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Sintering of piezoelectric PbNb²O⁶ with various dopants Chong, J. Edward 1991

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SINTERING OF PIEZOELECTRIC PbNb2O6 WITH VARIOUS DOPANTSbyJ. EDWARD CHONGB.A.Sc. (Engineering Physics - Materials Option), University of British Columbia, 1988A THESIS IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFMASTERS OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDepartment of Metals and Materials EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember, 1991© J. EDWARD CHONG, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Metals and Materials EngineeringThe University of British ColumbiaVancouver, CanadaDate ^DE-6 (2/88)iiABSTRACTThe effect of temperature, dopants, dopant concentration, and processing procedure onthe sintering kinetics of the piezoelectric ceramic PbNb 2O6 has been investigated. Theisothermal contraction behavior of PbNb 2O6 over the temperature range 1050 - 1200 °C withvarious dopants (SrO, Bi 203 , LiNbO3 , and KNaNb2O6) at concentrations from 0 to 1 weightpercent was studied. Because all mixing of the dopants with the PbNb 2O6 powder was doneby vibratory ball milling, the effect of milling on sintering kinetics was also studied.Vibratory ball milling of the undoped powder was found to produce specimens thatcontracted the most over the shortest period of time. A problem with 'milled' powder was theexcessive grain growth resulting in fragile specimens. SEM micrographs of fracturedspecimens showed that milling with the addition of any dopant hindered grain growth.Shrinkage curves of these doped powders were found not to contract as fast as the 'milled'powder with no dopants.Of the 4 dopants tested, Bi 203 was found to produce the largest grain size as well asexhibit the greatest contraction. This is in comparison to SrO additions which yielded theleast contraction. The relative effectiveness between LiNbO 3 and KNaNb2O6 could not bedetermined with certainty but were found to be between SrO and Bi 203 . Micrographs suggestthat increasing concentrations of SrO, LiNbO 3, and KNaNb2O6 decrease the grain size, unlikeincreasing additions of Bi 203 which led to a larger grain size.Initial stage sintering of PbNb2O6 was found to occur by bulk diffusion and bulk neckdiffusion. In general it was found that the sintering is dominated initially by bulk diffusion(e oc t0.8) before it was switched to bulk neck diffusion (e oc t0.4 ) with time. Additions ofBi203 , LiNbO3 , and KNaNb2O6 were found to encouraged bulk diffusion while the addition ofSrO encouraged bulk neck diffusion. Plots of p - In t for intermediate / final stage sinteringiiidata yielded linear regions which are indicative of bulk diffusion for these stages. Fromanalysis of these plots of p - In t an activation energy of 180 ± 18 kJ/mole was obtained forthe sintering of 'milled' PbNb2O6 .A mechanical model using viscoelastic elements, applicable to pressureless sintering ofPbNb2O6 , has been developed. Unlike previous viscoelastic models which used a unit stepfunction of stress for hot pressing conditions, this model uses a decaying stress function torepresent the decreasing contact or capillary stress as sintering progresses under pressurelessconditions. The model consisted of four mechanical components, one elastic, two viscous,and one mass term. The values of the elastic, viscous, and mass terms were determined fromthe experimental coefficients of the compaction curves. Only the values of the masscomponent and one of viscosity components were found to be significant and to decrease withincreasing temperature. This indicates that these two viscoelastic elements are related to thesintering process. However, more understanding is needed to correlate flow components withspecific mechanical elements.Table of ContentsABSTRACT^Table of Contents  ivList of Figures ^  viList of Tables  ixACKNOWLEDGEMENT^1 INTRODUCTION  11.1 Review of Piezoelectric Ceramics^  51.1.1 Development of Piezoelectric Ceramics ^  51.1.2 History of Ferroelectric Piezoelectrics  101.1.3 Applications of Piezoelectric Ceramics Piezoelectric Devices ^  121.1.4 Manufacture of Piezoelectric Ceramics^ Powder Synthesis Solid State Coprecipitation ^ Forming^ Sintering Finishing Poling  161.1.5 Lead Metaniobate ^ Crystal Structure Previous Sintering Studies ^  231.2 Driving Force for Sintering  261.3 Sintering Theories ^  311.4 Objective ^  352 EXPERIMENTAL PROCEDURE^  362.1 Materials  362.2 Sample Preparation ^  382.3 Isothermal Experimental Apparatus  392.3.1 Vertical Tube Furnace  412.3.2 Linear Variable Differential Transformer ^  412.4 Experimental Procedure^  442.4.1 Isothermal Contraction  442.4.2 Electrical Properties  462.4.3 Summary of the Experimental Procedure^  463 RESULTS ^  483.1 Powder Characteristics ^  483.2 Isothermal Contraction  523.2.1 Stages of Contraction  523.2.2 General Observations  553.2.3 Density^  563.2.4 Milling Dependence ^  58ivv3.2.5 Dopant Dependence^  593.2.6 Temperature Dependence  603.2.7 Concentration Dependence  633.2.8 Microstructure of Sintered Samples ^  653.2.8.1 Summary of Grain Size Data  753.3 Electrical Properties ^  794 DISCUSSION^  814.1 Sintering Theories  814.1.1 Analysis Using Establish Sintering Theories ^  824.1.2 Procedures for Calculating Activation Energy  1004.1.2.1 Calculations ^  1034.2 Viscoelastic Model  1044.2.1 Program ^  1114.2.2 Interpretation  1135 SUMMARY AND CONCLUSION ^  1186 RECOMMENDED WORK  1207 REFERENCES ^  121Appendix I  126Appendix II ^  127Appendix III  133Appendix IV ^  136Appendix V  140Appendix VI ^  143List of Figures1-1. World market for advanced and functional ceramics ^ 31-2. U.S. sales for advanced ceramic and those predicted for the year ^ 31-3. Scope of advanced functional ceramics ^  41-4. Direct piezoelectric effect ^  61-5. Converse piezoelectric effect  61-6. Interrelationship of piezoelectrics based on crystal symmetry ^ 71-7. Piezoelectric effect of BaTiO3 ^  91-8. Piezoelectric effect on a material (schematic) ^  101-9. Dimensional change and alignment as a result of poling ^ 171-10. Typical hysteresis loop for ferroelectrics  171-11. Structure of PbNb2O6 ^  211-12. Tetragonal cell structure of PbNb 2O6 ^  221-13. Pb0-Nb2O5 phase diagram  221-14. Dielectric constant vs. temperature for PbNb 2O6 ^  231-15. d33 and dielectric constant versus composition  251-16. Dielectric constant of PbNb 2O6 with various dopants ^ 251-17. AP to increase the radius of a submerges bubble  261-18. Expanded volume element due to transfer of mass  271-19. Schematic of contact between two sintered spheres ^ 281-20. Force balance for neck element during sintering  301-21. Tetrakaidechadron ^  342-1. Processing steps used to synthesize PbNb 2O6 powder ^ 372-2. Equipment used for isothermal compaction experiments  402-3. LVDT assembly and arrangement ^  422-4. Sample arrangement in the LVDT assembly ^  452-5. Summary of PbNb 2O6 specimen testing procedure  473-1. Particle size distribution before milling  493-2. Particle size distribution after milling ^  493-3. SEM micrograph of PbNb2O6 powders  513-4. DTA plot of PbNb2O6 powders  513-5. X-ray diffraction pattern of PbNb2O6 powder ^  523-6. Typical shrinkage and temperature curve  54vivii3-7. Typical normalized contraction curve of PbNb 206 ^  543-8. Contraction curves for 'as-received' and 'milled' PbNb 2O6 ^  593-9. Contraction curves for PbNb 2O6 - 1 wt. % dopant at 1100 °C  613-10. Contraction curves for PbNb2O6 - 1 wt. % dopant at 1200 °C ^ 613-11. Contraction curves for PbNb 2O6 - 1 wt. % SrO samples  623-12. Contraction curves for PbNb2O6 - 1 wt. % Bi203 samples ^ 623-13. Contraction curves for SrO doped PbNb 2O6 at 1100 °C for variousconcentrations ^  643-14. Contraction curves for Bi203 doped PbNb2O6 at 1100 °C for variousconcentration  643-15. SEM micrographs highlighting the porosity in the specimens ^ 673-16. SEM micrographs of fracture surfaces of undoped PbNb 2O6  683-17. SEM micrographs of fracture surfaces of SrO doped PbNb 2O6 ^ 703-18. SEM micrographs of fracture surfaces of Bi203 doped PbNb2O6 ^ 713-19. SEM micrographs of fracture surfaces of KNaNb 2O6 doped PbNb2O6 ^ 733-20. SEM micrographs of fracture surfaces of LiNbO3 doped PbNb2O6 ^ 743-21. SEM micrographs of fracture surface of unheat treated PbNb 2O6 ^ 804-1. Relative density curves for as-recieved PbNb 206 ^  844-2. In E - In t curves for as-recieved PbNb2O6 at 1050 °C  854-3. Relative density - In t curves for as-recieved PbNb2O6 ^  854-4. Relative density for milled PbNb2O6 ^  874-5. Relative density - in t curves for milled PbNb2O6 ^  874-6. Density curves for PbNb 2O6 - 1 wt. % SrO  884-7. Density curves for PbNb 206 - 0.5 wt. % SrO  884-8. Density curves for PbNb2O6 - 0.25 wt. % SrO ^  894-9. In e - In t curves for PbNb2O6 - 1 wt. % SrO  914-10. p - In t curves for PbNb2O6 - 1 wt. % SrO  914-11. In e - In t curves for PbNb2O6 - 0.5 wt. % SrO ^  924-12. p - In t curves for PbNb2O6 - 0.5 wt. % SrO  924-13. in c - In t curves for PbNb2O6 - 0.25 wt. % SrO  934-14. p - In t curves for PbNb2O6 - 0.25 wt. % SrO ^  934-15. Density curves for PbNb2O6 - 1 wt. % Bi203  974-16. Density curves for PbNb2O6 - 0.5 wt. % Bi203 ^  974-17. Density curves for PbNb2O6 - 0.25 wt. % Bi203  984-18. In E - In t curves for Bi203 doped PbNb2O6 at 1050 °C ^ 98viii4-19. p - In t curves for PbNb2O6 - 1.0 wt. % Bi203 ^  994-20. p - In t curves for PbNb2O6 - 0.25 wt. % Bi203  994-21. Activation energy plots for milled PbNb2O6 from intermediate stage data ^ 1034-22. Contraction - time plots showing the deviation between experiments ^ 1054-23. Comparison of curve fit with data ^  1064-24. 