r:; ."l MICROWAVE HEATING OF POLAR LIQUIDS AND SOLIDS BY Mark Christopher Finzel A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1985 DEDICATION To all soldiers, Raiders, and spaceheads past, present, and future, and to all symbiotic concepts of man and nature. "At one with the knowledge and magic of the source, attuned to the majesty of music, we march as one with the earth." - Jon Anderson, 1974 ii ACKNOWLEDGEMENTS I would like to thank Dr. Martin C. Hawley for guidance, financial support, and project oversight, as well as Dr. Jes Asmussen for an idea or two regarding electrical engineering. Copious thanks are also given to Mahmoud Dahmeine for designing and testing the experiment circuit and Bea Archambeault for striking selectric keys with precision. iii TABLE OF CONTENTS INTRODUCTION 0 O O O O O O O O O O O O C O O O I. II. III. Composite Materials, Thermosetting Resins, and Microwave Dielectric Heating . . . . . A. Composite Materials. . . . . . . . . . B. Curing of Thermosetting Resins and Composite Processing . . . . . . . . . C. Microwave Heating of Thermosetting Resins D. Desired Characteristics of Materials for Microwave Heating . . . . . . . . . . E. Advantages of Microwave Heating. . . . Dipolar Relaxation in Unassociated Liquids A. Debye Theory for Relaxation in Polar Liquids. . . . . . . . . . . . . . . . B. Activated Complex Theory . . . . . . . C. Debye Equations. . . . . . . . . . . . Dipolar Relaxation in Polymer Systems . . A. Dielectric Relaxation Processes in Polymers . . . . . . . . . . . . . . . B. Theory of Dielectric Relaxation in Polymers . . . . . . . . . . . . . . . C. The "Universal" Dielectric Response. . Page 10 11 13 13 16 18 21 21 22 26 D. Relaxation in Nylon 66 . . . . . . . . IV. Resonant Coupling and Cavity Diagnostics . A. Absorption of Electromagnetic Energy by Lossy Dielectrics in the TMO12 Mode. B. Significance of Measurements of Q and E 2 o o o o o o o o o o o o o o o o o rw V. Description of Experimental Apparatus. . . A. Microwave Circuit and Resonant Cavity. B. Low- and High-Power Measurements . . . VI. Modeling of Dielectric Relaxations . . . . A. Relaxations in -OH containing Liquids. B. Modeling of Relaxation in Nylon 66 . . VII. Results and Discussion . . . . . . . . . . VIII. Conclusions and Suggestions for Future Work . . . . . . . . . . . . . . . . . . . APPENDICES A. Calculation of AH? for H20 . . . . . . B. Finite Difference Equation for Laplace's Equation in Cylindrical Coordinates and FORTRAN Program for Temperature field. REFERENCES 0 C O O O O O O O O C O O O O O O O 31 33 35 35 37 47 47 56 62 70 72 76 Abstract Microwave Heating of Polar Liquids and Solids BY Mark Christopher Finzel A resonant microwave cavity was developed to couple microwave power to a.sariesof dielectrics. The cavity operated in the TM012 mode so that the maximum amount of power possible could be coupled to the dielectric. Coup- ling efficiencies (percent input power absorbed) ranged from 90% for ethylene glycol to l8% for teflon. Power absorbed was found to depend on both the dielectric con- stant and the complex permittivity. A model based on ACT Theory and internal molecular rotation successfully predicts the dielectric absorption of hydroxide-containing liquids, particularly small glycols. The dielectric spectrum of Nylon 66 was also modeled using ACT Theory and parameters based on cooperative relaxation of dipoles in polymers. Additionally, a finite difference method was used to determine temperature distributions in a nylon rod. Both models for liquids and solids predict the change in coupling efficiency with time and temperature correctly. INTRODUCTION: It is well established that radio and microwave fre- quency radiation is absorbed by polar liquids and solids. Large amounts of dielectric data have been collected for a wide variety of materials.[l'2’3'] This absorption of electromagnetic power by polar liquids and solids can be used to heat these materials. This is known as dielectric heating and is a method of affecting large increases in temperature in materials in a short period of time as any- one who has used a microwave oven for cooking purposes knows. The rapid heating possible using dielectric heating can be used for a variety of specialized applications. One such application is processing of composite materials. Com— posite materials are made of layers of ceramics, metals, fibers, and various polymers and have superior mechanical, chemical, and electrical resistance characteristics. The layers are joined together by adhesives (usually thermo- setting resins such as epoxies). Since many thermosetting resins include polar groups they are candidates for dielectric heating. The work included herein is an exploratory invest- igation of dielectric heating in liquids and solids in order to develop a data base and a fundamental understanding of the absorption of microwave radiation by liquids and solids, and to determine how much power can be coupled to various dielectrics. First, a description of composite materials, thermo- setting resins, and microwave heating is given in order to show the advantage of the concept as it relates to composite processing. A summary of the important theoretical models for dielectric absorption is then given for both polar liquids and polymer systems. A description of the experi- mental system and measurements is then given, followed by a description of the modeling used to describe the fundamental factors influencing the experimental results. Finally, the experimental and modeling results are compared. Since this work was carried out in order to develop a data base for a proposal to the Defense Advanced Research Projects Agency (DARPA), suggestions for future work based on the enclosed