to have various penetration. The screw will act as a shunt capacitance for 1/b < 3/4 and a shunt inductance for l/b>3/4 which is shown in figure 5.7. Therefore, the movable screw has the same effect as the short circuited stub used in the low power microwave circuit for the tuning purpose. The matching mechanism for a short circuited stub can be found in basic Figure 5.5 Impedance Analyzer 79 3/8 A,g Figure 5.6 A 4-Stub Tuner B' 0.5 Figure 5.7 Characteristics of One Stub 80 electromagnetic book. Theoretically, three stubs can match arbitrary load in the transmission line while four stubs are more than enough for all the matching purposes. 5.7 Iris For a metallic enclosure to act as a microwave applicator where microwave can resonant and materials can be heated in it, microwaves have to be coupled into the cavity. Basically, three coupling methods are usually used in the microwave engineering: (1) probe coupling, (2) loop coupling, and (3) iris coupling. In the high power microwave sector, iris coupling is often accepted to avoid high current in the loop or the probe. According to the shape and the position of the iris at the common boundary of the cavity and transmission line, the effect of the iris can be inductive or capacitive. Some of them are illustrated in figure 5.8. The effect of the iris can be to the first approximation equivalent to an electric dipole normal to the aperture and have a strength proportional to the normal component of the exciting electric field, plus a magnetic dipole in the plane of the iris and have a strength proportional to the exciting magnetic field as shown in figure 5.9. The constants of proportionality are parameters that depend on the iris size and shape. These constants are often referred to as electric and magnetic polarizability of the iris and characterize the coupling or radiating properties of the iris. For the iris used in this thesis, it is circular and located at the transverse wall of the transmission waveguide, its normalized inductive susceptance is = (5.1) 8 r 3 P„ where a is the dimension of the broad side of the waveguide, b is the short side of the waveguide, r is the radius of the aperture and (3g is the wavenumber in the waveguide. 81 Inductive Iris L b V '/////////////////a *------ a------ ► Capacitive Iris Figure 5.8 Three Coupling Iris for Material Processing Cavity 82 Electric Polarization i >i ■§) Magnetic Polarization Figure 5.9 Electric and Magnetic Polarization in the Iris 83 5.8 Resonator Cavity Essentially, a resonant cavity, or a heater, consists of a metallic enclosure into which a launched microwave signal of the correct electromagneticfield polarization will suffer from multiple reflections between preferred directions. The superposition of the incident and reflected wave is very well defined in the space. The precise knowledge of electromagnetic field configuration enables the dielectric material under treatment to be placed in the position of maximum electric field for optimum transformation of the electromagnetic energy to heat. Inside the cavity, large stored energy will be transformed to heat via displacement and conduction currents flowing through the dielectric material. In this thesis research, the rectangular single mode cavity is used. Its resonant mode is TE103. Therefore, the electromagnetic fields distributions in the cavity with no dielectric material inserted are Ey = r ? A K lQ3z Q a Sin 30L sin 3SZ. y 71 a d Hx = sin ^ cos H x = -2 jA s i n c o s a ( 5 .2 ) (5,3) d (5-4) The geometry of the rectangular cavity and its electric field distribution are shown in figure 5.10. Where a and b are the broad and narrow sides of the waveguide cross section and d is along the Z direction and represents the length of the cavity. An important parameter specifying the performance of the resonator cavity is the quality factor Q. In general, the quality factor is defined as q _ 0 ) (time-average energy stored in system) energy loss per second in system ^ ^ When a cavity is at resonance state, the time average electric and magnetic energy stored in the cavity are equal. The average stored electric energy is given by 84 TE103 Figure 5.10 Field Distribution in TE103 Cavity 85 W' = f L ‘ L j.EyEjdxdydz (5.6) =-^-aHdk5o3 4|A|2 The magnetic energy stored in the cavity is wrn= ^ | 0' J0l (hxH]i +HZHz)dx d y d z = We (5.7) In order to determine the Q of the cavity, the loss caused by the finite conductivity of walls must be evaluated. For small losses, the surface currents are essentially those associated with loss-free field solution. Thus the surface current is given by Js= n x H (5.8) where n is a unit normal to the surface and directed into the cavity. Hence the power loss in the walls is given by P, = ^ [ Js J s* ds = 5m I IH tan I2 dS J will* (5.9) / W illi where Rm= 1/ g 8s is the resistive part of the surface exhibited by the conducting wall having a conductivity a and for which the skin depth is 8 S = ( 2/o)(ia ) 1 /2 Htan is the tangential magnetic field at the surface of the cavity walls. Substituting eq. 5.2,5.3,5.4 to eq. 5.7, a straightforward calculation gives Pl = (A) Rm2a3b + 2d3b + adL+da3 (5.10) d2 By using eq. 5.5, the Q of the cavity is given by Q - “ (We + W.) (5 U ) When a ceramic rod, which has a relative permittivity e = e ’ - j e” is placed into the cavity, the lossy dielectric has an effective conductivity toe”, and hence the power loss in the dielectric is 86 Jfj Pid = dV = = i2 I J • E E **dV = I | E l 2 dV (5.12) Considering both dielectric and wall loss, the quality factor is then q > (W ^ W .) P| + Pid In practice, energy is coupled to a resonator structure by means of an iris which can be inductive and capacitive as discussed before. The iris used in this thesis research has circular shape and it is therefore inductive. The admittance of such an iris is given by eq. 5.1. For an empty cavity where only the circular iris and variable short are controlling its behavior, the input admittance at the plane of the iris is Yo + ----- j-1------ r tanh (ot + j (ij (5.14) When a cavity is in resonance, the imaginary part of the input admittance is zero. When it is matched, the real part of the input admittance will be unity. When both resonance and match are realized, the cavity is said to be critically coupled. At that time, there will be no reflection from the cavity and energy conversion efficiency is the highest. Hence, by solving equations in eq. 5.15, it would be able to find the required radius of the iris and the position of the variable short for critical coupling, i.e., I m a g .( ^ ) = 0 v Yo R e .( ^ ) = l (5.15) When the dielectric material is inserted into the cavity, the equivalent impedance of the inserted dielectrics usually cannot be expressed analytically. Such an impedance, however, can be obtained through a measurement of the reflection coefficient may using the afore mentioned reflectometer. A control algorithm can be designed for coupling and matching. Correspondingly, the related control equipment can be arranged so that an 87 automatic control system can be made for optimal transferring microwave to heat for material processing with best efficiency. 88 Chapter 6 IN SITU MICROWAVE HEATING, SINTERING AND CHARACTERIZATION 6.1 Introduction In developing microwave processing of ceramics, some problems need to be solved such as basic scientific studies on microwave-materials interactions and loss mechanisms. There is also a critical need for a broad data base on the dielectric properties of materials at high temperature over different frequencies. In situ heating and characterization of ceramics can be used to continuously measure material permittivity and offer unique approach for understanding the interactions between microwaves and materials. In this chapter, a method of using a high power single mode microwave heating cavity to simultaneously heat or sinter and characterize densified or green ceramics is described. Sintered ceramics rod was heated and characterized so that its dielectric property at high temperature was retrieved. A green ceramic rod, which was produced by an extrusion process, was simultaneously microwave sintered and characterized. 6.2 Historical Background One function of in situ microwave heating and characterization of ceramics is that it can be used to obtain the permittivity of materials. Knowledge of the permittivity of materials at elevated temperature is generally needed to develop industrial application. Such data is needed to develop process model that will predict the internal fields and the heating pattern and rates, so that optimum processing parameters can be developed to meet material and product requirements. Conventionally, dielectric properties of material at high temperature are often obtained through a free space measurement technique as done by Breeden [1969], Ho [ 1981], Gangnon et al. [1986] as well as Varadan et al. [1991]. In 89 the free space measurement technique, samples are positioned in the path of the incident beam and the complex transmission and reflection coefficients are measured by two identical receiving antenna suitably aligned with respect to the incident beam and the sample. The dielectric properties are then deduced from observed transmission and reflection coefficient. Through years of development, free space method seems to be a widely accepted technique. The advantage of using the free space method is that with one planar specimen, dielectric property of ceramic materials can be acquired for a wide range of frequencies. However, the free space method suffers from a long period of characterization time, due to the time needed to reach thermal equilibrium for a large planar sample. Such a long time heating at a high temperature may very well alter the microstructure of the material. More importantly, since the specimen being characterized was heated by conventional methods rather than by microwaves, the microwave-material interaction mechanisms were not revealed. Contrary to the free space technique with conventional method of heating of large planar specimen, in situ microwave heating and characterization method uses microwaves to heat thin ceramic rod and that microwaves are also used to detect the variation of the dielectric properties of ceramic rod with temperature. In some cases, another microwave source which doesn’t interfere with heating source is also used as detecting signal. The in situ microwave heating and characterization method has the advantages of obtaining a high heating rate and uniform heating pattern due to volumetric nature of microwave heating. Therefore, the required characterization time can be substantially reduced. Couderc et al. [1973] was the first to conduct the research on in situ heating and characterization. He used a cavity resonator which has two dominant resonating modes. One of those resonances is used for heating the sample at frequency of 2.45 GHz where the second resonances is used for measuring the dielectric