'System dynamics' relating output (strain) to input (unit stress) ^ 1084-25. Electrical and viscoelastic model of PbNb 2O6 sintering ^ 1104-26. Schematic of computer program ^  1124-27. g and Th for 'as-received' PbNb2O6  1164-28. p. and for 'milled' PbNb206 ^  1164-29. g and for 1 wt. % SrO - PbNb2O6 ^  1174-30. g and m for 1 wt. % Bi203 - PbNb2O6  117II-1. Relative density - In t curves for 0.5 wt. % Bi 203 doped PbNb2O6 ^ 12711-2. Relative density curves for 1 wt. % KNaNb 2O6 doped PbNb2O6 ^ 12811-3. Relative density - In t curves for 1 wt. % KNaNb 2O6 doped PbNb2O6 ^ 12811-4. Relative density curves for 0.25 wt. % KNaNb 2O6 doped PbNb2O6 ^ 12911-5. Relative density - In t curves for 0.25 wt. % KNaNb2O6 doped PbNb2O6^129II-6. Relative density curves for 1 wt. % LiNbO 3 doped PbNb2O6 ^ 13011-7. Relative density - In t curves for 1 wt. % LiNbO3 doped  13011-8. Relative density curves for 0.25 wt. % LiNbO3 doped PbNb2O6 ^ 13111-9. Relative density - In t curves for 0.25 wt. % LiNbO 3 doped  131II-10. In e - In t curves for KNaNb206 doped PbNb2O6 ^  132II-11. In E - In t curves for LiNbO3 doped PbNb2O6   132III-1. Activation energy plot for 'as-recieved' PbNb2O6 for intermediate stagesintering ^  133111-2. Act. en. plot for SrO doped PbNb 2O6 for intermediate stage sintering ^ 134111-3. Act. en. plot for Bi203 doped PbNb2O6 for intermediate stage sintering^134Act. en. plot for KNaNb2O6 doped PbNb2O6 for intermediate stage^135sintering ..III-5. Act. en. plot for LiNbO 3 doped PbNb2O6 for intermediate stage sintering^135IV-1. Equations for a) electrical systems b) mechanical systems. in impedenceZ(s) and admittance Y(s) format ^  137IV-2. Electrical and viscoelastic model of PbNb 2O6 sintering ^ 139List of Tables1-1. Properties of Piezoelectric Ceramics ^  181-2. Time Dependence of Various Sintering Mechanism ^ 332-1. Dimensions and Properties of Powder Compacts  382-2. Outline of isothermal shrinkage experiments conducted ^ 393-1. Melting point of constituent powder ^  553-2. Relative density (%) of PbNb2O6 specimens after sintering ^ 573-3. Dimensions of starting PbNb 2O6 powder  753-4. Grain size of selected PbNb2O6 specimens ^  76I-1. Dielectric constants of PbNb 2O6 specimens  126IV-1. Mechanical-Electrical Conversion Table ^  136VI-1. Viscoelastic coefficients (M,11, 1 1 ,11 2) of PbNb2O6 specimens ^ 143ixACKNOWLEDGEMENTThe author wishes to express his gratitude to his research supervisor, Dr. A. C. D.Chaklader for his advice, patience, and encouragement during this project. Thanks are alsoextended to the faculty, staff, and fellow graduate students in the Department of Metals andMaterials Engineering. The assistance of Dr. Eswar Prasad of B. M. Hi-Tech Inc.Collingwood, Ontario in determining the dielectric constants was especially appreciated.Financial assistance from B. M. Hi-Tech Inc. and the Natural Sciences and EngineeringResearch Council of Canada, is gratefully acknowledged.1 INTRODUCTIONCeramics can be grouped as either traditional or advanced. Traditional ceramicsinclude those ceramics commonly used for decades, by the general public and industry insuch areas as dinnerware, sanitary ware, insulation (electrical and thermal), glass, bricks, andrefractories. The composition of advanced and traditional ceramics can be very similar.What distinguishes advanced ceramics is a significantly higher performance level, as a resultof stringent controls over composition and processing, and a market value based on higherperformance that warrants their much higher cost.'The field of advanced ceramics can be further subdivided into two distinct groups:structural ceramics and functional ceramics. Into which category the ceramic belongs in isdependent upon the article's end use. As the name implies, structural ceramics are used in astructural capacity to withstand high temperatures, stresses, wear, and chemical attack whilefunctional ceramics are used primarily for their electrical, magnetic, and optical properties.Oxygen sensors, condensers, circuit board substrates, buzzers, fuel cells, opticalwaveguides, sensors, capacitors, and superconducting wires are a few applications thatemploy functional ceramics. 43 •4 '5 '6 Their brittleness is not a limitation to their use asfunctional ceramics, since the electrical, magnetic, or optical characteristics are the primeconcerns.Since structural ceramic components tend to be larger than functional ceramic units, thedollar value added when manufacturing structural ceramics is generally greater. Because ofthe greater potential profit, the majority of the current interest in advanced ceramics is in thearea of structural ceramics. Unlike structural ceramics that hold vast potential but are inlimited use presently, functional ceramics are presently used extensively and constitute themajor portion of the world market for advanced ceramics. This distinction can be clearly seenin Figure 1-1 a which shows the relative portions that functional and structural ceramics12occupy in the world market of $8.2 billion for advanced ceramics in 1987. 7 More recentfigures for the United States market (Figure 1-2) indicate that $3.6 billion worth of advancedceramic products were sold in 1990 with predicted revenue in excess of $9 billion by the turnof century. 8 Using U. S. sales numbers and if ceramic coatings are separated from structuralceramics, the anticipated annual growth rates for the various ceramic sectors are 18.1% forstructural ceramics, 8.4% for functional ceramics, and 9.7% for ceramic coatings. 8The field of functional ceramics can be further divided, according to their particularapplication, into groups such as piezoelectrics, capacitors, packages and substrates, ferrites,resistors, and sensors. This subdivision into the previously mentioned groups is presented forthe 1987 world market in Figure 1-lb while Figure 1-3 lists various functional ceramiccompounds and their applications. For the purpose of this study only piezoelectric ceramicswill be discussed, as this thesis deals with the sintering behavior of the piezoelectriccompound lead metaniobate (PbNb2O6).Packages andSubstrate-28%Capacitors-32%Resistor-5% Ferrites-21Piezoelec *cs-11 %3Total: $8.2 billion^ Total: $8 billionStructural-2%^ Sensor-2% Other-<1%a)^ b)Figure 1-1. World market a) for advanced ceramic and b) functional ceramics in 1987.[7]Total: $3.6 billion^ Total: $9 billionCoatingstructurala)^ b)Figure 1-2. U.S. sales a) for advanced ceramic in 1990 b) and predicted for the year2000. [8]a)4.)..5^5cjc4 0U•( 97E.• 0 o^f:4) 9as^csiNN  4. 4-474 2W4c.23O1.1 Review of Piezoelectric CeramicsThe following is a brief background of piezoelectric behavior.1.1.1 Development of Piezoelectric CeramicsThere are two effects when dealing with piezoelectric materials: the direct and theconverse. The direct effect, as shown in Figure 1-4, is characterized by the development ofan electric charge when a mechanical stress is applied whereas the converse effect, shownin Figure 1-5, is identified by a mechanical displacement caused by the application of anelectric potential.In order to understand the concept of piezoelectricity, it is necessary to study thestructure of materials and specifically the unit cell. Whether a material exhibits any signsof piezoelectricity depends on the symmetry of the ions that form the unit cell. Thesymmetry conditions necessary for the unit cell to be piezoelectric, extend upwards in scaleto determine if a single crystal of the material is piezoelectric and if it is possible forpolycrystalline aggregates to exhibit such behavior.All crystal structures belong to one of 32 different classes. 9 The different classes orsymmetry groups are classified according to symmetry elements. These symmetryelements are: (1) centre of symmetry, (2) axis of rotation, (3) mirror planes, and (4)combinations of these. As shown in Figure 1-6, there are 21 classes that do not possess acentre of symmetry, which is a necessary condition for piezoelectricity to exist. Of the 21classes, 20 are piezoelectric while one class is not piezoelectric because of other combinedsymmetry conditions.5PiezoelectricMaterialVoltmeterPiezoelectricMaterial---...^OpenSwitchI II ^BatteryPiezoelectricMaterialClosedSwitch1 IBatteryPiezoelectricMaterialVoltmeterK: -)6Figure 1-4. Direct piezoelectric effect.Figure 1-5. Converse piezoelectric effect.With the direct piezoelectric effect there is the generation of a charge or polarizationupon the application of stress. Thus, if the charge distribution is not symmetric about thecentre of the unit cell, deformation of the crystal can cause a shift in the polarization. 