information in this thesis are provided. I. Composite Materials, Thermosetting Resins, and Micro- wave Dielectric Heating. This discussion will focus on composite materials in military applications as well as the properties of the thermosetting resins used in the adhesive layer that are applicable to microwave heating. A. Composite Materials Composite materials are a broad classification of materialstxx>extensive to be discussed in detail here. Generally speaking, they are composed of layers of materials including metals, fabrics, and ceramics which are joined together by an adhesive layer. These composite structures exhibit mechanical strength, temperature resistance, and chemical and electrical properties superior to those of the component materials. One application of these materials is for ceramic com- [4] posite armor. This type of personnel and vehicle armor has been in use since the end of World War II. It involves the use of a ceramic (alumina, boron carbide, and silicon carbide have been used) backed up by a fabric layer (often a glass-reinforced plastic) Adhesives used include epoxy and polyester resins. If the fabric back up layer does not pro- vide adequate support, a projecttuaimpacting on the ceramic surface will cause the opposite surface of the ceramic to fail. The backup must deflect in order to prevent high sheer stress. A notable current use of composite armor is the Chobham armor used in the latest tank used by the U. S. Army, the M-1 Abrams. The exact composition of Chobham armor is of course a secret. Composite armor is also used in personal body armor ("flak jackets") as well as under- neath pilot seats in helicopters and aircraft for protec- tion against ground fire. Composites are also used in airframes of military air- craft due to their superior flexibility and lighter weight. Some of the fibre filaments used include boron and graphite fibres of various diameters, silicon carbide, alumina fibre, and even fiberglass. When used in conjuction with epoxy resins and aluminum honeycomb structures and titanium, light- weight airframes may be constructed. Parmley [5] provides several instances where composite structures are already in use, including the F—14 and F-lS fighters and the B-1 bomber. A great amount of the work in composites has occurred as a result of aerospace studies. The above is by no means a complete summary of the uses of composite materials. References [4, 5, 6] include a wide variety of materials used and applications of those materials. In addition to the above, composites have been used in space vehicles, transportation equipment, pressure vessels, and boat and building construction. B. Curing of Thermosetting Resins and Composite Processing Thermosetting resins are materials which change 1533: versibly when heated from a fusable, soluble material to an infusible, insoluble material. Heating these resins causes crosslinking to occur within the material forming a stable, covalently-crosslinked network which is somewhat more resis- tant to chemical and physical change. Therefore, heating these materials causes a change in the chemical nature of the system by producing cross—linked networks. A brief dis- cussion of some of these resins, their curing, and their use in composite processing follows. The most common thermosetting resins commercially used are the phenolic resins. Initial monomers for production of these resins are formed by ortho- or para- condensation of phenol with formaldehyde. Subsequent crosslinking and poly- merzation of these materials also occur by condensation. Crosslinks are either methylene or ester bridges. These materials are notable for their thermal and electricalresis- tance. Fillers are often used to improve properties and re- duce costs. Phenolic resins can withstand extremely high temperatures as they do not readily vaporize or melt. Instead phenolic resins carbonize forming a protective thermal barrier. In addition, they are used for laminating composites of wood (most plywood, in fact, is manufactured in this way), paper, glass, and others, and they are also used as bonding resins, varnishes, coatings, and adhesives, and as ion-exchange resins for chromatographic separations. Amino resins are formed by condensation between methylol compounds, which are formed by addition of formal- dehyde to amino compounds such as urea and melamine (I). These resins are colorless and have superior tensile strength and hardness when compared to phenolic resins, but their heat, moisture and electrical resistance are lower. These resins can be heated by high frequency electric currents due to their high polarity. These resins are used to im- prenate clothing to impart crease resistance, fire retard- ance, and stiffness control, and for laminating and other applications similar to phenols. Epoxy resins are polyethers which contain epoxide groups and are formed by reaction of diphenylol (bisphenol A) propane and epichlorohydrin. While other compounds can replace bisphenol A, the resulting epoxides are much more expensive. Crosslinking (or curing) occurs by addition of many types of materials, including urea-formaldehyde and phenol-formaldehyde resins previously discussed. The mechanism of curing includes esterification of the secon- dary hydroxyl groups of the epoxy or of the epoxide group itself, or condensation of secondary hydroxyl groups. They can also be cured by using Lewis acids in cationic poly- merzation, which forms polyethers from the epoxide groups. Epoxy resins are primarily used in surface coating applications, including highway surfacing and bridge build- ing, and are also used in composite processing with glass- fiber reinforcing to produce articles with superior mechan- ical strength and high chemical