properties. The shape of the test specimen could be either spherical or rod. The highest temperature reached in his experiment was 600°C. In 1984, Areneta et al. used an equivalent circuit to represent dielectric rod in the waveguide. The iris and the in-perfect movable short which are part of the cavity were represented by the impedances arrived from either analytically or 90 experimentally. As the temperature increases, the reflection coefficient will be changed due to the change of the dielectric properties of the ceramic rod. This change was monitored by a VSWR device. It is actually moving on a slotted line. During characterizing process, the VSWR meter need to be constantly moved mechanically along slotted line. The dielectric properties of the ceramic rod were deduced from measured reflection coefficient. The characterization errors may have been accumulated during experiment because of the movement of VSWR meter. Fukushima et al. [1987] used a rectangular cavity to heat and characterize ceramic material with microwave energy. A precise control of the iris size and the position of the variable short was adopted in their experiment to maintain the cavity in the critical coupling state. The dielectric properties of the sample was measured by detecting both variable iris size and position of the variable short which are controlled to give the resonance and critical coupling. The highest temperature achieved was 1800°C. Another application of the in situ microwave heating and characterization is to use the technique to actually sinter and characterize ceramics. In doing so, on line temperature and dielectric properties versus time profiles were measured during the sintering process. Therefore, the structure variation versus microwave absorptivity of the green sample during sintering can be determined. The mechanisms of microwave sintering of ceramics can be fully understood. However, such an important experiment has not been realized at present. Similar work has been done to the curing of resin and heating oil with microwaves. Jow et al. [1987], Hu [1983] observed fast changing dielectric constant in resin curing and oil shaking due to high power microwave irradiation. 6.3 Mathematical Model for the Characterization The capability of in situ characterization when sintering is very useful in providing insights into the various dynamic process associated with microwave sintering. In general, in-situ characterization is important in giving insights into the nature of physical properties and processing of the rod being heated. The approach used in analyzing the cavity and rod is to model them by their equivalence transmission line circuit representation and use circuit theory to determine the pertinent equation for characterization. In lieu of the above mentioned approach, it is important to have an accurate equivalent circuit model for the rod in the microwave cavity. In this section, a brief review of the development of such an equivalent circuit is given. Cylindrical obstacles in a rectangular waveguide are used in many microwave devices. Considerable efforts have been made by many authors to investigate such structures. Applying Schwinger’s [1968] variational formula for the equivalent circuit, Marcuvitz [1986] gave a model represent the rod by a T-equivalent circuit as displayed in figure 6.1. To improve the approximation give by Marcuvitz, Nelson [1969] described a numerical technique which removes the limitation on the rod diameter imposed by Marcuvitz’s formulation. By calculating the reflection and transmission coefficient of a rod in an infinite waveguide. The T-equivalent circuit is derived from those reflection and transmission coefficients. Araneta et al. [1984] took a further step in M arcuvitz’s derivation; he used more terms in the expansion which allowed more accuracy. The pertinent expression given by Schwinger’s variational approach is as follow: f r, . ry v Ml + Ml ) 2j ( { ET - l ) k 2 — — ( 6 . 1) | <J>o(x,z) dS - (e? - <J>e(x,z) Ge (x,Z x’,z) d>e(x', z') dS'dS Ceramic Rod Z ll - Z 1 2 Z ll - Z 12 Z12 Figure 6 .1 Equivalent Circuit of a Ceramic Rod in the Transmission Line 93 2j(eM)k2 _ ka(Z n - Z 1 2 ) (6.2) <I>o(x,z) c,z)dS dS -(ej - (ej -1) - l) kk2^ 21 JI (j)o(x,z) Go(x,z x',z')•J^x', z') dS'dS OoVodS where the integration is over the cross-section of the rod and, G: = - -r- sin sin ( sjn kI z- z'I ± sin kI z+z'l) 1 ka a a v 7 + j ^ i y sin 21®25-sin-11®24- / 3 (6.3) x 3 IgjlCol Z - Z I + gjKnl Z +z1ij I _______ ___ ______ ___ a n=2 K" where the plus is used when i = e and the minus sign is used when i = o; and <(>(x,z) = total electric field intensity (x,z) = incident electric field intensity k = 2 tiA k2 = k2 . (mLj2 , Kl = k (6.4) (6.5) e? = complex dielectric constant of the rod K= , X.g = wavelength in the waveguide A.* the subscripts e and 0 , respectively, denote even and odd symmetry about the reference located at z = 0. The coordinate used is shown in figure 6.3.1. The width of the broad side of the waveguide is “a” and the axis of the rod is the line ( x0, yo> 0 ). Eq. 6.1 and 6.2 can 94 be evaluated in conjunction with solution of the wave equation, (v 2 + e*k2) $ = (6 .6) 0 inside the rod. r , e ) = £ ( A 2m cos m0 J 2 m(lcr) + A2 m+i sin (2m+l) 0 J 2 m+i(k,r)}, even (6.7) m=0 <»o(r, 0 ) = X ( f i2m+i cos ( 2m+l)0 J 2 m+i(k'r) + B2 m sin 2m0 J 2 m(k'r)}, odd (6 .8 ) m=0 where (k ? = e,* k 2 when the rod is at the middle of the waveguide, its axis is at x=a/2, z= 0. In this case, Equation 6.7 and 6 . 8 may be simplified by symmetry consideration about the =a/2. Thus (6.9) r, 0) = X A 2 m cos m0 J 2 m(k'r) m=0 e)= X A 2m COS (6 . 10) m0 j 2 m(k'r) m=0 In the derivation of the Marcuvitz model, ( Zj 1 + Z 1 2 ) and ( Zj 1 - Z 1 2 ) were approximately by using only the first term of the <j>e and <}>0i in eq. 6.7 and 6 .8 . The results are j ( Zu + Z 12 ) = Ka c s c ^ f l X sin2 n2 - }2 -1 11=2 - i - In (kac sin - 1 dn27CX° + 2n \ k » > it a 4 (6 . 11) Y° ^ ' a J o(P)Y i(«) } a J 0(p )Ji(a )-p Ji(p )Jo (a ) _ 95 s in ® l + _La iM > I 4n Ji(ot) pj0((J)J1(ct) - otJi(p) J»(a) Z n - Z12 (6. 12) Considered the next term in the expansion of <J>e and (|>0,Areneta et al. [1984] obtained another formula for the equivalent circuit for the case where rod is atthe center of the waveguide. The form of <[>e and <f»0 used are as follows (r, 0) = A0 Jo(kr) + A2 cos 20 J2 (kr) <j>0 (r, 0 ) = Bi cos 0 Ji(k'r) +B 3 cos 30 J3 <kr) (6.13) (6.14) Substituting eq. 6.13 in 6 .1 and eq. 6.14 in 6.2, it is obtained 1 _ : (e? - 1 ) k2 Cq - C0 C2 (Dq2 - D 20 ) / D 22 + C2D 00/D 22 (ZH + Z 12) J ka D 2 2 - (D 0 2 D 2 0 /D 2 2 ) _ ;{e? - 1 ) k2 Ci - C 1C3 (D 13 - D 31 ) / D 33 + C3D 11/D 22 {Zll + Z i 2 ) = j D u -{D 13D 31/D 33 ) ka (6.15) (6.16) All the unknown coefficients “ C’s “ and “ D’s ” are listed in the appendix. 6.4 Model Description The model for in situ microwave heating, sintering and characterization is depicted in figure 6.2. As given by Marcuvitz [1986], the effect of the ceramic rod in a waveguide is modeled by the electric elements Z 1 2 andZ jj-Z ^- In figure 6.4.1, Z s is the impedance of the variable short, whereas Z l is the impedance of the iris. PI and P2 are the distances from the center of the ceramic rod to the position of iris and variable short, respectively. 96 Iris I I r,z Short Ceramic Rod -| Zll - Zl 2 - H . Z11-Z12 Zl Zs I J Figure 6.2 Equivalent Circuit of a Ceramic Rod in the Cavity 97 The measured reflection coefficient I \ and resulting impedance, Z, can be expressed in terms of ZI, ZS, Z ^ a n d Z ^ -Z ^ , i.e, ( 6 - 1 7 ) where z -Z o a n y i) Zo - Z tan^yPl) ^ v ' (6l9) Zl21 = Zn- Z12 + Z|2(f7“ ~2 ‘^ +r Sc) (Z ll + ZSC) + z otany 2) ZSc“ Z°Z0 + ZStanh(7P2) = z q z s (6-20) ^ 6 2 1 ) In the case when the dielectric constant and loss tangent of the ceramic rod are small, which is true for many ceramics, Zt l~Zl2 is also veiy small and can be neglected in the modeling. Hence Zc ZIc +^L_ + ^L_ ZSC Z 1 2 (6.21) By solving the complex equation, (6 2 2 ) the complex permittivity ( e'-je") can be deduced. 6.5. Impedance of the Iris and the Variable Short The impedance of a iris with circular shape made from a thin metal plate can be modeled by a inductive susceptance element in the transmission line as given by eq. 5.1. 98 Since the iris used here is made of an aluminum plate which is not a perfect conductor, its value has to be characterized. Ideally, the impedance of the short is zero and it reflects all the energy incident on to it. For the variable short used here, which is constructed in a way to prevent sparking at high microwave power, its impedance is not zero. It is necessary therefore that the impedances of both iris and variable short have to be characterized. To do so, a network analyzer system as shown in figure 6.3 has to be used for the measurement. To obtain the impedance of the iris, the set up in figure 6.4(a) is used. Its equivalent circuit is shown in figure 6.4(b). The measurement of the impedance of the short is accomplished by using the set up as shown in figure 6.5(a). The equivalent circuit is displayed in figure 6.5(b). The general procedures for obtaining impedance value of the iris and variable short is to get reflection first and then to convert this reflection coefficient to appropriate impedance value according to the given equivalent circuit. Before making any measurement, the network analyzer system has to be calibrated by using a TRL calibration technique as shown by Ghodgaonkar et al. [1989]. 6.6 Measurement of the Reflection Coefficient Analyzing the characterization model stated in section 6.4, it is easy to see that an accurate measurement of the reflection coefficient in the transmission line is crucial for the entire characterization process. Traditionally, the slotted line has been the instrument for measuring reflection coefficient. The measurement technique with slotted line needs to move detector mechanically, so it is not ideal for automatic system and also the measurement may suffer from such a mechanical move. In this section, a generalize multiprobe reflectometer is used so that direct connection between measuring devices and digital computer via analog-to-digital converter can be realized. The idea of using multiprobe reflectometer to measure reflection coefficient has HP Vectra PC-308 Technical Computer Synthesized Sweeper 0.01-40 GHz HP 8510B Microwave Network Analyzer HP 7440A Plotter S - Parameter Test - Set HP 82906A 0.045 - 40 GHz Printer Port 1 Coaxial to Waveguide Transition l i l t waveguide Coaxial to Waveguide Transition Figure 6.3 A Vector Network Analyzer 100 Iris Zo Zo Waveguide Transition (a) Iris Impedance Measurement (b) Equivalent Circuit Figure 6.4 Measurement of the Impedance of the Iris Zo Waveguide Transition Variable Short (a) Variable Short Impedance Measurement (b) Equivalent Circuit Figure 6.5 Measurement of the Impedance of the Variable Short 101 been studied by Caldecott [1973]. Here the multiprobe reflectometer is a modification of the impedance analyzer introduced in chapter 5. As stated in chapter 5, the impedance analyzer has four detectors evenly spaced with a spacing of 3A,g/8. The details on how to obtain the reflection coefficient by using the multiprobe reflectometer, as shown in figure 6.6, are discussed as follows. Since the operating frequency of the system is known, only the incident power and the phase and the amplitude of the reflection coefficient are not known. There are three unknowns and information obtained from the three probes is in general sufficient to evaluate them. Let V be the incident voltage and Vn the standing wave voltage on the transmission line at the location of the nth probe. Following Caldecott's [1973] treatment, V„ = V{ 1 + pexp 0(9-<}>„)]) (6.23) N 2 = V2 { 1 + p2 + 2pcos ( 0 - (J)„)) (6.24) where p and 0 are the magnitude and the phase of the reflection coefficient of the load and <|)n is the phase shift corresponding to the distance from the probe to the load and back, where <(>„ is taken as positive and must be accurately measured. The power at each probe position is linearly proportional to the I Vn I2. Therefore, P„= P { 1 + p2 + 2pcos ( 0 - <>„)} = P ( 1 + p2 + 2p (sin0sin <(>„ + cos0cos<t>n) } (6.25) let A = 2Pcos 0 B = 2P sin 0 D= P ( 1+p2 ) P n = |V „ P (6.26) Source load Figure 6.6 Three Detector Reflectometer. 