1°This is the basis of the direct piezoelectric effect. When such a material is mechanicallydeformed, an electric charge develops.32SymmetryPoint Groups721Noncentrosymmetric20Piezoelectric(Polarized Under Stress)10Pyroelectric(Spontaneously Polarized)Subgroup Ferroelectric(Spontaneously Polarized,Reversible Polarization)11CentrosymmetricFigure 1-6. Interrelationship of piezoelectrics and subgroups on the basis of internalcrystal symmetry. [9]8Figure 1-7 shows the unit cell of barium titanate (BaTiO3) from which thepiezoelectric effects may be better understood. In its piezoelectric form, the centre Ti4+ ionis displaced in the opposite direction with respect to the 0 2  ions, inducing an electricdipole. By the application of a stress, the dipole's length changes affecting the magnitudeof the electric dipole (Figure 1-7b). Conversely, an external potential induces an electricfield which effects the length of the dipole and therefore the dimension of the piezoelectric(Figure 1-8d).Devices made from single crystals are limited in terms of shape and size by the initialcrystals. Because ceramics are polycrystalline aggregates, their random orientation inhibitspiezoelectricity, so in order for a ceramic article to be piezoelectric it has to be oriented.This can be done via techniques such as extrusion and directional recrystallization. 1413What makes piezoelectric ceramics practical is the ferroelectric properties of certainpiezoelectric ceramics. Ferroelectricity is the ability of certain crystals with an electricallypolar structure to switch their direction of polarity under the influence of an electrical fieldbetween several crystallographical directions and to retain their new orientation afterremoval of the field. 14 This means that reorientation can be readily accomplished by theapplication of an electric field instead of the difficult task of aligning the crystals. Theterm, ferroelectricity, was used because of the analogous behavior observed with theferromagnetic nature of ferrites.As shown in Figure 1-6, ferroelectrics are a subgroup of pyroelectrics.Pyroelectricity is the ability of a material to be spontaneously polarized and to change theirdegree of polarization with temperature, hence the name pyroelectricity. The fact that aceramic can be readily polarized by the application of an electric potential means that itspiezoelectric, pyroelectric, or electro-optic properties can be easily controlled electrically,0.403 nm0.398 nm0.398 nma)0.008 nmT0.006 nm;0.006 nrn:Baz-• 1-14#dfacilitating their use. This has lead to the use of the term 'ferroelectics' to be associatedwith all useful pyroelectrics, piezoelectrics, and electro-optic materials so that they arecommonly but incorrectly used in place of one another.What makes piezoelectrics useful is the fact that the output signal is proportional tothe input signal, whether the device operates by the direct or the converse piezoelectriceffect. The electric charge is proportional to the applied stress in the direct effect devicesand vice-versa for converse effect devices.9b)Figure 1-7. Piezoelectric effect of BaTiO 3 , caused by an ions shift with respect to thecorner Ba2+ causing the unit cell to become noncubic. [11] 104-■•■■•••*1-144.41144.44dr (a)^ (b)^(c)^ (d)Figure 1-8. Piezoelectric material (schematic). (a) No external field: centre ofpositive and negative charges not coincident. (b) Pressure applied: compressionleads to voltage differential. (c) Pressure applied and electrical path present:compression leads to charge transfer. (d) External potential applied: dipolelengthening produces dimensional change. [11]1.1.2 History of Ferroelectric PiezoelectricsThe phenomenon of a charge developing upon the application of a stress wasdiscovered in 1880, by Jacques and Pierre Curie, working with quartz, zinc blende,tourmaline, and Rochelle salt (potassium sodium tartate). 15,16 This phenomenon was named"piezoelectricity" by W. Hankel in 1881. The prefix "piezo" comes from the ancient Greekword for pressure, therefore the literal translation is "pressure electricity".Piezoelectricity was a laboratory curiosity until 1916 when P. Langevin developedthe first major application of piezoelectricity for use in submarine detection. 16 Langevin'sdevice is the predecessor to modern-day sonar. At about the same time, the mechanicalresonance properties of quartz were discovered. This led to the use of quartz crystals ashighly stable frequency determining elements in radio communications.Ferroelectricity was first discovered in Rochelle salt by J. Valesek in 1920.' 5 Thediscovery of ferroelectricity suggested a way to create useful piezoelectric devices usingpolycrystalline materials. Prior to the 1940's, all commercial applications of piezoelectric11materials involved the use of naturally occurring single crystalline minerals. The first hintthat ceramic oxides might possess some unusual properties was suggested by the work ofThurnauer and Deaderick in 1941, on a series of barium oxide-titanium oxide (BaO-TiO 2)compositions. 15 When these materials were tested, a dielectric constant as high as 1100 wasobserved. This is in comparison to a value of 100 for rutile (TiO 2), which until then had thehighest known dielectric constant. Because of the second world war, the unusual propertiesof barium titanate were discovered independently by researchers in the United States,United Kingdom, Soviet Union, and Japan.There were three important steps in the discovery and understanding of piezoelectricceramics. The first step was the fact that these new piezoelectric ceramics also seemed topossess a high dielectric constant. The second step was the realization that the cause of thehigh dielectric constant was ferroelectricity. The third step was the discovery of the"poling" process to impart piezoelectric properties into polycrystalline ceramics.Prior to the discovery of piezoelectric ceramics it was not understood how arandomly oriented polycrystalline material could exhibit piezoelectricity. Knowledge ofthe poling process, in addition to the inherent ferroelectric properties of certainpiezoelectrics, made it possible for a select group of polycrystalline ceramics to exhibitpiezoelectricity.The first commercial piezoelectric ceramic devices were BaTiO3 turntable cartridgesmarketed in 1947. Through various additions to BaTiO 3 , improvements in temperaturestability and voltage output were realized. In 1952, the structurally different, leadmetaniobate (PbNb2O6) was discovered by Goodman. 15 '17 Of great practical importancewas the discovery, in 1954, of lead zirconate titanate (PZT) with its strong and highly stablepiezoelectric characteristics.1512There are several reasons why PZT compositions have dominated piezoelectrictransducer applications. PZT compositions (1) possess high electromechanical couplingcoefficients, (2) have a high Curie point, which permits higher temperature operation orprocessing, (3) can be easily poled, (4) possess a wide range of dielectric constants, (5) arerelatively easy to sinter, and (6) form solid solutions with many chemical compositions,thus allowing a wide range of achievable properties. 91.1.3 Applications of Piezoelectric CeramicSince this is the first thesis in this department on piezoelectric ceramics, a briefoutline of their uses was thought to be desirable. Piezoelectric devices allow for the directconversion of electrical energy into mechanical energy and vice-versa without any movingparts. In theory, they could compete with other electromechanical transducers includingelectromagnetic motors and generators. In practice, piezoelectric devices are limited tovery small mechanical displacements (-1-100 gm) and small amounts of electric charge percycle (-0.1 mC). 19 Because of these limitations, they are not useful for low frequencyapplications and are essentially useless for static forces and fields. Piezoelectrics becomeincreasingly useful with higher frequencies yielding substantial acoustic or electric power,even into the 1-10 horsepower range. Piezoelectric DevicePiezoelectric ceramic devices can be broken down into three different groupsaccording to how they operate. Depending on the intended use of the device, the directpiezoelectric effect, the converse, or a combination of both are used. Some applications of19,,21piezoelectrics include the following: 10,15,16,1820Direct piezoelectric effect: microphones, spark pumps, gas ignitors, cattle prods, flashbulb actuators, phonograph cartridges, accelerometers, hydrophones, flawdetectors, touch controls;Converse piezoelectric effect: underwater sound generators, sonar, ultrasoniccleaners, ultrasonic welding/machining, pest control devices, loudspeakers,tweeters, headphones, buzzers, alarms, motors, pumps, fans, positioners, printerheads, flaw detectors; andCombination of converse and direct effects: flaw detectors, IF filters, delay lines,surface wave filters, piezo-transformers.1.1.4 Manufacture of Piezoelectric CeramicsOne of the major advantages that ceramic piezoelectrics have over naturallyoccurring piezoelectric materials is the ease with which ceramics can be fabricated intoalmost any desired shape and size. The fact that natural piezoelectrics are single crystalslimits their use since forming is confined to cutting and lapping.The manufacture of piezoelectric ceramics follows that of conventional