and electrical resistance. Unsaturated polyester resins include dibasic acids and glycols containing unsaturated carbon-carbon double bonds. Crosslinking occurs via radical chain polymerzation involving these double bonds and a vinyl monomer, usually styrene, in the presence of a fibrous filler, usually glass. Curing is accomplished by using a radical-generating initiator, such as a peroxide. Depending upon the degree of unsaturation of the acid or glycol, the exothermicity of the cure and the stiffness of the product can be fixed to a controlled value. A large degree of unsaturation implies a high exothermicity and an extensively cross-linked rigid product. Applications include lamination and contact molding to produce reinforced plastics for use in boat hulls, autoparts, and for other uses. Epoxy resins are generally superior to unsaturated polyester resins in mechanical, chemical and electrical properties although they are more expensive. Other types of thermosetting resins include urethane foams, silicone polymers, alkyd resins, and several additional varities. [7] Curing of laminates or composites with these resins is a multi-stage process. First, the material to be laminated is impregnated with either a liquid or dissolved resin, at which point the adhesive characteristics of the resin are used to assemble the sheets. Finally, the sheets are com- pressed and heated to affect the cure. One way of doing this has been to hold the sheets in place with an inflatable rubber mattress, which is filled with steam in order to provide both heat and pressure. It is proposed that micro- wave heating is an alternative method of providing heat in- put for thermosetting the impregnating resin. Following is a discussion on microwave heating mechanisms in epoxy systems. C. Microwave Heating of Thermosetting Resins The following discussion focuses on epoxy resins al- though the principles are generally consistent for systems of other thermosetting resins. Epoxides have been chosen as a basis for the discussion due to the large amount of current interest in the literature. First, the molecular structure of an epoxide resin is examined, and then the mechanism and heating at differing stages of curing is discussed. Thermosetting resins are similar to one another due to the large number of polar side chain groups at which condensation, esterification, and other cross-linkingsoccur. Bearing in mind that relaxation occurs at polar side groups as well as over the length of the entire molecule, the following relaxation processes have been identified. Maxima in the loss tangent vs. temperature curve occur for the CHZCH(OH)CH20 group, the epoxy group, supermolecular molecular structure reorientation and reorientation of associations of macromolecules. The CHZCHOHCHZO group could be excited by rotation of the group about the chain axis or of the hydroxyl group itself. Systems that absorb in the microwave region generally tend to be highly polar and small in size. It is indeed fortunate that thermosetting resins contain large numbers of polar groups, and the fact that these groups are generally where the esterifications or condensations associated with crosslinking occur has led to investigations of microwave curing of epoxy systems. Gourdennels’gl has investigated curing of a glass-filled epoxy system, and WilliamsIlo] has investigated curing of epoxy-impregnated pipe at 2450 MHz. Gourdenne has also calculated thermal phenomena in microwave heating of both epoxies and polyesters and has calculated temperature vs time curves for these systems. Heating occurs slowly at first with no crosslinking as the activation energy for crosslinking has not been reached. Then, temperature increases rapidly as the resins undergo crosslinking since the condensation reaction is exothermic. The temperature increases until crosslinking is completed and a maximum equilibrium temperature is reached. Then, the temperature decreases slowly with heat being transferred to the system by microwave energy and to the surroundings by convection. D. Desired Characteristics of Materials for Microwave Heating. The most important characteristic is, of course, that the medium to be heated have a dipole moment, either locally (on the molecular chain) or for the entire molecule. The resins discussed do indeed have this property, ranging from phenolic resins containing hydroxide groups to epoxy resins containing epoxide groups, amino groups in amino resins, and others. Systems will not absorb in the microwave region with- out having a dipole moment.[ll] A more subtle characteristic concerns the behavior of the dielectric loss with respect to temperature. As micro- waves are absorbed the temperature of the dielectric increases. If it so happens that the dielectric loss of a material in- creases with increasing temperature, than such a system would be expected to exhibit an exponential increase in absorption with respect to time. This enables heating to be localized to an extremely narrow slot or interface without too much microwave power being absorbed by the surroundings. Variations of loss tangent with frequency and temperature are available in the literature for a variety of materials including resins, various fillers, and numerous other dielectrics.