103 After some mathematical manipulation, the final results are 0 =arctanfiA (a 2 " p B2) on 2P 4- (6.27) 2 = A-( d + ( d 2 - a 2 - b 2 £1 (6.28) (6.29) The DC voltage outputted from detector and input microwave power satisfies the square law relationship when input microwave power is small. For the experimental work performed here, the detectors are not operated in such a square law region because high microwave power has to be used to heat the ceramic rod in the cavity. It is therefore necessary that the relationship between output DC voltage from detector and large input microwave power has to be established. To do so, a dummy load which has the same impedance as the transmission line is attached to the reflectometer. Since the attached load is matched, there would be no reflection, i.e. p = 0 and the power at each detector position is same. By adjusting input microwave power to the transmission line and measuring the corresponding DC voltage form each detector, it is easy to establish corresponding relation between input microwave power and DC voltage from each detector. Although it is necessary to carry out such a calibration every time the characterization process is carried out, the general trend of the DC voltage and input microwave power is depicted in the figure 6.7. It shows that the relation is cubical. To acquire desired reflection data, the measured DC voltage output results will be transformed to the power by using that established relation. Subsequently, those transformed data will be substituted to the equation 6.28 to get reflection coefficient 6.7 Effect of Iris on the Reflection Coefficient Measurement In the in situ microwave heating and characterization set-up, the impedance analyzer is located immediately ahead of the rectangular cavity which is excited by a circular iris. To 104 DC Voltage From Detectors { V ) 0.4 0.3 0.2 u— •— **" 0.1 detector 1 detector 2 detectors detector 4 0.0 0 1000 2000 3000 Input Microwave Power ( w ) Figure 6.7 Input Microwave Power vs. Output DC Voltage from Detector 105 satisfy the boundary condition, higher microwave modes have to present in the region of the iris. Those modes are the so-called evanescent modes, i.e., they will be attenuated along transmission line. Since impedance analyzer is just ahead of the cavity, the effect of those modes need to be considered. According to the microwave theory, the cutoff frequency for for both TE and TM wave is (6.30) the wavelength at the cutoff frequency is t, - A. (6.31) The corresponding propagation constant is therefore Yg = Vkc2 ' k2 ■ w (6.32) w So the electric field will have the form E = Ebe'Y«z (6.33) and the time average power has the form p = p0 e-2Y*z (6.34) Few higher modes which could effect the measurement are listed in table 6.1. According to the dimension of the impedance analyzer, first crystal detector is 8A,g/3 at 2.45 GHz. So the possible effects of the higher modes are minimum. Hence, the value obtained from the crystal will not be biased from those degenerated modes. 6.8 G reen C eram ic Rod P reparation Green ceramic rod is needed for performing in situ microwave sintering and characterization. Such a green ceramic rod is rarely available in the market due to its fragile 106 nature in the green state. On the other hand, the desired size and particular composition make it even harder to obtain green ceramic rod samples. Therefore, it is necessary to find a way to make such a specimen. In general, green ceramic rod can be made with various processing methods, such as pressing, extrusion, slip casting and injection molding, etc. Among those processing methods, extrusion has proven to be an economical and often a necessary way to produce a large piece of ceramic ware of either regular or irregular cross section. With this consideration and the availability of the processing machine, the extrusion method is adopted to make the green ceramic rod. Extrusion is the shaping of the cross section of a cohesive plastic material by forcing it through a rigid die. Products formed by extrusion include structural refractory products, hollow furnace tubes, honeycomb catalyst supplies, transparency alumina tubes for lamps and flat substrates and tile products. In the extrusion process, the feed material is usually plastic which is commonly formed by directly batching and mixing the raw material and additive in a high shear mixer. Since the green alumina rod is needed in the experiment, alumina powder with small particle size is selected. The alumina powder used is Baikalox CR30 alumina with average particle size of 0.1 micron. The powder has a major alpha phase above 85%. The content of sodium, potassium, silicon, iron and calcium are less than 40PPM, so the purity of alumina is better than 99.99%, Alumina powder doped with 0.3% wt. MgO was ball milled in methanol for two hours in a plastic jar. The mill ball used is made of zirconia. The mixed slurry is then dried in a glass tray for about 12 hours at a temperature of 70°C. To make the plastic body for the extrusion processing, the alumina powder must be mixed with certain binder so that the slurry would have desired rheological properties. In this study, the composition of the binder and the ceramic powder is taken from the research results of Bruch [1972]. In their research study, similar particle size of alumina powder were used. Therefore, the 20% wt. PVA ( polyvinyl Alcohol) is first mixed with distilled water in a glass beaker. Then 50 vol.% binder and 50 vol.% of alumina powder are mixed in a shear mixer. The mixer has two mixing heads. The rotation speed of the mixing 107 Thble 6.1 Characteristics of Non-Propagating Mode in the Waveguide Frequency k (m m ) Power Field Y(l/mm) Attenuation (db) Attenuation (db) TE20,TEoi 109.2 0.02599 38.47 19.23 TEn.TM n 97.67 0.03877 25.79 12.89 TE21.TM21 84.28 0.05406 18.49 9.24 72.8 0.06938 14.41 7.20 TE30 108 heads is 20 rpm. To reduce the heat generated by the friction between the powder and the mixing heads which are made of stainless steel, the mixer is also cooled by pressured air. After the powder and binder are mixed together for about two hours, the mix can be fed to the extruder. Samples are collected through a glass tray. The relative density of the green alumina rod is about 41%. 6.9 Experimental Set* up and Characterizing Procedures A schematic diagram of the automated dielectric characterization system is shown in figure 5.1.1. As a main controller, the combination of a HP9826 personal computer and a HP3494 data acquisition system is used. The output voltages from crystal detectors are related to the input microwave power. The voltage from the temperature controller is linearly related to the temperature. The voltage from the linear voltage displacement transducer indicates the position of the variable short. Those voltages will be collected by a HP3494 data acquisition system controlled by an algorithm written in HP BASIC. The geometric data of the ceramic rod, waveguide dimensions and operating frequency are imputed in advance to the data file. To increase the experimental accuracy, the EIP578 frequency counter is connected to the characterizing system to monitor operating frequency. Upon turning on the microwave generator, the variable short is adjusted to the resonating position so that the microwave power can be absorbed by the ceramic rod efficiently. The output voltages from three detectors, the voltage from the temperature controller and the voltage from the linear voltage displacement transducer were acquired by the data acquisition system at a prescribed temperatures. These raw data were then transformed to the input microwave power at each detector. The magnitude and the phase of the reflection coefficient were found by solving the nonlinear eq. 6.28. Eq. 6.23 is used to set up a complex nonlinear equation to find real and imaginary parts of the complex dielectric constant. When solving a complex equation, the real and imaginary parts of the function F(xj, x2) are assumed to be fj and f2. The complex equation therefore becomes 109 ftfxi, x2) = Re( =0 (6.35) f2(*i, *2 ) = Im ( F{x1? x2)) = 0 (6.36) F ( x i , x 2 )) Let X denote the vector of value ( xl, x 2 ) then, in the neighborhood of X, each of the function f j can be expanded by using Taylor’s series 2 "jr fi( X + 8X) =fi( X )+ £ 8x: + O (8 X2 ) (6.37) j=l dxj By neglecting terms of order dX2 and higher, a set of linear equations for the corrections dX that moves each function closer to zero can be obtained. They are X otij 8xj = f t i=i (6.38) where aij ^ (6.39) 9xj Pi = -fi (6.40) Matrix equations can be solved by Krammer's rule. The corrections are then added to the solution vector, xjiew = x9ld+8xi , i= 1,2 (6.41) and the process is iterated to convergence. 