ceramics.This generally takes place in four steps: (1) powder synthesis and processing, (2) forming,(3) sintering, and (4) finishing.' For piezoelectric ceramics an additional fifth step, poling,is required before the material exhibits its piezoelectric properties. Powder SynthesisPowders made via solid state synthesis or coprecipitation are used to make variouspiezoelectric compounds. Factors such as purity required and costs determine thesynthesis route.In the synthesis of piezoelectric compounds, the raw materials are mixed andcalcined to a sufficiently high temperature to cause the reactants to form a homogeneoussolid solution. A problem with lead oxide (PbO) containing piezoelectrics is itssublimation which can affect the stoichiometry; this is commonly remedied by theaddition of excess PbO. 19 Organic binders may be added to the ceramic powders as aforming aid. Solid StatePiezoelectric ceramics are traditionally made from powders formulated fromindividual oxides. This technique is the most economical method of forming ceramicpowders since industrial quantities of the starting materials are readily available.Appropriate quantities of the constituent powders are thoroughly mixed and calcined. Amajor drawback with solid state synthesis is the inconsistency of the starting powders.Composition variation in the initial powders, make tight control in later processing stepsredundant. CoprecipitationThere are several advantages with the coprecipitation route of powder synthesis.These advantages are:(1) The purity of the powder can be more closely controlled;(2) The particle size of the precipitate can often be made sufficiently small thatgrinding is not necessary; and(3) The homogeneity of a mixed precipitate may be very good. 22It is possible to coprecipitate various piezoelectric powders such as BaTiO3 , 15PZT,23.24 '2526 '27 and PbNb206. 23,24,28,29,30,31 Although purity levels are higher than with thesolid state synthesis route, there may be some residue of the precipitating agent that maybe difficult to remove.14151.1.4.2 FormingMethods of forming used in other areas of the ceramics industry can be applied topiezoelectric ceramic powders. The ability of the ceramic powder to flow, allows it can beconsolidated into complex shapes unlike single-crystal piezoelectrics. The principleforming methods are:isostatic pressing, cold pressing, slip casting, extrusion, injection molding, tapecasting, dipping and evaporation. 13,l5.2"2Organic binders or lubricants added during this step are "burned out" so as not tocompromise the piezoelectric's properties. If slip casting was used, controlled evaporationis necessary to prevent shrinkage cracks. After this stage, the powder is loosely heldtogether but has no strength and is highly porous. SinteringTo densify the compact, the material is sintered. With HIPping (hot isostaticpressing) or hot pressing, the forming step is combined with sintering.Depending on the piezoelectric compound, various sintering temperatures and firingroutines are used. For BaTiO 3 , the sintering temperature is in the 1350 - 1450 °C range.'5PZT articles are normally sintered in the 1200 - 1300 °C range while PbNb2O6 matures atabout 1300 °C. 15 Some piezoelectric ceramics require a relatively low, slow firing, othersrequire a short time at a high temperature.The two steps that have the greatest effect on the end product are powder processingand sintering. Concerns during sintering include:1. Rapid grain growth that could trap pores within grains,2.Excessive grain growth which weakens the ceramic, and3. Partial sublimation of the constituent powder that may change the stoichiometry. FinishingOnce sintered, the ceramic articles are extremely hard and difficult to machine. Inorder to get articles with a good tolerance, conventional ceramic grinding and cuttingmachinery is used with Al203 , SiC, or diamond as the abrasive mediumA set of conductive electrodes, used in poling, are attached to all piezoelectricceramic elements. Electrode materials include silver, gold, nickel, copper, palladium, andplatinum. PolingTo impart the piezoelectric properties onto the ceramics, it is necessary to apply anexternal electric field, which aligns the domains. This is only possible in piezoelectricsthat are also ferroelectrics. Although there is a certain variation in the direction ofpolarization between individual grains, macroscopically there is a net charge distributionon the article as shown in Figure 1-9.The most noted feature of ferroelectric ceramics is its hysteresis loop which is a plotof polarization versus electric field as shown in Figure 1-10. Until the electric field issufficiently high to switch the dipoles in the crystallites, there is little effect. At higherfields, the polarization increases sharply until it is saturated and then begins to level off.With a reduction of the field to zero, the material still has a net permanent polarizationknown as the remnant polarization PR. As the field is reversed, the polarization is initiallyreduced to zero and eventually becomes saturated in the opposite direction. If the field ischanged again, the value of the electric field at which the polarization is zero is called thecoercive field E c. This cycling of the electric field generates the hysteresis loop. Polingconsists of applying the initial one-quarter of the hysteresis loop, leaving the material witha remnant polarization of PR.16(a) ( b)VIRGIN CURVE• +EFigure 1-9. Dimensional change and alignment of a ceramic as a result of poling (a)before poling (b) after poling. [9]17(a)Figure 1-10. Typical hysteresis loop for ferroelectrics. [9]181.1.5 Lead MetaniobateThis study deals primarily with the sintering of PbNb2O6 and will be the onlypiezoelectric ceramic that will be dealt with in any great detail. There are several reasonsfor the selection of this material for study. The sponsor of this research is B.M. Hi-TechInc., a manufacturer of piezoelectric underwater transducers, who has an interest in thisparticular material. Secondly, since piezoelectric properties are material dependent, theunique properties of PbNb 2O6 make it well suited for specialized applications.Although lead metaniobate has been known to be a piezoelectric since 1952, it hasnot been used extensively in commercial applications. As mentioned previously, PZT wasdiscovered in 1954 but because of its greater versatility it has dominated the market forpiezoelectrics since its discovery. 15 Due to the unusual properties of PbNb2O6 there isrenewed interest in this material.Table 1-1. Properties of various piezoelectric ceramics. [14]Dielectric Piezoelectric Constant^Young^Q^Coercive Field E, Curie Pointconstant^(10.12 C/N)^Modulus (103 V/m)^(°C)(1010 N/m2)K3 d31 d33 Y Q E,BaTiO3 1700.0 78.0 190.0 11.0 400 400 115Ba,3.9Pb0.,TiO3 500.0 23.0 70.0 12.0 800 500 150Na1 .6C42Nb206 2000.0 70.0 175.0 11.0 300 1000 220PbTi0.45Zr05503 500.0 56.0 130.0 7.5 300 >1000 350PZT-4 1200.0 105.0 250.0 8.1 600 >1000 340PbNb2O6 225.0 11.0 80.0 3.5 11 >1000 570Quartz crystal 4.5 2.3 2.3 8.0 106 Not ferro.19Table 1-1 lists some properties of various piezoelectric materials. The dielectricconstant is the ability of a material to hold a charge and is the important property forcapacitor use. Although BaTiO3 is not presently used for its piezoelectric properties,because of its high dielectric constant it is still used extensively in capacitors. The highestpiezoelectric constants are those of PZT which means that it generates the largest output fora given input. Both PZT and quartz have large values of Q which is a measure of howmuch energy is stored compared to how much is dissipated when resonating.To indicate the directional characteristics of the piezoelectric, subscripts are used.The subscript 3 is the direction of the poling field and subscript 1 is perpendicular to thefield. The first index refers to the direction of the applied electric signal, and the secondindex to direction of elastic stress or strain. Looking at the two piezoelectric constants ofPbNb2O6 it can be seen that the values of d 33 and d31 are not large when compared to thoseof other materials which is a disadvantage. However, the d 33/d31 ratio is approximately 8and this ratio is larger than that of the other piezoelectric ceramics meaning that PbNb 2O6exhibits greater directionality. This material property of PbNb2O6 would be advantageousfor transducers used in determining the location of objects.Since piezoelectric elements could be considered to be a structural component of adevice, they must withstand high stress levels, which necessitates a high Young's modulusof the material. In this respect, the use of PbNb 2O6 with its low Young's modulus would bea disadvantage. In addition, PbNb2O6 's rapid grain growth and the associated trappedporosity further compromises the strength of PbNb 2O6 elements during poling and inservice. 