[l'2'3] The materials from which the composite is constructed would have to exhibit little dielectric loss in the microwave region or else they would interfere with local absorption by the resin to be crosslinked. While highly absorptive filler 10 materials would supply heating, energy transfer would occur with less efficiency. An advantage of microwave heating is the selective input of energy into the resin, and strongly absorbing fillers or composites removes this advantage. E. Advantages of Microwave Heating. Consequences of the dielectric loss increasing with temperature have been previously discussed. If the dielectric exhibits this property, heating at interfaces without too much dissapation to the surroundings can be accomplished. This is useful for fusing two sheets into one, either surface- to-surface (composite processing) or edge-to-edge (sealing). Whitellz] has discussed sealing plastics in this manner, and has sealed seams up to 24 meters long of a polyvinyl fluoride film, while Williams[10] has sealed epoxy-impregnated pipe at 2450 MHz. Composite processing with microwaves is a compari- tively new area, although sealing applications have been in use for a longer time. Microwaves are able to heat these systems extremely rapidly. White carries out a brief calculation of the thermal time constant of a typical polymer film, and estimates it to be 20 to 100 milliseconds. This short time constant enables heat to be concentrated in a small area without heat transfer to the surroundings if the heating time is less than that of the value of the time constant. Longer heating times will heat the area surrounding the interface. 11 The basic advantage of microwave heating, then, is the ability to be able to concentrate heating of polymers or composites on a very small area. In addition, heating is expected to be much more rapid due to the high frequency of the radiation. 12 II. Dipolar Relaxation in Unassociated Liquids This discussion will focus on the basic elements of dipole relaxation as presented in the Debye Theory and the ACT Theory. While intermolecular interactions in liquids are less complicated than in solids, the basic principles presented here are common to all dielectric materials. Results for an "ideal" Debye response (ethylene glycol) are included. A. Debye Theory for Relaxation in Polar Liquids The first modeling effort toward the explanation for absorption of electromagnetic power by polar liquids and solids was that of Peter Debye, who sought an explanation for the observation by Drude (1897) that dielectric con- stant decreased with increasing frequency for a variety of substances, particularly those containing hydroxyl or [13] amino groups. Debye postulated that it was the polarity of the molecules that caused this effect. Debye also noticed that maximum absorption occurs at the frequency w = % whose T is defined as the relaxation time. The relaxation time is an indication of the amount oftimerequired for a collection of dipoles in an electric field to revert to a random orientation once the field is removed. Debye sought to model the randomizathmiof dipoles in 13 a liquid by considering the randomization mechanism as thermal Brownian notion of a rotating sphere (the molecule) in a viscous medium (the liquid). By determining the time dependency of the angular distribution function for the rotating sphere, the time constant for the decay was found to be _5_ 2kT where g is the friction factor for the sphere,]< is Boltzmann's T: constant, and T is the absolute temperature. Substituting the friction factor found by Stokes for a rotating sphere in a viscous medium. E = 81Tna3 where g is the viscosity and a3 is the molecular volume, the relaxation time is found to be _ 41ma3 l"??— A medium will have a short relaxation time at high temper- atures, low viscosities (low intermolecular attraction) and small molecular size (or small dipole size). Absorption can be described by first breaking up the dielectric constant into a real and complex part as follows: 8* = e' - js" with 6' being the dielectric constant associated with electro- magnetic conduction and e" (the complex part) governing the strength of the absorption.[ll] The electric displacement can be written as -> -> -> D = E + 4n P 14 + where P is the polarization per unit volume and is a func- tion of the motion of the dipole moments of the system. If a sinusoidal field (E.= E 6 cos mt) has persisted for a time, the displacement will also be oscillating at a frequency but at a phase angle¢> behind the motion of E. This can be expressed by writing -> -> —> , = Do cos (wt - ¢) = Dl cos wt + D2 Sln wt 0% 0+ . '* . l- 50 cos ¢. D:2 - Do sm ¢ Since it is also true that .+ D = g E (5 static zero-frequency dielectric constant) S S the above can be used to define a pair of dielectric con- stants. a l The e' and e" are, therefore, functions of frequency. = €(w)EO and 52 = E" (MEO Debye developed relationships to describe the frequency dependency and these equations are included in a subsequent section. Absorption can be thought of as a function of how the dipoles respond to the oscillating electric field of the incident radiation. At frequencies w<<1/T, the motion of the dipoles is more rapid than the motion of the electric field and absorption will not occur. At w = 1/1 the oscillatory motion of the dipoles is of the same frequency as the electric field although the motions are not in phase with one another. The response of the dipoles lags behind the forcing electric field causing energy transfer 15 from the field to the system. Finally at high frequencies (w>>l/T), the dipoles can no longer respond to the field quicklyenough and absorption does not occur. The ratio of D to 61 is 2 32... 39 SHE‘S" (E) Eo= 23¢ = tan ¢ = €"(wl 51 DO cosq> E' (w) B0 COS ¢ s'(w) The ratio of the absorbing part to the non-absorbing part is often called the loss tangent of the system for a par- ticular frequency and absorption is often referred to as dielectric loss. A high loss tangent gives high absorp- tion.