6.10 R esults and Discussion 1. Simultaneous Microwave Heating rod Characterization A Coors AD-998 alumina rod with a diameter of 0.8cm was used in the characterization experiment. Figure 6.8 gives the thermal historical during microwave heating and characterization. To exclude any microwave susceptor effects, the alumina rod was heated with microwave alone. Temperature was measured with a pyrometer. Since the 110 work range of the pyrometer is at 500-1 SOOT, the thermal history under SOOT was not given. Since the pyrometer can only measure surface temperature, the highest temperature measured is only 1200T. During the characterization process, the position of the variable short has to be adjusted so that the microwave cavity can be always in resonant status. Figure 6.9 gives the relationship between position of the variable short and the time. It is easy to see that the position of the variable was gradually moved towards the iris plate with increasing time of heating. The energy was absorbed by the ceramics by properly adjusting the variable short. This is an expected result. As the temperature goes up, so do the dielectric constants of the alumina rod, which essentially shorten the length of the cavity for resonant. The magnitude of the reflection coefficients measures by the impedance analyzer are plotted against the time as shown in figure 6.10. It is seen that the magnitude of the reflection coefficients are very close to 0.3 during entire characterization process. The energy reflected was only around 10%. The dielectric constants, the real part and the imaginary part, are given in figure 6.11. In figure 6.11, the results from Fukushima et al. [1987] were also plotted. The material used here is essentially 94% alumina while Fukushima’s is 92% alumina. It is easy to see that the results obtained here are compatible to that of Fukushima’s. Generally speaking, both the real and imaginary parts of the dielectric constants of AD-998 have smaller value at high temperature than that of 92% at same temperature. At room temperature, AD-998 has a larger value that that of 92% alumina. 2. In Situ Microwave Sintering and Characterization Alumina green ceramic rods made from the extrusion process as stated in 6.8 are used in this experiment. Unlike densified ceramics, green ceramic rod which is not densified will be sintered or consolidated in the microwave heating process. Sintering is usually finished in such a way that the temperature of the green ceramics is raised to half or two-thirds of the melting point of the material and kept there for a certain amount of time; the green ceramics will be densified during this dwell time. In order to be able to get the 1200 1000 u TJBT - 800600- 12 400- I £ 200- 0 10 20 30 Time ( min.) Figure 6.8 Thermal History of Microwave Heating and Characterization Short Position (m) .012^ . 011- .007- Time (min.) Figure 6.9 Position of the Variable Short vs. Time 112 1.0-r* Reflection Coefficient 0 .8- - 0.0 Time ( min.) Figure 6.10 Reflection Coefficient vs. Time 113 2.0 1.5 e " 1.0 0.5 0.0 0 250 500 750 1000 1250 1500 1750 2000 18 AD-998 92%ALUMINA 16 14 e' 12 10 8 0 1000 2000 Temperature ( 0 C ) Figure 6.11 Measured Dielectric Properties of Coors AD-998 114 information about microwave sintering, the whole experimental procedure can be divided into two stages, similar to the conventional process. The first stage is the initial heating where the green ceramic rod is heated to 1200°C. The temperature is measured by using a pyrometer described in section 7.2, the measured temperature is actually the surface temperature. The inside temperature of the ceramic rod would be much higher. A higher surface temperature is found to be harmful to the sample. Figure 6.12 shows the history of the initial heating and subsequent sintering time. At the preheating stage, no densification occurs. The second stage is the sintering stage, where green ceramics will start to sinter. Since the information about the diameter and the density of the ceramic rod is needed, ceramic rods are heated to 1200°C and sintered according to a prescribed schedule. Figure 6.13 gives the densification history of the ceramic rod after it reaches 1200°C. Figure 6.14 indicates the diameter of the ceramic rod at different sintering stages. The variation of the diameter of the ceramic rod will be coupled in to the computer program in retrieving dielectric data. Figure 6.15 demonstrates the variation of dielectric properties of the green ceramic rod during microwave sintering process. The variation of the dielectric constants reflect the microstructure change in the microwave sintering.The real part changes very little in the initial heating stage, where only natural physical property changes of the ceramics are displayed. In the sintering stage, the ceramic material is becoming densified and therefore the dielectric constant is also changing more or less according to the mixing rule. For the imaginary part of the dielectric constant, it changes rather rapidly in the initial stage where it is believed that intrinsic conduction increases loss rapidly with increasing temperature. In the sintering stage, the variation of the imaginary part of the dielectric constant was quite slow compared with the initial stage of the heating. This is due to two mechanisms. On the one hand, the mixing rule is still valid so the dielectric loss would increase with densification. On the other hand, the reduction of the pore size and number would decrease the multiple scattering loss in the ceramics. It is observed in this characterization process that microwave sintering and characterization is more complicated than microwave heating and characterization. In the 1400-r- Temperature ( ° C ) 1200— 1000- - 800— 600— 400 Time ( Min.) Figure 6.12 Microwave Sintering of Alumina Ceramic Rod History 116 Density of the Green Ceramic Rod ( %) 0.60 0.55 - 0.50 0.45 0.40 0 10 20 30 TIME ( m in.) Figure 6.13 Density vs. Sintering Time 40 50 117 Diameter of the Green Ceramic Rod (cm) 0.94 0.92 0.90 0.68 .86 ° Column 3 0.84 0.82 0.80 0 10 20 30 40 50 TIME ( min. ) Figure 6.14 Variation of the Ceramic Rod Diameter with Sintering Time 118 5.0 4.5 i 4.0 3.5 3.0 0 10 20 30 40 50 40 50 TIME ( m in.) 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 10 20 30 TIME ( m in.) Figure 6.15 Variation of Dielectric Properties of the Green Ceramics During Microwave Sintering 119 sintering process, the ceramic rod not only densities in the axial direction but also in the longitudinal direction such that green ceramics are continuously brought to the cavity which creates densification gradients. The heat loss at both ends of the ceramic rod will attribute to the nonuniform densification. Hence, the density of the ceramic rod could not be brought to a higher value. The obtained dielectric data is an average one. There are many factors could effect the accuracy of this characterization technique such as the mathematical model, the impedance of the iris and variable short, the imperfection of the cavity which act as a transmission line and measurement of the reflection coefficient, etc. Although every effort has been made to enhance the accuracy of this characterization technique, there is still room for improvement To be able evaluate this procedure, some standard samples were measured for comparison with known results from Ghodgaonkar et al. [1989]. Table 6.2 gives the results for both Teflon and Quartz rods. It is seen that the results obtained from this technique are quite close to the known results. For the real part of the dielectric constant, it is observed that the error is about 6.8%. The imaginary part of the dielectric constant was not able to compare. 6 .1 1 Conclusions A system has been established for microwave heating, sintering and characterizing of both densified and green ceramics. Mathematical model and experimental procedures are presented for such a characterization. The system is useful in revealing the interactions between ceramics and microwaves. The method is convenient and the speed of characterization has been dramatically increased. The established method is especially promising for characterizing microwave sintering of ceramics. Because of heat loss at surface and both ends of the sample, the temperature measured by the pyrometer at surface is less than that of inside. Therefore, the measured dielectric values are average properties of the materials. The appearances of the temperature gradients at surface and both ends of sample made the characterization process very difficult. The relationships among diameter of the 120 Table 6.2 Real Part of the Dielectric Constant of Teflon and Quartz ^ ' s N Material Methoa'V^ Teflon Quartz This Research 2.158 4.01 Published 2.02 3.82 121 rod green sample, density and time are difficult to obtain. A sophisticated experimental set-up has to be developed, which would be able to rotate and move the sample longitudinally, to guarantee no temperature distribution gradient A good insulation material which is microwave transparent has to be used to reduce the temperature distribution in axial direction. On the other hand, the resonator, iris and variable short have to be finely made so that their properties are closer to that of the ideal elements. 122 Chapter 7 APPLICATION OF MICROWAVE ENERGY TO CERAMIC PROCESSING 7.1 Introduction The use of microwave energy for the processing of ceramics is a relatively new concept which is still very much in its infancy. Preliminary investigation has shown that it is applicable to drying, slip casting, calcination, sintering, joining, plasma assisted sintering and chemical vapor deposition. Its potential is considerable since a number of very distinct advantages have been claimed for the technology. The volumetric nature of power deposition and the subsequent inverse temperature profile appears to offer four main advantages over the conventional sintering. Those may be summarized as i) the possibility of heating large shapes very rapidly and uniformly; ii) a reduction in thermal stresses which lead to cracking during processing; iii) the potential for sintering ceramics “from the inside out”; and iv) since the absorption of microwave energy varies with the composition and structure of different phases, selective heating is possible, creating the possibility of generating new structure with useful properties. A further possible benefit, which still requires verification, is the possibility that microwave radiation results in “ non-thermal” effects during the processing of materials. With respect to the sintering of ceramics, this has manifested itself as apparent reduction in activation energies for the atomic movement processes involved. The evidence for this phenomenon consists of reduced time and/or temperature requirements; however, the results are as yet inconclusive since there are still many difficulties associated with temperature measurement during microwave processing. The two principle reasons for this are i) the effects of microwaves on thermocouple-they induce currents, and ii) the existence of the inverse temperature profile which needs to be corrected by setting up proper insulation scheme. Such an inverse temperature profile means that the surface temperature, which most temperature measurement devices have recorded, is lower than the internal temperature, which controls most diffusive processes. 