34The Curie point of PbNb2O6 is much higher than that of other known piezoelectricceramics, allowing PbNb 2O6 to operate at higher temperatures. When piezoelectrictransducers are oscillated, some of the energy is converted to mechanical motion and some20into heat for each cycle. For a given potential, transducers at higher frequencies attainhigher temperatures. Because of PbNb 2O6 higher Curie point it can operate at higherfrequencies than those of other piezoelectrics.At shorter wavelengths or higher frequencies it is possible to resolve smaller objects.In an analogous example, a scanning electron microscope has greater resolving power thanan optical microscope because the wavelength of the electrons is smaller. This also extendsto acoustics so that a PbNb 2O6 transducer operating at a higher frequency can be used toresolve smaller objects. The upper operating frequency of PbNb 2O6 is 25 MHz while theupper operating frequency for PZT is 10 MHz. 35 A limitation to this greater resolution isthe fact that high frequencies are attenuated more, making detection of deeply recessedflaws difficult. 21,361.1.5.1 Crystal StructureLead metaniobate was the first piezoelectric discovered that did not have theperovskite structure of barium titanate (Figure 1-7). PbNb 2O6 exists as two phases at roomtemperature, the orthorhombic ferroelectric phase and the nonferroelectric rhombohedralphase.37The powder x-ray diffraction pattern of the orthorhombic phase indicates that it hasunit cell dimensions of a=17.65, b= 17.91, and c=7.736 A with a space group of Cnam2containing 20 molecules per unit ce11. 38 Above its Curie point of approximately 570 °C,the orthorhombic phase converts to a tetragonal structure having lattice constants a=12.56and c=3.925, with a space group of P4/mbm. 14 The orthorhombic phase arises as a resultof a slight distortion of the tetragonal phase.Figure 1-11 shows schematically the projection of the tetragonal structure along the[001] axis. A "cross" represents a Nb0 6 octahedron that projects into and out of the page.21These octahedra link at their corners forming rings of 3, 4, and 5 members, leaving holesat the centre that the Pb 2+ ions occupy. In PbNb2O6, the Pb2+ ions occupies five out the sixsites in the unit ce11. 38 '39 Figure 1-12 shows a perspective view of this structure.Appropriate heat treating was chosen to ensure the absence of the rhombohedralphase of PbNb2O6 in the end product. The rhombohedral phase is stable up to atemperature of 1200 °C. To form the ferroelectric orthorhombic phase the PbNb 2O6 has tobe rapidly cooled from temperatures in excess of 1250 °C.' 5 '37.4° These temperatures areapproximate, for example some work has shown that 1200 °C was high enough for theorthorhombic phase to form, but required 2.5 hours until the transformation went tocompletion. 31 The equilibrium phase diagram of Pb0-Nb2O5 (Figure 1-13), indicates thatit is possible for the transformation to occur at a temperature as low as 1150 °C.41Figure 1-11. Structure of PbNb2O6. Possible locations for Pb 2+ ions are shown ascircles. [43]0 Lead^°Oxygen^• NiobiumFigure 1-12. Tetragonal cell structure of PbNb 2O6 showing pemvsldte typesub-units. [37]L+P 3 N 2^L. r_aN ss L+T - PNss L -N,^__ —,,_ 7 ---^7L+ P2N ^T \-734s3s.^13, ,/ L + PN, s s'^/^ -^-+1400 —^7\^c7° , 7 N ss33°^---:-.....-2. I^‘---^310.7 -1--1-;--....- - -4--L^5 2^2 N ss --,,,,,... \t/i G-,33*N.,^,d,, )  ^334' I7 \1220 °-- .--------tp,N, +1200 .-^ 225° J '7 -PN/ L"  115C°/^4-/ P5 N 21000—^/ 985* /-*P2 N•P 3 NP 3 N 2R-PN 60^80Mot.Figure 1-13. PbO-Nb 205 phase diagram. P PbO; N Nb205 ; T-PN -- tetragonalPbONb205 ; R-PN rhombehedral PbaNb 205; ss -- solid solution; L -- liquid. [41]22k",jL P3N835° 800 i \ L + PY ssPY ss + P 3 NI P3 N+P 5 N^600—t^zo ^Pb0^20s szcsj z^Nz0-40R-PNssPN 2 ss.L R-PNssIT-RN ss+ PN 2 ss75° ssI +R-PN ssPN 2 +NssPN 2 ssa_N ssNb 2 05,A 8 C 0TAN. 6 (300' C.) 0.040 0.030 0.035 0.035CURIE TEMP. (.c.) 575 575 HIGH ELECT. LOSSESK c.I. 6280 42 70 HIGH ELECT. LOSSESK 25 219 171 82 80POROSITY (*/•) 13 18 8 4LEAD LOSS^( '/..) 3.6 2.5 0.7 1.2A80400O1^_L^t 50^100^ISO^200 250^300^350^400^450TEMPERATURE (*C. )5001.1.5.2 Previous Sintering StudiesThe majority of the previous work on PbNb 2O6 has not dealt with sintering.Emphasis has been on the electrical properties and atomic structure, with sinteringconducted to make test specimens. The effect of firing temperature and time on weightloss, porosity, and dielectric properties has been investigated by Nelson et al.34 Theyfound that the weight loss of PbO as a percent of available PbO varied linearly with time,for temperatures between 1250 and 1350 °C.Trapped porosity from excessive grain growth at 1300 °C and partial melting at1350 °C were explanations used by Nelson et al. to account for the decreased porosity atthe higher temperature. When coprecipitated PbNb2O6 powder is used, the small size ofthe powder produced excessive grain growth when sintered at 1250 °C making specimenunusable 2A Figure 1-14 shows that high temperature firing yields specimens with a higherdielectric constant and a lower dissipation factor in spite of the higher PbO loss andgreater porosity.23Figure 1-14. Dielectric constant versus temperature for PbNb 2O6 sintered at varioustemperatures. [34]24Small compositional changes (+/- 4 molar % Pb0:Nb0 5) were found to have noappreciable effect on the electrical properties 34 Fickert et al. studied various compositionsin the Pb0-Nb2O5 system. Figure 1-15 shows the dielectric constant and piezoelectricconstant d33 as a function of Nb2O5. To explain the high piezoelectric and dielectricproperties of off-stoichiometric compositions Fickert et al. proposed the presence of twophases to account for its high activity. 28Various studies have looked into the effect of additions to PbNb206. In general,dopants cause a decrease in the Curie point but broaden the dielectric constant versustemperature curve as shown in Figure 1-16. If only small quantities of dopants are added,this effect assists in creating a material with a higher dielectric constant at roomtemperature.Cadmium additions lessen the temperature variation of the dielectric constant andimprove the fired density. 34 Strontium additions made sintering easier and yielded aferroelectric material at a temperature of 1170 *C. 24 The substitution of Bi3+ and Ti4+ forPb2+ and Nb3+ respectively, decreases the Curie point. 39 The electrical properties areenhanced when Pb2+ is partially replaced by Mg 2+, Ca2+, Sr2+, and Ba2+.37 Other dopantssuch as TiO2 and ZrO2 have been used to reduce the tendency of the PbNb 2O6 to absorbwater!' Dopants such as Pb 2Ti2O6, Pb2Zr2O6, Pb2Sn2O6, Lac.33Nb206 , Sm0.33Nb206 , andY0.33Nb206 were used by Subbaro and Shirane 42 to see their effect on lattice parametersand Curie temperature. The addition of Ta 5+ as a replacement for Nb5+ was found tohinder the formation of the ferroelectric orthorhombic phase.4350^ 51Nb 205 Content [mole %]1201008060 CL4M"cr402005349600500400300200100025Figure 1-15. d33 and dielectric constant versus composition. [28]Temp (°C )Figure 1-16. Dielectric constant of PbNb2O6. Curve (A) PbNb 2O6-ZrO2(98.63%-1.37%); (B) PbNb2O6 corrected for specimen porosity; (C) PbNb 2O6measured. [17]261.2 Driving Force for SinteringThe driving force for sintering is the decrease in surface area and therefore the totalsurface energy as the solid-vapor interface is replaced by the lower energy solid-solidinterface. The net decrease in the free energy resulting from sintering 1 gm size powders tofull density is in the order of a few J/gm." Because there is only a small decrease in totalfree energy, this fact does not satisfactorily account for the sintering of a loose compact tofull density. The essential step in sintering is the interaction at the interface betweenparticles.Stresses during sintering arise from difference in surface curvature. This effect may bestudied by considering the expansion of a bubble from the end of a capillary tube submergedin a liquid bath as shown in Figure 1-17. If the density difference is negligible, the workdone by the expansion of the bubble goes into increasing the surface area and the totalsurface energy. Equating the work from the expansion (APdv) to the increased surfaceenergy (ydA) yields:APdv = ydA.^ (1-1)1APdr1Figure 1-17. Difference in pressure AP necessary to increase the radius of asubmerged bubble. [44]— — -z‘ Expanded Surface.I 44.713\\ r =Radius of Curvature .Figure 1-18. Expanded volume element due to transfer of mass.Now, consider the expansion of a unit surface area as shown in Figure 1-18 where thesurface area is ABCD, the angle between AB is 0 1 , the angle between BC is 02, and the radiifrom the common centre of each segment is r1 and r2 respectively, therefore:02= 1r2(1-2)If each side is increased by dr the expanded area becomes:= (r1 + dr)0 1 • (r2 + dr)02= (r i + dr)—xr1 • (r2 + dr) —r2ry rir2 + ridr + r2dr + dr 2]= —[r1r2and neglecting the squared differential term (dr2) and noting that xy = 1,dr dr1= + — + —r2 r1therefore the change in area is:dA = Aafter -Abefore = 1 + —dr +—dr - 1r2^r 127(1-3)(1-4)dA = dr(-1 + —1).r2Substituting into equation 1-1 yields:1^1AP& = 7dr(— + —r2but dv = xy dr = dr1 + 1(3=AP =7r2The pressure or stress which develops across a curved surface may be substantial forsmall particles. A similar equation relating the pressure arising from a curved surface due tocapillary forces of liquid phase sintering was developed by Kingery.45 The configuration atthe neck is slightly different for spherical particles as seen in Figure 1-19. The radii ofcurvature between the neck (p) and the particle (a) have opposite signs with respect to thesolid surfaces.Figure 1-19. Schematic representation of contact area between two partially sinteredspheres.28(1-5)(1-6)29Since p < a, the stress takes a negative sign as shown in equation 1-7. This means that thestress acts outwards from the neck causing material to move to the neck. When the neckinitially forms p << a with the stress being mainly affected by the neck radius. As the neckgrows the stress associated with surface curvature decreases.