[ll] B. Activated Complex Theory. The Debye equation for dielectric relaxation of a spherical molecule V = molecular volume has been discussed previously. Since the relaxation time T is defined as the time necessary for the dipole correlation function to decay to l/e of its initial value, the inverse of the expression for I could be interpreted [16] as a first order rate constant for dipole randomization. - k = kVT relax 3Vn HIE-J Comparing to the expression for the rate constant given by 16 activated complex theory (ACT), the same dependence on (kBT) is noted. Additionally, the (Vn) term in the de- nominator can be interpreted as proportional to an activation energy. An increase in either term leads to a greater amount of friction exerted on the rotating a species, leading to an increase in activation energy AG kBT — AG*/RT —h e krelax = C k = O kBT e As‘/Re —AH*/RT relax AGat = free energy change of activated complex e . AS = entropy change of activated complex AHI = enthalpy increase of activated complex 0 = transmission coefficient (generally set equal to one due to lack of better information). Reinverting, the following expression for T is obtained 1. with O _ h - AS*/R Ant/RT T——e e kBT This is the equation which will be used to interpret the experimental results. Debye's equation was the first to attempt to correl- ate macroscopically observed dielectric phenomena with the actual molecular motions associated with relaxation. These 17 motions are extremely difficult to characterize theoret- ically, and a large amount of work has been done in this area without a conclusive theory being developed. The idea that dipole relaxation can be described as a rate process is an old one, and the literature abounds with results fitted to the Arrhenius rate law, with T-1 as the rate constant 1-1 = A exp (-Q/RT) and where Q is the activation energy for dipole reorient- ation. Activated complex theory was developed to describe chemical reaction rates but was also applied by Erying )[14,15] (1936) and Kauzmann (1942 to viscous flow, ionic conduction and dielectric relaxation. C. Debye Equations. Once the relaxation time is estimated from the activ- ation energy for a particular relaxation, the real and imaginary parts of the dielectric constant, 6' and e", associatedwith conduction and absorption can be calculated using the famous Debye equations.[17] e - e €r=€m+(o co) 1 + w2 T2 static (zero frequency dielectric constant) to II 18 w = excitation frequency (radians/sec) e = static (zero-frequency) (dielectric constant e = infinite frequency dielec- tric constant (n ) where n is the index of rgfraction. For systems with more than one relaxation mechanism, the following equations can be used [17]: n e' - e co _ C l e - e i 0 0° 1+(n2'lt.2 1 i=1 n e" wt 6 - e = Ci 0 w l + wz Tiz i=1 The subscript i denotes a particular relaxation mechanism, and the constant Ci is the weighting factor for the i-th mechanism with Ti the relaxation time associated with the ith dipole reorientation. 19 If a plot of e' and s" with frequency m at constant'T conforms even approximately to the above equations, the dipole relaxation is referred to as a Debye Relaxation. A typical Debye relaxation for ethylene glycol is shown in Figure 1518] Materials exhibiting this behavior are gen- erally either small molecules or molecules containing small polar groups in the liquid state where intermo- lecular attractions are relatively low. Water, various glycols, small-chain organic alcohols and others show dielectric relaxations conforming to the Debye equations.[23] Using ACT Theory, the AH‘ was determined to be 3.9 kcal/mol. Highly viscous liquids containing high degrees of molecular association or solids do not exhibit normal Debye relaxations.[25] While there is still a peak in the 5" curve, the plot of e" versus w for these materials gives a much broader curve than that of a Debye Relaxation. Modeling of these systems is described in the following section. 20 I ETHYLENE GLYCOL ‘ (25°C ) Y I T I Figure 1. Example of Debye Relaxation. The dielectric spectrum of ethylene glycol shows a clear absorption peak around 1.5 GHZ. (Ref. 18). 20a III. DipoknrRelaxation in Polymeric Systems This section describes theories of dielectric relaxa- tion in polymeric systems. After a brief discussion of dipole relaxation mechanisms in polymers, the concept of a dipole correlation function is introduced with several examples of these functions as developed by Williams and [20] Watts. The idea of a "universal" high-frequency dielec- tric response as put forward by Jonscher and Ngai[21'22] is discussed. Finally, relaxation mechanisms of a particular polymer, Nylon 66, are discussed in greater and more speci- fic detail. A. Dielectric Relaxation Processes in Polymers Polymers often exhibit several distinct types of relaxa- tion mechanisms.[19] The first type of mechanism is called the a process in the literature. This mechanism randomizes the dipole moments through random Brownian motion of large segments of the polymer chain. These relaxations take place at low frequencies and therefore high relaxation times. The characteristic relaxation time is increased over that of the liquid state due to reduced molecular motion in the solid state. Another type of mechanism occurs at higher frequency and is called the 8 process. This mechanism randomizes the dipole 21 moments by reorientation of individual dipoles in the chain, involving segmented motion of a small section of the polymer chain than the a relaxation. If absorption is caused by motion of dipolar groups in the chain backbone (polyvinyl chloride, then reorientation occurs by movement of the chain with respect to itself. On the other hand, dipolar side groups can reorientate essentially independently of the chain. A plot of the loss tangent versus temperature would yield a broad, flat distribution due to the fact that there are a number of different dipoles involved in a B-process each with its own characteristic absorption. These processes are higher frequency processes and as such might be expected to occur in or near the microwave region. As the relaxations occur at higher and higher frequencies, they are denoted y, 6, etc. in order of increasing characteristic frequency. These relaxations involve successively smaller sections of the polymer chain in the relaxation. B. Theory of Dielectric Relaxation inPolymers The theory of dielectric (dipole) relaxation is based heavily on a quality known as a dipole correlation function the form of which for an ith dipole is given aslzo] r..