123 Nevertheless, there is now evidence from a growing number of sources that activation energies might be affected in some ways. In lieu of the practical needs in applying microwave heating to ceramic processing, this chapter will be devoted to the discussion of some fundamentals in microwave processing, such as insulation and temperature measurements, and to giving some practical applications. The insulation schemes used in microwave processing to guarantee uniform temperature distribution are reviewed first. The principles of temperature measurements with pyrometer and thermocouple are stated. Special attention is paid to the temperature measurement with thermocouple in a single mode cavity used in this thesis research for microwave heating. In doing so, the effect of microwave on the temperature measurements is revealed. Finally, microwave energy is applied to binder burnout for ceramic tape and sintering alumina/c-zirconia composites. The results show that microwave processing has many advantages over conventional processing. 7.2 Insulation Considerations in Microwave Processing For microwave processing of ceramics, uniform electric field distribution, or practically no temperature gradient, is needed to insure uniform microstructure in the entire ceramic product. By analyzing the mechanism of microwave and conventional heating, it is not difficult to see that using microwave alone will not guarantee a uniform temperature distribution throughout the sample because of the heat loss due to radiation and convection at the boundary. Conventional heating will slow down the heating rate because ceramics are not good heat conductors. Naturally, the hybrid heating technique is considered to provide initial and uniform heating during microwave processing. Levinson [1972] suggested a method for heating ceramic article that is not self heating in a microwave furnace by surrounding the articles with finely divided particles of materials that do couple strongly with microwave fields. In particular, he cited the use of finely divided carbon powders, iron ore, magnetite, and radio-frequency ferrite powders. Paterson [1975] dealt explicitly with the problem of “runaway” heating of material when 124 using microwaves. In his case, the material used is nylon, which exhibits “a rapid change in the rate of heating takes place beyond a specific temperature with the rate of temperature rise being exponential.” The poor thermal conductivity of nylon coupled with the “runaway” heating behavior leads to localized melting and generally poor control of thermal processing. The solution to this problem proposed by Peterson was to coat the nylon article being heated with a second material. Initially, the coating is heated preferentially to the nylon by microwave. Because the coating does not undergo thermal runaway, heating is accomplished in a controlled manner. Methods specific to the heating of ceramics were also proposed in the last decade. Nishitani [1979] reported incorporating lossy, conductive particles in a dielectric material to effect microwave heating. The only requirement he cited is that the bulk conductivity of lossy material must be > 103£}/cm at high temperature and that the diameter of the particle must be within S to 10 time the skin depth of the bulk conductor. Sutton and Johnson [1980] reported additional findings using “noncoupling” oxides and “strong” coupling materials. They claimed a range of additive between 1% and 90%, which covers a much large range of additive than that claimed by Nishitani. Holcombe and Dykes [1990] worked specifically with zirconia, yttria and alumina and combined both external and internal indirect heating methods. They described the heating of the ceramic article as occurring in three stages. Initially, the microwaves couple to the zirconia insulation surrounding the article, which transfer the heat to the part by conduction; Holcombe and co-workers called this the “electric blanket effect” and referred to the zirconia insulation as a microwave “Pump.” At 700°C to 1000"C, the second stage of heating is initiated. The third stage of heating starts when the major-phase material begins to couple, which for most ceramics occurs above 1000*C. Recently, Janney et al. [1992] used a so called “picket fence” arrangement for sintering zirconia. They argued that a “picket fence” provides uniform field distribution and uniform heating. The arrangement of the picket fence consists of five SiC rods surrounding the zirconia part to be microwave sintered. Zirconia bulk fiber is placed between and around the parts, and the entire construction is enclosed in an alumina fiber crucible. They further said that such an arrangement would facilitate 125 controlled indirect microwave heating at low temperature and controlled direct heating at high temperature. The application of a hybrid heating technique was extensively investigated by Arindam et al. [1990]. The arrangement used by Arindam is similar to Janny’s. The ceramic sample is surrounded by the microwave transparent insulation material. Several SiC rods are placed around the whole structure. By using this kind of construction, they found that microwave hybrid heating results in an improved parity in temperature across specimen cross-section vis-a-vis conventional fast firing and stand alone microwave sintering. This enhanced parity in temperature is said to be responsible for the better microstructure homogeneity and improved mechanical properties relative to the conventional fast firing. They also found that for large ceramic specimens, microwave hybrid heating has many advantages over conventional fast firing and stand-alone microwave sintering in terms of microstructure uniformity and mechanical properties relative to smaller sample. Following the above analysis and modeling results of Chapter 3 and Chapter 4, the insulation scheme used in this thesis research is shown in figure 7.1. 7.3 Temperature Measurement 7.3,1 Introduction Temperature measurement is an important issue in microwave processing of ceramics. An accurate measurement of temperature is vital for correctly assessing the effects of microwaves on ceramic processing such as enhanced diffusion, lowering sintering temperature. Currently there are two measurement techniques employed by researchers in the microwave processing research community. One is to use a pyrometer, which is primarily dependent on the radiation of the infrared light of the ceramic body at high temperature. Another is to use a regular thermocouple with proper electromagnetic shielding. Because both radiation and microwaves could affect those measurements, it is important to understand those measuring mechanisms so that proper correction can 126 Insulating Material Microwave Susceptor Ceramic Sample Thermocouple Insulating Material Figure 7.1 Insulation Scheme for Microwave Processing 127 be made for obtaining accurate temperature measurements. 7.3.2 Pyrometer Measurement The energy radiated by any object at a particular wavelength is strongly dependent on the absolute temperature, a dependence described by Plank’s equation. A measurement of emitted radiation from a target at certain defined wavelengths allows one to make a calculation of absolute temperature. Typically, an optical filter is chosen which limits the radiation collected by the detector to a certain selected range of wavelengths. This allows for tailoring the characteristics of the detector and surface properties of the target material. The pyrometer sensor is essentially a small telescope, designed to gather light from an incandescent target at greater distances and with finer spectral resolution. Major advantages also include simplicity of installation, greater ease of high temperature measurement and again, no microwave interference in the measurement. However, the variation of surface structure during microwave processing of ceramics could change the emissivity of the matter to be measured which in turn effects the temperature measurement. To solve this problem, the emissivity of the ceramics during different processing stages needs to be stored so that correct measurements can be made. 7.3.3 Thermocouple Measurement 7.3.3.1 Introduction In practice, sparking is often encountered when a thermocouple is used directly to measure temperature in the presence of the electromagnetic fields. In this thesis research work, it is found that sparking is more likely to happen in the multimode commercial microwave oven than in the single mode cavity where the thermocouple is inserted into the cavity through a non-radiating hole in the short side of the waveguide wall as illustrated in figure 7.2. The detailed effect of the electromagnetic field on measuring temperature with 128 thermocouple is still quite unknown. Here, some qualitative analysis is given to explain often observed phenomena which may provide some insights to temperature measurements with a thermocouple. 7.3.3.2 The Principle of the Thermocouple A thermocouple is a device that converts a temperature difference into an electromotive force called Seebeck voltage. Thermocouples are usually made of two dissimilar metal wires connected so that one junction is held at a reference temperature and the other junction serves as the temperature sensing device. Thermoelectric measurements then require (1) a sensing element connected through a reference junction by (2) electrical lead wires to a (3) voltage measuring instrument. The operation of the thermocouple measuring temperature relies on the Seebeck voltage, which was discovered by Thomas Johann Seebeck in 1821 in Germany. He discovered that when two wires of different compositions were connected at their ends only, an electrical current would flow in the circuit if one of the connections was heated. Efforts were also made by Jean Charles Athanase in 1824 in France, A.C. Becquerel in 1826 in Paris, and Lord Kelvin in 1847 in England. Henri Le Chatelier, in 1885 in France, proposed a thermocouple with pure platinum as one leg and platinum 90, rhodium 10 as the other, making it possible for a thermocouple to be used for practical applications. In 1927 it was adopted as the sensing element for Range III of the IPTS and is still used for that purpose. 