(1-7)a= p aAnother approach of determining the stress associated with pressureless sinteringinvolves the force balance at the neck. Figure 1-20 shows the neck element ABCD withforce F x and F p in nonequilibrium. The forces on element ABCD are:F = x +^ (1-8)The magnitude of these forces are:Fx = yAD^F = yAB^(1-9)where y is the surface tension. AD=p0 and AB x0 for small 0 thereforeFx = —yp0^Fp = yx0.^ (1-10)The resultant force on element ABCD is:= 2.(Fx sin; + Fp sin;e ebut sin -2 - -2 for small 0_ 027(p _x).Since the area of ABCD is xpO2 the above equation becomes/ —1 1= p xas previously calculated.(1-12)(1-13)Figure 1-20. Force balance for neck element during sintering.As an estimate of this stress, consider 10 µm diameter SiO2 glass spheres with a neckradius of 10% of the particle diameter, and a surface energy of 300 ergs/cm 2. Substitutinginto equation 1-7 yields a stress of 540 kPa." At elevated temperatures this stress can havean appreciable effect on densification.This stress causes vapor and vacancy concentration gradients between the convex andconcave regions. For the case of vapor concentration, the use of the gas law yields:PP K7'— dPPa PI V dP =JV AP =V (P p — P ,i)= K7' 1n7,--^ (1-14)where P, is the vapor pressure over the neck, P. is the vapor pressure over the particle, K isBoltzmann's constant, T is the absolute temperature, and Pp -Pa <0. Multiplying equation1-13 by unit area to get AP and substituting into equation 1-14 yields:Pp^1-1 1 )^ (1-15)K7' In--- = + —"a^p a3031where p < a. The relationship for vacancy concentration and vapor pressure is PplPa =CalCpwhere Cp is vacancy concentration at the neck and Ca is the vacancy concentration of theparticle. Substituting this into equation 1-15 yields the relationship between vacancyconcentration and surface curvature shown below.CD^I —1 1K7' In Ca = —V77 ±—p a(1-16) Since the right hand side of equation 1-16 is positive this indicates that there is a higherconcentration of vacancies at the concave surface (neck) than the convex surface (particle).The curvature difference causes a vapor pressure and vacancy concentration gradient thatleads to material transport to the neck region. Depending on the mass transport mechanism,densification of the powder compact may occur.1.3 Sintering TheoriesSintering refers to the consolidation and densification of powder at temperaturesnormally above one-half the melting temperature T., where T m is in degrees Kelvin. Initiallythe particles coalesce and as the fused region increases there is densification. Sintering canoccur in the solid state, in the presence of a liquid phase, in viscous bodies, or under anapplied stress.Sintering can be broken up into 3 different stages. The initial stage is characterized bythe fusion of particles resulting in the formation of necks. When bonds between particleshave grown to an appreciable size relative to the particle diameters and the pores are stillinterconnected the pore geometry enters the intermediate stage of sintering. The final stageis characterized by the presence of isolated pores and grain growth.Sintering is a complex process that is further complicated by simultaneous processessuch as grain growth, particle rearrangement, phase transformation, and chemical reactions."32Many attempts have been made to develop an understanding of the processes occurringduring sintering, with varying degrees of success. Investigations on the sintering of twoindividual particles have been fairly successful but extension of these findings to thesintering of compacts composed of a large number of particles have met with questionablesuccess. It should be noted that the various stages and mechanisms occur throughout thesintering process, and that subdivision into dominant mechanisms at different stages is donein order to make the subject more understandable.Most of the current initial stage sintering theories are based on studies of the neckgrowth between two individual particles as a result of various mass transport mechanisms.Kuczynski48 derived and experimentally verified neck growth equations between twoparticles based on mass transport mechanisms such as evaporation/condensation, volumediffusion, and surface diffusion. Subsequent WOrk45.49.50 '51 '52 yielded the same general form ofKuczynski's neck growth equation:y)n^F ayogamKT(1-17)where y is the surface tension, co is the molar volume, D is diffusion coefficient, K isBoltzmann's constant, T is the temperature (Kelvin), t is the time, y is the neck radius and ais the particle radius while constants n, m and F vary according to the neck growthmechanism and the diffusion path.Using Kuczynski's general neck growth equations, similar equations for the sinteringof spherical monosized particle compacts were developed relating linear shrinkage (AL /Lo )to time (t) (Table 1-2). The generalized shrinkage equation for compacts was determined tobe:(1-18)33where D, y, co, K, T, and t were previously defined. In general the constants G, n, and m donot correspond to those for the neck growth equation given the same transport mechanismand diffusion path.By plotting in e versus In t a linear relationship with a slope of m is predicted. Fromthe slope the mass transport mechanism can be determined. Johnson and Cutler in their testswith alumina found an initial curved portion, then a linear portion followed by a curvedportion 52"53 The initial curved portion was attributed to particle rearrangement andequilibration with respect to temperature while the final curved portion was attributed toneck impingement.Table 1-2. Time dependence of various sintering mechanisms.Mechanism for the Shrinkage ofCompactsStage 1 Sinteringe ocAuthorsViscous Flow t Frenkel47Volume (Bulk) Diffusion 0 8.t Kuczynski48Neck Volume Diffusion P.4 Kuczynski48Grain-Boundary Diffusion 10.33 Kuczynski48Liquid Phase Sintering (LPS) t0.33 KingerymLPS boundary reaction ratecontrollingtOS Kingery51In all derivations it was assumed that the number of contacts per particle remains thesame, although it has been shown that particles do rotate and that the number of contactsincreases as sintering progresses. 51 '56 Grain growth was also assumed not to be present. Theequations do not take into account the impingement of the necks with one another, therefore,are only applicable to a few percent shrinkage (— 7%) during the initial stages.34Figure 1-21. Tetrakaidechadron with edges where cylindrical pores develop (stage 2)and corners where spherical pores develop (stage 3).For the derivation of the intermediate stage sintering model, it was assumed that thepores are continuous and cylindrical. Porosity decreases as the cylindrical pore shrinks.Coble54.55 used diffusion mass transport mechanisms and started with a shape capable ofattaining full density: a tetrakaidechadron. The porosity was assumed to be located along theedges of the tetrakaidechadron presented in Figure 1-21. Coble derived and verified theequation:rtD ^jrnInICT^t)m (1-19)where F, D, K, T, t, and y have been previously defined. P is porosity, SI is the vacancyvolume, 1 is the length of the cylindrical pore, t1 is the time to reach full density, and m, n aredependent on the sintering mechanism whether it be boundary diffusion or bulk diffusion.Similar equations to Coble's were developed by Beere 56 to take into account a more realisticpore shape and by Gessinger et al. 57 for liquid phase sintering.Final stage sintering involves the removal of spherical pores from the grain corners(Figure 1-21). Coble54.55 derived and experimentally confirmed the equation:P — (F 7r.D Sly(tf — t)1 3 KT ) (1-20)35where the variables have been defined. Previous work has suggested that trapped gaseswithin pores may cause densification to decrease 55'56,58The models discussed could not independently explain the sintering of PbNb 206, as thedensification ranged from 5% to 30% from an initial density of —65% of theoretical. Thismeant that multiple stages of sintering were occurring at different times with the possibilityof multiple mechanisms operating simultaneously. The PbNb 2O6 particles were also smoothparticles of irregular shape which exhibited significant grain growth. Because the PbNb 2O6particles were not spherical monosized particles, there is some question of using initial stagesintering models with PbNb2O6 .1.4 ObjectiveThe objective of this project was to study the sintering kinetics of lead metaniobatewith various dopants, some of which have a low melting temperature. This was done inorder to lower the sintering temperature which may help in reducing grain growth normallyencountered when sintering PbNb2O6. The dopants