(t) = (“1(O)°U;(t)> ui + vector quantity 11 U2 u + magnitude squared, scalar 22 This function is meant to describe how the motion of the dipole moment and its orientation ui(t) at a given time t=t is correlated to the initial orientation of the dipole ui(o) at t=0. Since all directions are considered to be equally probable initially = ui(o)-. The quantity is the average dipole orientation at time t given the dipole orientation ui(o) at t=0. The Fourier transform of the time derivative of the dipole correlation function Tii(t) is related to the complex permittivity in the following manner. 00 6*(jw) -eco dPii(t) em = - —-—————- exp (-jwt)dt e - 6 dt 5 oo 0 O O €(w) 8(0) If the dipole motion is strongly correlated than Fii(t) is essentially constant in time and the complex permit- tivity is approximately zero, while if Fii(t) changes strongly with time then a strong absorption would be expected to occur. Pblymers exhibiting multiple relaxation processes would be expected to exhibit different decay regions at different times. Polymers exhibit dipole‘correlations along the polymer chain and as a result the correlation function contains cross terms involving correlations 23 between an i-th dipole and a j—th dipole in the same chain. For flexible chains only two or three nearby dipoles give terms which contribute significantly to the correlation function. The further apart the dipoles are separated, the less correlation between them. A suitable correlation function is given as <. . t )3 . .11- Mt) = 111(0) u( )> +1=1 > 2> ) and cross terms. This function is used in the same manner as Pii(t) described earlier in order to find the complex permittivity. The Debye Theory of relaxation in polar liquids assumes an exponential decay in time for the value of the correlation function. Tii(t) = EXP (-t/T) annfleuxmotions in solids are different than in liquids due to the fact that there is more "cooperation" required in the reorientation of the dipole, and the decay process is therefore non-exponential. An empirical form uses 8 adjustable parameter. ¢(t) = exp [(-t/IO)B] with O PS gas is the term dealing with relaxation of the environment "s" via the a relaxation with PS being equal to the probability of being in the environment "s" and gas being equal to (s>u2) where s is the net dipole moment obtained for environment "s" by averaging over a time scale long compared with the local B-processes but short compared to the a-processes. The second term on the right. (Donut) gPs qu¢Bs(t) qu = l - q0Ls contains both decay terms, while the first term only con- tains a decay term for the a-process. This model predicts two separate decay regions for low temperatures since ¢Bs(t) decays much more quickly than ¢a(t). The high fre- quency end is characterized by qBS and the range of values associated with this quantity will produce a broad B-peak. 25 At high temperatures the ¢a(t) decays so quickly that the effects of the motions arising from the local dipole motions are dominated by Brownian relaxation. [20] Discussion of the Williams-Watts model is included for the sale of completeness. Due to its greater simpli- city, the model proposed by Jonscher and Ngai (next section) will be used to predict the complex permittivity e" as a function of frequency and temperature. In spite of the fact that the Williams-Watts model was not used to interpret the experimental results, a description of it is useful as it serves to show how fundamental processes contribute to dipolar relaxation. C. The "Universal" Dielectric Response It may be recalled that the Debye Equation for the absorptive part of the dielective constant is given by (80 - em) wt C" (0.)) = ' l + w2 Tl where all terms have been defined previously. If T is held constant it can be seen that 1 -1 E" a — = w for a Debye relaxation at high frequencies. While Debye relaxations are rare, it has been noted by Jonscher and Ngail21’22] (1977, 1979) that the high- frequency behavior of e" for practically all materials 26 follows the form 6" “ wn-l with 0 ofiuuowamwo 2H mm mo uon mOHImOA .OH wusmflm "w o.ooa o.oH o.H .mHIOH J «Hioa MHIOH NM ; 1 OH / NH- / / mpflswflq b // m H O O i e.H m )/ Hausa 58a The partial differential equation used is of the form: k2T+P =P 93 )7 {/V] Cpat (conduction (absorption)(accumulation) The set of grid points in this solution can be greatly reduced in size by invoking symmetry and using the fact that E2 is symmetric about the axial center of the cavity and Ea = 0 everywhere. A schematic is shown in Fig. 11. The rod to be modeled has a radius of 0.6 cm, with a length of 7.2 cm inside the cavity and 9.0 cm outside the cavity. The grid points were selected to be 0.2 cm apart. 59 .6 mo uanCOQOCCH we com .o n u com A N\H n N uzonm UHuquE>m we wufl>mu CH pawflm ofluuowam .ponqu coflusaom Hmoaqusz mo oflumEmnom .HH wusmflm 3 ND up; / a >DH>MO w>m3ouoflz i. >DH>mU wo umucwu Hmflxfl Io 4/ \ ()1 I ll ., / Ofluuowawflo . x / I , . x a \ , ii . ‘ \J.o fl s‘I/a \44/ m . L 5 S. .. I K a . i . i L x \\ / kONC......5HOw / \. OHHOEDC MOM // \ HGHOQ MO “mm 59a \ ‘0..-— P'" a--- I I \ s_--’ ‘P-..