7.3.3.3 Theory o f Thermoelectricity Most of the thermocouples are based on metal conductors, although ceramic conductors are also used. The theory of metallic conduction is a reasonable base for the qualitative discussion of thermoelectric measurement. The band theory of metal is based on the concept of metal nuclei arranged in a 129 periodic crystalline array in such a way that the outer electrons of each atom come so close together that the Pauli exclusion principle requires them to be arranged in a quasi-free electron cloud around the nuclei. The energy levels of these electrons are degenerate; that is, they are collected together in bands of energy so that any electron in the band can have a particular allowed energy state. Because every metal or alloy has a unique electronic and crystalline structure, the allowed energy states and their electronic population will also be unique. Therefore, when two metals come in contact, the electrons in the metal with higher energy will flow into the one with lower energy. This will occur at the junction of the two metals until the excess electrons in the metal of lower energy build up a reverse EMF which, if current is allowed to flow by making a second connection at a different temperature, can deliver heat at the first junction equal to the product of current and EMF; that is the heat equals the electrical energy, or dQ = 7t I d0 (7.1) where dQ is the heat delivered at a junction in time dt by a current I. The Peltier EMF depends only on the temperature and the two junction materials. It is fundamental potential responsible for the Seebeck voltage. Thus the Petilier voltage for two materials, A and B, with junctions at temperature T1 and T2 when current flow is zero, is «ABT i -T2 = {*AB>ri - (7.2) The Seebeck voltage is the net voltage for such a junction. It includes another term, which also depends on the electronic structure of A and B. It depends on the two Thomson voltages, which are fundamental reversible thermodynamic quantities that can deliver heat if current is allowed to flow. The Thomson voltage depends on the way the Fermi energy of each conductor varies with temperature. In simplest form it can be written lomson where a is an empirically determined Thomson coefficient. Then the first law of the thermodynamics requires that 130 dQ r = vTT dO = adT ID 6 (7.3.b) 1 . The direction of the heat flow depends on the direction of current flow, being delivered in the direction the electrons travel. The Seebeck voltage, then, is the sum of two reversible thermodynamic electropotentials at open-circuit conditions: the Peltier potential and the Thomson potential. The latter is the net Thomson potential depending on the difference in Ef for the temperature range imposed. In equation form, VS= VP + VT (7.4) The Seebeck coefficient a s is often called thermodynamic power and is determined, for materials A and B, as dVs = ocab dT (7.5) 7.3.3.4 Thermocouple in the TE103 Cavity The arrangement of a thermocouple in the cavity for measuring temperature is shown in figure 7.2. The effect of the electromagnetic field on the function of the thermocouple can be analyzed as follows. The field distributions in a TE103 mode cavity are (7.6) (7.7) 131 Iris v CeramicRod Variable Short Non-Radiating Hole Thermol Couple Figure 7.2 Measuring Temperature with Thermocouple in the Cavity 132 E y = Ei03y sin ( ^ ) sin (3az) (7.8) according to the boundary condition at an interface, the electrical field which is perpendicular to the conductor could generate electric charge and the magnetic field which is parallel to the metal conductor surface could create surface current These currents may prevent normal measurement of the temperature. In the case of TE103 mode cavity, the magnetic field which is parallel to the thermocouple direction is very close to the minimum position and will create negligible currents on the thermocouple. On the other hand, the thermocouple can be seen as a coaxial probe which will couple electromagnetic energy to the cavity. The thermocouple wire that is the conductor forms the center conductor of the coaxial line. According to the waveguide exciting theory of Montgomery et al. [1948], such a probe can only excite the TE01 mode where the electric field is parallel to the wide side of the waveguide rather than to the short side. However, such a mode cannot exist in the cavity because of the frequency of the microwaves and the shape of the waveguide. By using the reciprocity theory, it is easy to see that such a setup will not couple any energy to the coaxial line because the mode of the cavity is TE103. From another point of view, any waveguide mode that has a nonzero electric field along the probe will excite currents on the probe according to the coupling theory. Since the resonant mode used in the cavity is TE103, there is no electric field parallel to the thermocouple direction. The coupling of microwave energy to the coaxial line is therefore minimum. On the other hand, the electric field distribution has to be disturbed by insertion of the thermocouple which could be seen as a metal wire. Such a disturbance could generate some other modes different from the TE10 mode and have electric fields parallel to the thermocouple direction. Those parallel electric fields could excite electric current on the thermocouple which would effect the temperature measurement Some of the electric fields other than TE10 mode which could be generated in the cavity are listed in the table 6.1. The small leakage observed in the practice suggested that some high mode of microwaves which favor the coupling of the energy to the coaxial line occurs in the cavity to satisfy the boundary condition created by the insertion of the thermocouple. The leakage 133 is relatively large when the cavity is not in resonating mode and will be negligible when the cavity is in resonance. 7.3.3.5 Summary In practice, the single mode cavity will not be suitable for processing large pieces of ceramics which are embedded in certain crucible insulations. In order to avoid the effect of electromagnetic fields on the temperature measurements, a metal foil sheath is often used. Some researchers have suggested the use of ceramic material to cover the thermocouple, so that sparking could be prevented, for it reduces the magnitude of the electromagnetic fields. In conclusion, both the pyrometer and the thermocouple have advantages and disadvantages. On the other hand, according to Meek [1991], correct temperature, measurement can never be made during microwave sintering of ceramics because of the temperature difference between ceramic grain and grain boundary after some experimental observations. However, such an argument was rejected by Johnson [1991] after some simple calculations. At the same time, Johnson’s calculation was not adequate for calculating the field distribution around the contacting region of ceramic particle during microwave heating. 7 . 4 Binder Burnout of Tape-Casted Ceramics by Microwave Energy 7 .4.1.In trod u ction Ceramic materials are good for their ability to resist corrosion and maintain strength at high temperature, and for their varied electrical properties. Recently, plastic forming methods applied to the making of the complex-shaped green ceramic product are receiving a lot of interest. Plastic forming methods include extrusion, injection molding, slip casting and tape casting. In those forming methods, a large volume of binder, up to 50%, has to be 134 added to the ceramic powders to achieve appropriate rheologic properties for processing. This large volume of binder has to be removed before final sintering can take place. Methods of binder burnout include evaporation by Wei et al. [1988], thermal by Calvert and Cima [1990], degradation by Mutsuddy [1987], and solvent extraction by Watson and Smith [1989]. The conventional furnace heating method requires a very low heating rate to complete the binder burnout process. If a high heating rate is used, it will cause a large thermal gradient which may generate cracks in the green sample. A long period of time is often required to remove the last traces of residue. Reduction in time and energy required by the debinding step could significantly enhance the economics and productivity of the manufacturing process. Microwave heating, known to be a volumetric heating process, is the subject of this investigation for binder burnout. In this investigation, both conventional and microwave fired tapes are characterized for their dielectric properties in a waveguide. These dielectric properties are effective for determining completeness of firing and understanding the firing process. The measurement of the dielectric properties can be easily expanded to a free space characterization method for online process monitoring, as shown by Varadan et al. [1991]. 7.4.2 M icrowave Processing Use of microwaves as a heating source was developed after World W ar II. Applications of microwaves to material processing started in 60’s. In the last few years, tremendous interest has been generated in the material research community in the use of microwave energy to process materials such as ceramics, polymers and their composites, as surveyed by Sutton et al. [1988]. Applications of microwave processing of ceramics have shown good promise in such areas as fine microstructure, lower sintering temperature and reduced sintering time, as well as energy savings. However, use of microwave power for the binder bumout process has not received much attention. Harrison et al. [1988] is the only one who tried to use microwave to binder bumout, but the study was not systematic. 135 7.4.2 Ceramic Tape Preparation To make ceramic tape for the microwave and conventional binder bumout experiments, a tape casting technique is used as shown in figure 7.3. Barium strontium titanate was mixed with 28 w t% of binder (B73210 from Polamar-MSI) and ball-milled in a plastic jar with zirconia media for 24 hours. The slurry was then deaired and tape casted. The tape was dried, cut and pressed at a pressure of 35 MPa and at a temperature of 6070°C for 15 minutes. 7.4.3 Characterization Method To understand both microwave and conventional binder bumout processes, the ceramic tapes were characterized at room temperature by measuring their dielectric properties at 15GHz after different stages of firing at room temperature. The samples were heated to various temperature by exposing them to the microwave fields as shown in figure 7.4. Samples were also heated in a conventional furnace and the time history is as recorded in figure 7.5. The tapes are then sized to fit into a rectangular waveguide for characterization. To guarantee intimate contacts between sample and waveguide wall, the edges of the specimen were coated with a thin layer of silver paint. The measurements were done using a HP-8510 network analyzer and the data acquisition system as depicted in figure 6.3. Prior to the measurement, a 2-port TRL calibration procedure as shown by Ghodgankar et al. [1989] was performed. The complex reflection coefficient from the metal-backed sample was measured. The dielectric constant is obtained from the reflection coefficients and the permeability of the material is assumed to be unity. 7.4.5 Results and Discussion The values of dielectric constants of both microwave and conventionally fired ceramic tapes at different stages are plotted in figure 7.6. The change in dielectric constant OROANICS BALL MILL FOR 24 HOURS DE-AIR TAPECAST LAMINATE 35MP» 00-70 «C BINDER BURNOUT Figure 7.3 Ceramic Tape Manufacturing Flow Chart 137 700. 