used were namely SrO, Bi203 , LiNbO3 ,and KNaNb2O6. It was hoped that these compounds would have a minimal effect on theferroelectric properties of PbNb2O6 and act as grain-growth inhibitors.The sintering behavior was studied by measuring the linear dimensional change ofcylindrical compacts under isothermal conditions in the temperature range 1050 to 1250 °C.The effect of varying amounts of dopant, up to 1 weight percent, was also investigated. Thechoice of dopants was based on previous work as well as on the selection of dopants thatwere known to be ferroelectric.362 EXPERIMENTAL PROCEDURE2.1 MaterialsLead metaniobate powders used in this study were prepared by B.M. Hi-Tech Inc. Astoichiometric mixture of reagent grade PbO and Nb 205 was ball milled for 8 hours. Tohomogenize and to form PbNb 2O6, the mixture was calcined at 800 °C for 8 hours. It wasthen furnace cooled to room temperature and vibratory ball milled for a further two hours.Two 500 gm. batches of PbNb2O6 powder were supplied. Intensity peaks from x-ray diffrac-tion tests of each batch matched known 2e values of PbNb 2O6. It should be realized that the10 hour total milling time with Al 203 balls used in preparing the PbNb2O6 powder must haveintroduced some Al203 . It is not known low much alumina is present and what could be theeffect of it's presence in the system.The dopants added to the PbNb2O6 were Bi203 , SrO, LiNbO3 , and KNaNb2O6 .Reagent grade Bi203 was available, unlike SrO which was made by the decomposition ofreagent grade SrCO3 while LiNbO3 was made by reacting a stoichiometric mixture ofreagent grade LiNO3 and Nb205 at 1000°C for 6 hours. KNaNb2O6 was made by reacting astoichiometric mixture of reagent grade K2CO3 , Na2CO3 , and Nb205 at 1000 °C for 6 hours.X-ray diffraction patterns were taken to ensure that the reactions went to completion.Dopants levels of 1, 0.5 and 0.25 weight percent were added to the PbNb 2O6 powders.The doped powders were then vibratory ball milled twice for 5 minutes in an alumina con-tainer with alumina balls. Figure 2-1 shows a flow chart of the steps used to produce thefinal powder. The particle size distribution of the 'as-received' and the vibratory milledpowders were determined using a Horiba CAPA-700 particle size distribution analyzer. Ascanning electron microscope (SEM Hitachi S-570 or S-2300) was used to study the frac-tured specimens and powder morphology.Ball milled, 8 hoursFurnace CoolAttrition Milled 2 hourstiDopant AdditionSr0, Bi 203 ,LiNb0 3 ' KNaNb 206Attrition MilledAl 203 balls and container5 minutes twiceCalcined 800 CAir, 8 hoursiPbNb 20 6PbODoped and UndopedPbNb 206Figure 2-1. Flow diagram of the processing steps used to synthesize PbNb 206powder.372.2 Sample PreparationTo study the sintering behavior of the PbNb2O6 powders, cylindrical pellets having thenominal dimensions shown in Table 2-1 were prepared as follows. A quantity of doped orundoped powders was weighed and cold compacted in a double-action floating die to apressure of 50 MPa (7000 psi). The first group of compacts weighed approximately 2grams but due to difficulties when measuring their electrical properties thinner compactswere prepared. Later compacts were made using approximately 1.45 grams of PbNb 2O6 .Table 2-2 lists the parameters (dopants, dopant concentrations, sinter temperature, andmilling) of the specimens used for the isothermal sintering experiments.From the literature it was found that platinum had been used successfully to containPbNb2O6 at high temperatures 12,17,34,39,40,42,43.^Following these recommendations, all surfacesin contact with PbNb2O6 , at elevated temperatures (>1000 °C), were made of platinum.Table 2-1. Dimensions and properties of powder compacts.Sample diameter^9.5 mm (0.375 in.)Sample weight^—1.4 gramsSample height^—4.5 mm (0.18 in.)Initial density^—0.6 P theoreticcd3839Table 2-2. Outline of isothermal shrinkage experiments conducted listing the wt. % dopantused for a given temperature and dopant.Sintering TemperatureDop ant 1200 °C 1150 °C 1100 °C 1050 °CSrO 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25Bi203 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25LiNbO3 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25KNaNb206 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25 1, 0.5, 0.25'milled' done done done done'as-rec.' done_done done done12.3 Isothermal Experimental ApparatusA schematic diagram of the apparatus used for the isothermal sintering study ofPbNb2O6 is shown in Figure 2-2. It consists of a vertical tube furnace, AC linear variabledifferential transformer (LVDT), an input module, and a chart recorder.Power SupplyTemperature ControllerFigure 2-2. Equipment used for isothermal compaction experiments.RollerTrackQT 0 CQ40LVDTAssemblyType KThermocouplQuairtz TibeTyr e SThermocoupleSuperKanthalTubeFurnaceI^IInput ModuleChart RecorderPower412.3.1 Vertical Tube FurnaceThe vertical tube furnace is composed of superkanthal heating elements with a powersource/temperature controller capable of reaching 1600 °C. This furnace was limited to anoperating temperature of —1300 °C because of the 3.9 cm diameter quartz containment tube.To record the sintering temperature, a chromel/alumel thermocouple (type K) was attachedto the LVDT apparatus and positioned next to the compact. As shown in Figure 2-2, thefurnace controller was connected to a platinum/platinum-10 % rhodium thermocouple(type S) which measured the temperature at the exterior of the quartz tube. The controllerattached to the Pt/Pt-10 % Rh thermocouple was adjusted to correspond to the temperaturegiven by the chromel/alumel thermocouple near the sample.The intermittent nature of the power supply meant that the temperature varied byapproximately +/-3 °C. Because of this, the midpoint of the variation was used as thesetpoint for the experiments.2.3.2 Linear Variable Differential TransformerAn expanded view of the LVDT assembly used to measure isothermal compaction asa function of time is shown in Figure 2-3. It consists of an alumina tube with an outsidediameter of 26 mm and an inside diameter of 20 mm. At the bottom of the tube a15 x 30 mm window was cut, across which a 3 mm thick alumina platten was placed tosupport the sample. A 40 cm long, 1.5 mm diameter alumina rod was inserted into thelarger alumina tube. This rod extended from the alumina platten to a 30 cm brass extensiontube that supported the iron core of the LVDT. A 7 mm hole was core drilled 130 mmfrom the bottom of the large alumina tube so a guide could be inserted for the 1.5 mm rod.LVDT Assembly(Cross Section)WaterCooling04—Brass— AluminaConnectionInput Module"NI-- Type KThermocoupleRod GuideChart RecorderFurnace42Roller^Brass TubeTrack^and Iron CoreFigure 2-3. LVDT assembly and arrangement used for isothermal compactionexperiments.43Around the top of the alumina tube a water cooled copper collar was wrapped toprevent heat buildup of the metal and electrical parts. The LVDT body was connected tothe assembly by a threaded sleeve allowing for 5 cm of adjustment.The LVDT was connected to a Daytronic type-70 differential transformer inputmodule. A setting of 0.127 cm (0.050 inches) full scale range was selected for allexperiments. Before each experiment was conducted the input module and the chartrecorder were zeroed and calibrated using a metal strip of known thickness. The inputmodule was connected to a two-pen Kipp & Zonen BD-41 strip chart recorder. Both thedimensional change and the temperature of the specimen were recorded.This assembly was mounted on a vertical roller track allowing it to be raised andlowered into the tube furnace. The assembly was designed so that when lowered, thespecimen window corresponded to the designated hot zone of the furnace. The total weightsupported by the pellet during sintering was 13.920 gm. This represents a pressure of196.4 Pa on the pellet face. The pressure present on the pellet was so small that it wasassumed to be negligible and have no effect on sintering.442.4 Experimental Procedure2.4.1 Isothermal ContractionTo study the effect of temperature, dopant, and dopant concentration on PbNb 2O6 ,specimens produced as described in Section 2.2 were isothermally sintered. The samplearrangement used for studying isothermal contraction is shown in Figure 2-4. The PbNb2O6specimens were placed between two alumina plattens covered with platinum foil to preventthe PbNb2O6 from reacting with the alumina. The alumina tube on the top of the plattenwas used to measure any linear change in the height of the sample during sintering.Isothermal contraction experiments were conducted at 1050, 1100, 1150, and 1200 °Cfor PbNb2O6 with various dopants at concentrations of 1.0, 0.5, or 0.25 weight percent. Allexperiments were repeated several times. Because of the possible adverse effect of dopantson the Curie point of PbNb2O6, dopant concentration was limited to 1 weight percent.The cylindrical test specimens described in Section 2.2 were placed into the LVDTassembly as shown in Figure 2-4 and lowered into the preheated furnace. The assemblywas lowered so the type K thermocouple initially read —400 and then —800 °C before beingmoved into the hot zone. The specimens were