___-h '9‘” Ofluuowawflo m0 uwucwu HcHx¢\\\ Aconvective heat transfer coefficient of 2.00 BTU/hr ftz‘fiiwas arbitrarily chosen to estimate heat transfer from the rod to the air. Results are given in Table 10 for four different power levels and include (1) percent absorbed at t = 5 min and (2) number of minutes the rod takes to reach steady state. Additionally, Figures 12 - 15 show the evolution of temperature with time at varying positions in the rod at each power level at t = 5 min and at steady state. It should be noted that after ten minutes in a 10 watt cavity, the nylon rod would have already exceed- ed its melting point of about 4100 K. Power absorbed increases sharply once parts of the rod are heated above 3230K (Tg) because of the onset of the a-relaxation. 60 Table 10. Selected Results of Numerical Calculation of Temperature % Power Absorbed Power Level w (t = 5 min) Steady State, min 1.5 x 10’3 49.4 13 1 51.7 77 3 56.8 117 10 90.2 97 61 "H .0 umBom mcflmcmno Ham umwoxw .mHimH mwuhmam 0w mwmawww wusumaocwfioc wane .muumz maoo.o we Ho>oa uu3om .awa>oo asp mpflmcfl who mafia Hobauuo> cognac one mo umoa on» o» mucflom .coflusaom mumum xpmmum muocmp mucflod .cHE m n u ouocwp muCAOQ .pom coH>z ca coflusofiuumflo wuswmnwmaoa .NH wusmam Eu .>0H>mu mo kucwo Eoum wocmumflo Hmflxd N ma o.m wa m.m o oo.mmm 1 / 8.9mm mo.mm~ l 61a .Amuum3 o.H u Hmv pom COsz CH coflusnfluumflo wusumuodEwB .MH wusmflm EU .wocmumflo Hoax< m.mH o.m N.n m.m 0 III. . oom , xo.B . cam . omm 61b . . H Amuum3 o m n my mom ceamz CA :Oflusnfluumflo ousumumdfiwe .Eo .wocmuwflo Hmflx< .vH ousmflm iomm oom omm can i own 0mm oov 61c .Amupm3 o.oa u Hmv pom COHmz cfl coflusnfluumflo mnsumuwmawe .mH musmflm EU .mocmumflo Hmflx< N 0 KO H 0 O'\ N O I" \D m 0 ---....--_‘ 1 0mm 1 oov r oom » cow 61d VII. Results and Discussion Coupling efficiences for various materials are presented in Table 11 and 12 along with values of their complex permittivties at room temperature. The results for nylon 66 and teflon are of course not directly comparable due to the larger diameters of those dielec- trics (1.2 cm) as Opposed to those of the liquids (0.45 cm). The most notable results are the coupling efficiencies for the solid systems. Coupling effi- ciencies for teflon range from 4% to 19%, which is higher than one would expect for such a lossless mater— ial (8" = 0.0002)!31 It is the large electric fields at the center of the resonant cavity in the TMO12 mode which enable this high coupling to be accomplished. Using the measured values of power absorbed per unit volume along with the literature values of e" in the equation. P/V= 1; w e" E2 The values of E2 at the cavity center for each material examined could be estimated. The fact that there is an obvious connection between these two parameters was the basis for a correlation developed and discussed in the previous section. Again, power absorbed depends on both the electric field in the dielectric and the 62 complex permittivities e". A large value of one of the other leads to a high coupling efficiency. Since ethylene glycol and propylene glycol have similar pr0perties, a comparison of the experimental results for these two materials is in order. Both systems exhibit Debye relaxations and Operate at approx- imately resonant conditions (w = l/r) , and both have similar values of 6' so that the ratio of their coup- ling efficiencies would be expected to correspond to the ratio of their complex permittivities. The diff— erence in the ratios is well within experimental error. 63 mooo.o ca x 5mm.H wha.ma Ioa x mmh.~ m.mH coamoa : mmmo.o ca x mmm.e wha.ma IOH x vwo.> H.Hm mm ceamz : mom. OH x 5mm.a mmm.m loH x emm.m b.m~ Hoomau s mcwamdoumhaom m.oa oH x moa.m mmm.~ OH x mmH.H n.5h Hoowaw ml mamaamoum m.va OH x mam.m mmm.~ mica x omm.a 0.0m H00>H0 mamaxnum =w mEO\3 mEo 3.©mnuomn< pmnHOmnd Hmwumumz oEdHo> .oESHo> uo3om Hm3om \Hw3om unmoumm HA OHQMB 64 coxmu no: ucosoucmmoe monocooil Lama i 0H3 oao.o .lI iii oeo.o sonms Asoav mmm.o oom.o mmm.o mmm.o so soflmz Issac Aa~.o i ka~.o Aa~.o Houses mcoamoudmaom Asmac ~mm.o mam.o ~mm.o mom.o gonads mamassous mam.o Hmm.o «Hm.o Ham.o Hoomam mamassum 3 NH I ca 3 m 3 H 3 led x m.a Hmflumumz m mHo>wq uozom mcfihuo> um mmflocwfloammm mcwamsou .NH magma 65 Looking at the values of the cavity Q for ethylene and propylene glycol at differing incident power levels, it can be seen that these species have resonant absorp- tion peaks at or near the frequency of our apparatus (w = 1.539 x 1010 rad/sec). By interpreting increasing power level as increasing temperature one can see how absorption first increases than decreases with increasing temperature indicating the nearness of a resonant peak. Values of wt for each system were estimated at various temperatures for each liquid using the ACT model and the calculated activator energies at 3940 kcal/mol for ethylene glycol and 4100 kcal/mol for prepylene glycol. The results of this calculation are shown below. (w = 1.539 x 101° rad sec-1). 66 Table 13. Values of wT O T, K Et (OH)2 l, 2-Pr (OH)2 298 1.92 2.52 310 1.43 1.85 325 1.01 1.30 340 0.74 0.94 360 0.51 0.63 67 These values indicate that increasing temperature slightly can affect resonance sharply. The Q values for polyprOpylene glycol do not change as much with changing power levels. The resonance peak occurs at a much lower frequency so that the magnitude of change in wt with temperature is slight at these frequencies. The temperature field for the nylon 66 dielectric at differing power levels was determined numerically using an explicit method solution. Power absorbed increased sharply as temperature increased. The experimentally determined coupling efficiency for Nylon 66 ranged from approximately 50% for minimal power levels to around 80% for the highest power level used. Theoretically determined percent power absorbed is plotted versus time and compared to the measured power levels in Figure 16. It can easily be seen that the calculated values conform to the experimentally deter- mined values within experimental error. As it turns out, the Q measurements for the nylon rod at differing power levels were probably not taken at steady states, as was implied earlier. All measurements were taken within ten minutes or so after the field was turned off,while it is estimated that the rod takes about 90 68 10.0 watts 90 ~ 80 Coupling . Efficiency, % 70 r 60 ’ 3.0 watts r— 1.0 watts ++ 50 T4; 3. 