600500o 400- z s 300- <5 o. § 200 - H 100 - 0 30 20 10 Time ( Minute ) Figure 7.4 Temperature as a Function of Time for Microwave Binder Burnout 1000 800 600 400 0 , 200 100 200 300 400 Tim e ( M in. ) Figure 7.5 Temperature as a Function of Time for Conventional Binder Burnout 138 60 conventional m icrow ave Dielectric Constant 50- 40- 30 - 20 0 200 400 600 800 1000 Temperature ( ° C ) Figure 7.6 Dielectric Properties of Ceramic Tapes at Different Temperatures 139 with temperature explains the phenomena during the binder burnout process. Before Bring, the green tape can be considered as a homogeneous mixture of ceramic powders and binder. The binder can be considered as the matrix phase where no connectivity between the ceramic particles exists. The dielectric constant of the green tape can be easily calculated by using a mixing rule. The value of the dielectric constant was around 55. When the temperature of the tape reaches around 200°C, the binder starts to leave the sample gradually. The dielectric constant is the result of the coexistence of ceramic powders, binder and air pores. Since the dielectric constants of the binder and air pores are equal to 2.5 and 1, respectively, the dielectric constant reduces when the binder leaves the tape. At the end stage of binder burnout, the tape contains no binder and the dielectric constant is the compositive property of the ceramic powder with weak connectivity and pores. The weak connectivity between ceramic particles contributes to the large variation of the dielectric properties. After exposure to high temperatures in both the microwave and the conventional heating process, the dielectric constants of the tapes are increased. Comparison of dielectric properties of microwave and the conventionally heated tape shows that microwave heating completes the binder burnout process much faster and earlier than that of the conventional process with no damage to the sample. It is not clear at this moment whether this difference in the results is due to different heating mechanisms or the different temperature at the ceramic boundary and inside the ceramic grain, which is suggested by Meek [1987a] in observation of microwave sintering of ceramics. Meek has suggested that the enhanced diffusion in microwave sintering of ceramics is due to the different temperature distribution at the ceramic grain boundary and inside the ceramic grain. 7.4.6 C onclusion It has been shown here that green ceramics made from the plastic forming process can be prefired by using a microwave heating technique. Dielectric characterization is a useful method of monitoring binder burnout process. Microwave assisted binder burnout 140 was completed at a lower temperature and in a shorter time than that of conventional heating. 7.5 Microwave Sintering o f Al20 3/ c-Z r02 C om posites 7.5.1 Background It has been the object of much research that the tetragonal to monoclinic Z r02 phase transformation can be utilized to increase the fracture toughness of ceramics. In zirconia toughened alumina (ZTA), retention of >10 vol.% metastable, tetragonal Z r0 2 in the alumina matrix is the key to obtain increased room-temperature fracture toughness. However, to retain the metastable tetragonal Z r0 2 it is essential that the Z r0 2 grain size be less than some critical size which is reported to be in the range of 0.5 to 0.8|im. Above the critical size, the Z r0 2 grain size transforms to the monoclinic form. Garvie [1984] has discussed the benefits of having Z r02 present in both the monoclinic and tetragonal forms for cutting tool applications. One method of controlling the volume retention of tetragonal Z i0 2 grains is to partially stabilize the zirconia with MgO or Y 20^. However, the high temperature required for ZTA densification usually results in significant grain coarsening and the inability to control the desired zirconia volume concentration and the phase form. To meet the requirement, numerous processing approaches have been investigated to achieve lower densification temperature and homogeneous distribution of Z r0 2 in an alumina matrix. These methods have included attrition milling by Claussen and Ruhle [1981], colloidal processing by Aksay et al. [1983], chemical vapor-co-deposited aluminaZr02 powders with a surfactant by Hori et al. [1981], co-pyrolyzed solutions by Sproson [1984], hydrothermal reaction of Al-Zr alloys by Somiya et al. [1984], hydrolysis of a zirconium alkoxide in the presence of A12O j particles by Fegley et al. [1985], polymer/powder flocculation by Moffatt et al. [1988], and sol-gel processing by Becher [1981], Most of these powder processing approaches require hot pressing because there is excessive Z r0 2 grain growth at the elevated temperature (>1600) which is required for 141 densification. Recently, Sproson and Messing et al. [1984] used seeded boehmite gels for the controlled synthesis and relatively low-temperature sintering of zirconia-toughened alumina was realized. 7.5.2 Microwave Sintering Sintering of zirconia toughened alumina with microwave energy has been studied by several researchers. Kimrey et al. [1990] used both 2.45GHz and 28GHz microwave furnaces to sinter alumina/zirconia system containing 10-70wt.% Z r0 2- It is found that microwave enhanced diffusion was more easily observable at higher frequency. Microwave sintering needs as low as 500°C compared with the conventional sintering process. No mechanical properties were reported with both conventional and microwave sintering results. They concluded that the microwave effect is the function of the frequency and that the reduction in sintering temperature results in a significant decrease in the grain size in the microstructure. Patil et al. [1991] have used microwaves to sinter alumina powders mixed with 15 vol.% zirconia in a single mode cavity. In their study, the general results indicate that microwave sintering is superior to the conventional sintering; sometimes it is even comparable to the high gas pressure sintering. However, the temperature measurement was very questionable in his study and the mechanical property comparisons between results of both conventional and microwave sintering were not given. Hence, microwave sintering can be used to substitute for the gradually popular hot isostatic press sintering method which is effective but expensive for densifying materials that are usually difficult to sinter with conventional methods. In the following study, both conventional and microwave methods are used to sinter alumina/zirconia composites containing Owt.%, 4wt.% and 10wt% zirconia. The densified samples are characterized with scanning electron microscope to observe microstructure. The density, hardness, elastic properties and diametral compression strength are obtained for comparison. 142 7.5.3 Sample Preparation Ceramic materials in the Z r0 2/ A12O j system containing Owt.%,4 wt.% and 10wt.% Z r0 2 were chosen for this study. Proper combination of alumina, zirconia and 0.2wt% MgO powders were mixed together by the ball milling process in a plastic jaw using zirconia ball for 3 hours in the isopropanol. The slurry was sonicated for 10 minutes and then dried in a glass plate. The dried cake was crushed by using a food processor. Specimens were uniaxially pressed in a cylindrical die, which has a diameter of 1.27-cm, at a pressure of 30 MPa. The green specimen has a cylindrical shape and has a diameter of 1.27cm and a height around 0.4cm. The specimens were prefired in a electrical furnace at 600°C for two hours to bum out binders. The conventional sintering was performed in an electrical furnace according to the sintering schedule described in figure 7.7. The conventional sintering time and temperature were taken from French et al. [1992]. The microwave sintering was performed in a single mode high power microwave heating device introduced in chapter 5. The microwave sintering schedule is shown in figure 7.8. Such a sintering temperature was taken to compare with a commercial product. 7.5.4. Mechanical Properties In this section, mechanical properties were obtained to compare conventional and microwave sintering results. Table 7.1 gives sintering time, total cycling time, sintering temperature, and final density, as well as elastic properties. 1. Density Sample density is obtained by using Archimedes’ rule. It is obvious that microwave processing needs lower temperature, and shorter times and reaches a higher density than that of conventional processing method. Therefore, microwave processing enhances diffusion and lowers sintering temperature. Since there are only a small quantity of samples used in both microwave and conventional processing, the economic aspects of 143 Temperature ( ° C ) 2000 1000 - 0 10 20 Time ( hour ) Figure 7.7 Conventional Sintering Schedule Temperature ( 0 C ) 2000 temperature 0 100 200 Time (min.) Figure 7.8 Microwave Sintering Schedule 144 Table 7.1 Comparison of Results from Microwave and Conventional Sintering Samples Sintering Method Sintering Time( hour) Sintering Temp. ( °C ) Relative Density (%) Total Cycle Time (hour) A A mw conv. mw conv. 2 3 2 9 1530 1600 1530 1600 97.91 96.63 2.66 16.6 2.66 2.66 16.6 365.2 387.3 393.5 277.9 282.6 .228 0.262 0.302 Young's (GPa) 376.8 Modulus Poisson's 0.199 Ratio ZTA4 ZTA4 ZTA 10 ZTA1C mw 2 conv. 9 1530 1600 99.99 99.01 99.98 99.89 16.6 0.242 0.305 145 the two processing techniques is not given here. 2. Elastic Properties Longitudinal and transverse elastic wave velocities were measured from polished samples made from either conventional or microwave sintering. The elastic properties (Young's modulus and Poisson ratio) were then calculated from these values using the standard relationship from Papadakis [1967]. Table 7.1 shows the behavior of Young’s modulus for the composites. It can be seen that Young’s modulus decreases with increasing c- Zr02 content. It can be seen that composites with 0 % wt.of Z r0 2 from both microwave and conventional processing techniques have a lower value, which is probably due to a slightly higher value of porosity content in those samples. For ZTA4 and ZTA 10, the samples produced from microwave processing have slightly smaller values of Young’s modulus and Poisson ratio. 3. Hardness For each sample, 8 to 10 room-temperature hardness measurements were made using Vickers indenter with 2.0392kg load and 10-s dwell time. Hardness values were determined from the equation H = 2PSin(e/2)/d2 (7.9) where H is the hardness number, P is the indentation load (kg), 0 is the angle between opposite faces of the indenter which is 136° here, and d is the indentation diagonal length (mm). An indentation load of 2.0392kg was chosen to produce impressions significantly larger than the grain size, thus providing an adequate sampling of the microstructure. Table 7.2 shows the behavior of the composite samples. It is seen that in general products made from microwave processing have produced higher hardness than that of conventional methods which may be due to a slight higher density of the composites produced from microwave processing. 4. Diametral Compression Test The diametral compression test is based upon the state of stress developed when a cylindrical specimen is compressed between two diametrically opposite surfaces. This ideal 146 loading, shown in figure 7.9, produces a biaxial stress distribution within the specimen. The stresses at any point in a cross-section can be calculated by the elastic theory. Of the primary interests here are the maximum tensile stresses, which act across the loaded diameter and have the constant magnitude: where: a = maximum tensile stress P= applied load D= specimen diameter, and t = specimen thickness During the course of testing, proper load is obtained by placing a narrow pad of suitable material between the specimen and the loading plates. The pad will be soft enough to allow distribution of the load over a reasonable area and yet narrow or thin enough to prevent the contact area from becoming excessive. The average maximum tensile stresses for different specimens are shown in figure 7.10. It is seen that microwave processed samples of ZTA4 and ZTA 10 have a lower diamentral compression strength than that processed from conventional methods; even microwave processed samples have a higher density. The differences may be caused by high heating rate or local densification due to non-uniform temperature distribution, which could cause internal crack during microwave processing. S.Characterization - SEM Observation Both conventional and microwave sintered samples were polished to lum finish using standard metallographic techniques. Polished surfaces were thermally etched in air at 1450°C for 90 minutes to reveal the grain boundaries. Scanning Electron Microscope(SEM) was used to observe the microstructure. The specimens were sputter coated with Au-Pd to prevent surface charging during SEM observation. Figure 7.11 to figure 7.14 gives the microstructure of the both conventional and microwave sintered 147 Table 7.2 Vickers Hardness of the Sintered Ceramics Sample Processing Method Vicker's Hardness A MW 1961.97 A CONV. 1883.22 ZTA4 MW 1953.06 ZTA4 CONV. 1867.22 ZTA10 MW 1786.92 ZTA10 CONV. 1607.45 148 1 T Diametral Compression Strength ( M P a) Figure 7.5.4 Diametral Compression Test 400 300 Microwave Conventional 200 100 Weight Concentration of Zirconia (%) Figure 7.10 Diametral Compression Strength 149 results. It is easy to see that microwave processing yields a more fine and uniform microstructure than that of conventional processing. 7.5.5 Conclusion It is easy to see that microwave sintering of ceramics is applicable to the ceramic processing. Although it is still not known how much microwaves have effected ceramic densification and microstructure development during ceramic sintering, it is shown here that microwaves have helped to reduce the sintering temperature, shorten sintering time and yield fine and uniform microstructure. Since microwave sintering of ceramics is a new field, the effect of microwave sintering on the strength of the final product is not known and there are not many data available for comparison. Therefore systematic research is needed for providing the right schedule to achieve required product quality. 150 Figure 7.11 Micrograph of Conventional Sintered Alumina-4%Zirconia [ 00F 0 11 Figure 7.12 Micrograph of Microwave Sintered Alumina-4%Zirconia 151 Figure 7.13 Micrograph of Conventional Sintered Alumina-10%Zirconia Figure 7.14 Micrograph of Microwave Sintered Alumina-10%Zirconia 152 Chapter 8 CONCLUSIONS AND FUTURE WORK Microwave processing, where microwave energy acts as an energy source, has been successfully applied to paper manufacturing and food processing, etc. Its application to the ceramic processing is still new. Although preliminary research has shown that it provides some nonthermal effects when it is applied to the sintering of ceramics, the mechanisms of microwave heating, especially the nonthermal effects, are still not fully understood. Research work has been undertaken in many laboratories, but a systematic research effort is still lacking. Most of the research results which have been published are on very small scale. It is necessary for the entire process to be modeled, characterized and subsequently applied to practical application. The focus of this study is therefore directed to those three areas. The achievements and conclusion made in this thesis research are summarized as follows: 1.The equations to describe microwave processing of ceramics are derived. In the description, the ceramics under microwave sintering are treated as deformable dielectrics. The treatment o f ceramics in the microwave sintering as deformable dielectrics enables considering the effects of moving charges and currents. Therefore, a complete description of interactions between microwaves and ceramics is obtained. To predict temperature distribution in the ceramics, a thermal diffusion equation has to be added the picture. To account for the sintering process where ceramics are densified, a mass diffusion equation is also needed. It is concluded that those equations can be solved by using a finite element method once all parameters are determined through either experiment or postulation. Those experiments will be very time consuming. It is hoped that such a formulation can be used in future studies. 2. In order to have some insights about microwave heating, the microwave heating of a ceramic slab is modeled. The modeling results show that ceramics can be heated with microwaves to the desired temperatures for processing. It is also found that nonuniform 153 heating can be resulted from nonuniform power absorption and heat loss at boundaries. Therefore, it is important to design a microwave applicator which would distribute microwave power evenly into the samples. Insulation is also extremely important to ensure that heat losses at boundaries are minimum. More work is also needed to model microwave heating of samples of complicated shape and structure where insulating materials are used. To do so, a Finite Difference Time Domain method is used to model a single mode cavity used for ceramic processing and characterization. The results show that the FDTD method is applicable to the modeling of complicated structures comprised of the ceramic sample, insulation material and microwave susceptor. The electromagnetic field distribution in the cavity and the power absorption by the ceramics sample, insulation material and microwave susceptor are obtained. Those results are useful for the designing of industrial microwave processing. 3. For the purpose of experimental microwave processing and characterizing of ceramics, a single mode high power microwave heating system is established. It can be used not only to process ceramics but also to do in situ microwave heating, sintering and characterization. The acquisition of the data is computerized with a computer and a digital and analog converter. This system can be further improved so that the operation of the variable short and variable iris would be automated. In doing so, critical coupling that is essential to the energy efficiency can be readily obtained. 4. To understand both microwave heating and sintering, the established single mode microwave heating system is used to characterize microwave heating and sintering of ceramic rods. It is found that material properties can be revealed by using such a system. The technique of in situ microwave sintering and characterization is extremely interesting to investigate the microwave sintering process. The results tell that the real part of the dielectric constant indicates the densification more than the imaginary part, while the imaginary part of the dielectric constant mainly shows that the material properties change with temperature. The mechanisms of dielectric property changes during sintering can be generally explained by mixture theory and multiple scattering theory. However, more rigorous work is needed to detail those mechanisms. On the other hand, it is concluded that 154 the microwave components, such as cavity, iris and variable short, have to be carefully made so that they will behave like the ideal elements. 5. To demonstrate the applicability of microwave heating to the ceramic processing, the single mode microwave heating system is used for binder burnout and sintering of ceramics. The results show that microwaves bum out binder in less time and lower temperature than conventional ones. In using microwaves to sinter alumina/c-zirconia composites, it is found that microwave sintering results in lower sintering temperature, higher density and shorter time than conventional sintering. The initial intent to model, characterize and apply microwave energy to the ceramics processing is accomplished, although there are problems to be solved before this technique can be fully applied to the industrial application. 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Yee, K.S., “Numerical Solution of Initial Boundary Value Problems Involving Maxwell Equation for Isotropic Media,” IEEE Trans. Antennas Propagat., Vol. AP-14, pp. 302-307, May, 1966. 172 APPENDIX EXPRESSIONS FOR C ; AND Du The expression for the Cj’s and Dy’s are p _ -Qo (A.l) ( r f ■l)k» _ _ - Qi(K/k) (A.2) 02 (A.3) (e?- l)k a C3 = ■~ ^3 3 ( e r - l) k a [> ' trfl (A.4) where (A.5) p j„ (a )J„ .,(p ) • a Jn-l (a)Jn (p) for n= 1,2,3,4. The Djj’s are given by Doo = -(e; - l ) '1 <X-2 { C^r-(0,0) + i-Q o t P Y0 (ocPi (p) - a Y, (oc)Jo(p)] ) (A.6) D „ = - (ej - 1)'•' o J {(fij* [ | + ( f f ( log, [ ( 2 w ) -2] + )] + £ (A.7) |,-(ri2 -w 2 P -3 p n=3, odd +1 q , [ p Y! (o)Io (p) - a Y0 (a(J, (p) ] | D33 = -(e? -1} 1 a-2 ( ( ^ ) ? [ r„ (0 ,0 ) + (A ) r ^ o , o ) ^ IV ^O .O ) (A.8) -^•Q 3 [ p Y 3 (ot)J2(p) - a Y2( a p 3 ( p ) ] ) 173 (A.9) 0,3=031=Ernwlr“’(0’0)+©w H (A. 10) 0,3 =°31 =(e M ^ [r“ (0'0) +© W (A. 11) H f e ) 4 < 0 ,0 ) = W'2 [ I + ( f f ( loge [ ( 2 w ) -2] + (A. 12) + X n - (n 2 - w 2)1/2- 2 £ ) n=3, odd ^ , , ( 0 . 0 ) = w '2 [ 1 - ( if C log, [ ( 2 w ) -2] + > (A. 13) + £ n - n2(n2 - w2)"122 + - ^ 11=3, odd 11 W I W 0 , 0 ) = » - 4 ( M . + »d<log.2 -2 )-1 7 /1 6 (A. 14) + ^ n^n2 - w2)-122 - n3 + w3 + ^ d ) n=3, odd ( * ) W O , ® = w •M § - ^ 2 ( S f ( 2- log,2 ) (A. 15) + X n5_ n^n2 - w2)'1/2 - 111 w2 - n & i n=3, odd 2 8 3w6i 48n VITA Xiang Dong Yu was born in Nanton, China. He graduated from Dong An High School, QiDong, China, in July 1977. He received a Bachelor of Science degree in January 1982 in the Mechanical Engineering Department at the East China Institute of Technology, Nanjing, China. After spending three and a half years as an instructor in his Alma Mater, he came to the United States to pursue further education in August 1985. He studied in the Mathematics Department for one year and subsequently transferred to the Civil and Engineering Mechanics Department at the Southern Illinois University at Carbondale, IL. He obtained his Master of Science Degree in May 1988. He came to Penn State in August 1988 and joined the Research Center for the Engineering of Electronic and Acoustic Materials. Since then, he has been a research assistant in the Department of Engineering Science and Mechanics. Xiang Dong Yu is a member of the American Ceramic Society, the American Society of Mechanical Engineering, the Society of Advanced Material Processing Engineering, and the Society of Engineering Education.

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