4 0.0015 watts 0 1 2 3 4 5 Time, min Figure 16. DependenceofCoupling Efficiency on time for varying incident power levels. 68a minutes to reach a steady state. These estimated steady state times were presented in a previous section. It should be commented that for the 10 watt measurement, it is fortunate that only ten minutes or so elapsed as the rod probably would have melted before too much longer. 69 VIII. Conclusions and Suggestions for Future Work Several factors influence the absorption of micro— wave power by liquids and solids. The size of the dielec- tric, the temperature of the dielectric and its influence on 6' all affect the amount of power absorbed, along with the electric field strength. The temperature (and freqency dependencyiof s" can be interpreted using the Arrhenius activation energy and actived complex theory. Coupling to materials of low E" is possible since the electric field at the cavity center in the TM mode is large. 012 Future investigations of these phenomena could in- clude measurements of temperature outside the cavity at different axial and radial positions using thermocouples and inside the cavity using fluoropticthermometry on the dielectric surface. The modeling already developed would be used to compare these measured temperatures to experi- ment. The modeling will be expanded not only to include a more detailed consideration of the electric fields in the cavity but also to determine the effect of more than one dielectric on the absorption. Also, a reaction term will be included to describe heat generation resulting from the curing of a thermosetting resin along with a material balance for the extent of curing of the resin. It is hoped that the modeling can be used to monitor 70 the extent of curing in order to be used to develop an automated, intelligent microwave processing system for the curing of thermosetting resins and composites. 71 APPENDIX A Calculation of AHI for H20 The following data was used to model the dielectric relaxation of H20 at various temperatures and frequencies. Table A1. 5' and e" for H20 Temperature e e e" O C 1 0 e _ 9 1 f==3 x 10 sec 3 x 10 1 x 10 1.5 85.0 2.0 2.80 25.00 39.00 25 76.2 2.0 1.25 12.26 30.0 55 66.2 2.0 0.63 6.00 23.87 85 56 2.0 0.42 3.10 14.0 (Source — Ref. 3) This data was used to find I under various conditions using the Debye equation for s" with w = 2 n f. ( II_ _ e — (60 em) wt 1 + w2 T2 Finding a T for each temperature and frequency listed above, and averaging them, the following results for 1(OC) were obtained along with the corresponding value of AHI from the ACT expression. 72 Table A2. t and AH:F at various temperatures Temperature, 0C T X 1012 sec AH= (kcal/mol) 1.5 15.05 2.43 25 8.34 2.34 55 5.32 2.34 85 3.73 2.37 These values of AH# average out to 2.37 kcal/mol (: 0.04 kcal/mol). 73 APPENDIX B Finite Difference Equation for Laplace's Equation in CylindriCal Coordinates and FORTRAN Program for Temper- ature Field. Laplace's equation for the temperature field in cylindrical coordinates without heat generation is:[29] 2 _ 1 3T 82T 32T VT-srttsz'+a-zr The finite difference form of this equation is: 1(T.+. . _ T. 1 .) (T.+1 . T._l . 2T. g.) + V2T=- 1 1:3 1" I] + 1 13 + 1 13 " 1 L1 r 2Ar (Ar)2 (Ti.j+l + Ti,j-l - 2Ti,j) (A2)2 If Ar = A2, and r = iAr, than 1 va = ii (Ti+1,j Ti-iij) + (Ti+1jj I Ti-1-,j I T.. +T.. +T.. -4T..) lLl-l 1,3-1 i,3+1 1,3 (Ar)2 This equation will be used to evaluate how conduction affects the temperature field. This will be added to a term describing how power is absorbed and numerically integrated using a time increment of 5 seconds. Larger time step sizes were not effective as they resulted in oscillatory 74 solutions. Electric field (Ez) dependency was included 2wz L by using a (cos2 ( )) term. The- program determines in turn: (1) Fraction power absorbed in reference volume. (2) Power absorbed in reference volume (PABS (I,J)) (3) Conduction contribtuion (T COND (I,J)) (4) Absoprtion contribution (T NEXT (I,J)) (5) AT/At for each volume, and finally AT Program reads as input data: (1) Freqency in meghertz. (2450 mHz) (2) Initial temperature (usually 250C) (3) Incident power and s' (4) Time step (5 sec) and heating time (60-120 min) Program outputs at one minute intervals: (1) Percent power absorbed (2) Time in minutes (3) Temperature at intervals of 0.2 cm radially 1.8 cm axially A c0py of the program is included for reference. 75 ) TI U C Tl) U5— O.D. : 9 1h» .5( EC.. LIE Iris. ‘1 TP \l)) ’9 ))) TI) ..U)\J U»... 9JIU Fr 7.. ’9 “NJ .. (7‘7. :( ITT 74.0 C// .....n..' . o o a» ... , . .... punv _. 7 IT 1...». I9 -..? a. TI) ((Il “hurrfi Lir— n. Ir) XV...» T. 9 ......r. U“ ‘l‘( Cl. ... t a an IT )\.I)—.. T, Y/ 7 + )))H U: + . . .. lulu... r . .. 9 9 9 “In 7 . H. . T. v .3...) T.. 97 I c o ..l. . ((l\ 0 ()1 I .( 9).. .-.... PITT: . . ~ » . g . U ' l/lr . a. 0.... ... \I. . .V. My HI)\I\I n- 9...... ...H). ) 9.. T o... + .i... T a... ,. c L... _. U C.....L.TA .....4 v r '7! .... 1 7.44? ..LI To. a.) or... ..- ... : .T. «:1 .9; ). . . I. 7,.Ti1).i. .. . . o I.-. .. .... o. 9 ...Lrt: a P— . 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