held at each intermediate temperature for—10 minutes. Heating the specimen to —800 °C for this amount of time resulted in nocompaction. The temperature setting of the control unit as read by the type S(Pt/Pt-10%Rh) was then adjusted to give the desired temperature on the type Kthermocouple.This heating procedure was used to reduce thermal gradients within the specimen andthe assembly that may lead to fracture. The specimens were held in the hot zone forapproximately 4 hours. Subsequently they were raised from the hot zone in 2 stages eachlasting 10 minutes then allowed to cool to room temperature. The weight and cylindricalAlumina Rod45dimensions were then measured.In order to obtain the ferroelectric orthorhombic phase a further heat treatment wasrequired. The PbNb2O6 specimens were heated in the LVDT assembly at 1250 T for 1hour using the previous lowering and raising procedures. Afterwards the specimens wereweighed and measured again. Grain size calculations were done on specimens after thisheat treatment. As an estimate of the average grain size, the number of grains along threelines drawn on SEM micrographs was determined.LVDTTranducer(Cross Section)Water Cooling AluminaTubePlatinum FoilFigure 2-4. Sample arrangement in the LVDT assembly for isothermal compactionexperiments.2.4.2 Electrical PropertiesSince the eventual use of PbNb 2O6 is in transducers, some electrical properties of thespecimens were measured. Although this thesis deals mainly with the sintering ofPbNb2O 6 , for completion the dielectric constant of the sintered specimens are included inthe Appendix I. In order to pole and test the specimens for their dielectric strength, silverelectrodes were applied. A silver/toluene solution with 3 wt. % glass frit was applied to theflat faces. To consolidate the silver suspension and ensure good adhesion with thePbNb2O6 the silver specimens were fired to 600 °C at a rate of —200 °C/hr. They were heldthere for 1 hour before being furnace cooled to room temperature overnight.All dielectric strength tests on the specimens were conducted by the B.M. Hi-TechInc. A poling voltage of 15,000 volts DC (25-30 kV/cm) was applied for 30 minutes acrossthe silver electrodes while the specimens were heated to 120 °C in a silicone oil bath. Thespecimens were aged for 3 days after which the capacitance was measured and thedielectric constant calculated.2.4.3 Summary of the Experimental ProcedurePresented in Figure 2-5 is a summary of the testing procedure conducted on thePbNb2O6 specimens.46ICold Pressing of SpecimensiIsothermal Sintering4 hrs. at various TemperaturesCooled to Room TemperatureSpecimens Weighed and MeasuredHeat Treated, lhr. 1250 °CCooled to Room TemperatureSpecimens Weighed and MeasuredTElectrodes applied, Fired 1 hr. 600 °CCooled to Room TemperaturePoled in 120 °C silicone oil, 25-30 kV/cmCapacitance MeasuredSEM Study on Fractured SpecimensPowder SynthesisFigure 2-1Figure 2-5. Summary of PbNb206 specimen testing procedure.47483 RESULTS3.1 Powder CharacteristicsThe 'as-received' powders from B.M. Hi-Tech Inc. were characterized with respect tocrystallography, particle size, particle size distribution, and morphology. Because vibratoryball milling during the mixing of the PbNb2O6 with the dopant could have changed theparticle size distribution and therefore the powder's sintering characteristics, the effect ofmilling was also studied. Milling was assumed to have yielded the same particle sizedistribution for all milled powders, both doped and undoped.As shown in Figure 2-1, all the doped PbNb2O6 powders were milled in order tothoroughly mix the PbNb2O6 with the various dopants. When doped powders are referred to,this automatically means that they have been milled, unless otherwise stated. The'as-received' powder refers to the powder obtained by B.M. Hi-Tech Inc. The 'milled'powder refers to the 'as-received' powder that was further milled in-house but to which nodopant additions were made.The particle size distributions of the 'as-received' and 'milled' powder are shown inFigure 3-1 and 3-2, respectively. The particle size distribution plots of both the 'milled' and'as-received' powders represent the average of three separate tests. The figures clearly showthe effect of milling which leads to the formation of smaller particles. While the'as-received' powder exhibited a median particle size of 1.90 p.m, the 'milled' powder had amedian particle size of 1.49 pm. It was therefore assumed that milling done during themixing of the PbNb2O6 with the dopants resulted in a change in the particle size distribution.4920. 100, 16 : 80,>, 12 -. 4; 60C.)C -.51.1111111111 "9. 1.‘1111111111(1) . 11co=0.) 8 "'S 40Liti4a 20()^. 0^•7.0^6.0^5.0^4.0^3.0^2.0^1.0^0.0^a)^Diameter (microns)7.0 6.0^5.0^4.0^3.0^2.0^1.0^0.0b)^Diameter (microns)Figure 3-1. a) Particle size distribution b) cumulative percent before milling.10024 .- 20 -^ .--. 8016 - ;. 60 -t.)^.4.) 12- a^-=- ^40 -cr2 8-40 :. . . . IIIII ^0 •^41III I ° 2°  . . . *.011111 tz.^-7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0a) Diameter (microns) b) Diameter (microns)Figure 3-2. a) Particle size distribution b) cumulative percent after milling.No difference in morphology was noticed between the 'milled' and the 'as-received'powders from the SEM micrographs. Figure 3-3 shows the typical morphology of thepowders prior to sintering. It can be seen that the individual particles are rounded andirregularly shaped. Even though care was taken to disperse the powders by ultrasonicagitation and by the addition of sodium hexametaphosphate as a deflocculant,agglomerations were still present as evidenced in the micrographs. Closer examinationshows necks between some of the grains indicating the presence of hard agglomerates.Partial sintering is believed to have occurred during the calcination of the PbO and Nb 2O5leading to the formation of PbNb 2O6 (Figure 2-1). Individual grains can be seen in themicrographs. It was determined that the 'milled' and 'as-received' powders have initial50grain sizes of —0.5 gm and —1.0 gm, respectively. Because of the hard agglomerates,individual particles do not necessarily correspond to the starting grain size. This should bekept in mind when studying the results from the Horiba CAPA-700 since it considers hardagglomerates as large particles.Differential thermal analysis (DTA) results of PbNb 2O6 with and without the variousdopants are presented in Figure 3-4. Because of the small amount of dopant added to thePbNb2O 6 (1 wt. %) any reaction between the PbNb 2O6 and the dopants was difficult todetect. In order to emphasize any reaction, powder mixtures consisting of —50 volume %dopant and PbNb2O6 were tested. The use of this powder mixture to represent the reactionsoccurring with the lightly doped powder used in the sintering experiments can be justified.Since mixing of the dopants with the PbNb2O6 was mechanical, the resulting mixtureconsisted of individual particles of the dopant and the PbNb 2O6 which resulted in localizedhigh concentrations of the dopant.The 'as-received' and 'milled' powders did not show signs of reaction prior to themelting of the PbNb2O6. The DTA curve of the SrO doped PbNb 2O6 exhibited someendothermic reactions occurring in the range —500 - 650 °C due to the removal of H 2O orCO2 although the SrO did not react with the PbNb 2O6. Figure 3-4 shows reactions for Bi 203doped PbNb2O6 at —740 °C, LiNbO3 doped PbNb2O6 at —1200 °C, and KNaNb2O6 dopedPbNb2O6 above 1200 °C.An x-ray diffraction pattern of the initial batch of powder from 29 values of 20 to 60degrees is shown in Figure 3-5. The second batch of powder was similarly tested and foundto be identical. All peaks were found to correspond to the known 29 values for PbNb2O6.6°a)^ b)-2.5-351Figure 3-3. SEM micrograph of PbNb 2O6 powders a) 'as-received' b) 'milled'.1--- Sr00.5•\ 'milled'\^/0-3.5 ^400. •• ----- • '...^••! LiNb0 3KNaNb2 O 3600^800^1000^1200^1400Temperature ( °C)Figure 3-4. DTA plots of PbNb 2O6 doped and undoped powders tested at 10 °C/min.„va A5230I^I^I40 50^100 ^90 -80 ---- 70 -60C50a.)407)c4 30 -20 -10 -^0 ^20 60Figure 3-5. X-ray diffraction pattern of PbNb2O6 powder.3.2 Isothermal Contraction3.2.1 Stages of ContractionA typical compaction-temperature curve from the beginning of the specimen'sinsertion into the furnace to the end of the experiment is shown in Figure 3-6. Thethermocouple used to measure the temperature of the specimen (Figure 2-3) requiredapproximately 10 minutes before a constant temperature was observed. Previous studies bySunderland6i in which he introduced thermocouples at the centre and outside of specimens,of comparable dimensions, indicated that the centre temperature was —97% of the surfacetemperature within —10 minutes.53From the compaction curve shown in Figure 3-6, three distinct regions can be seen.Region I is characterized by negative values of compaction or in other words, expansion.Over this region the expansion of the assembly and the specimen exceeds the shrinkage dueto sintering. The start of region II begins when the shrinkage of the specimen exceeds theexpansion of the system. This region is characterized by a nonlinear increase in shrinkagewith time. The third region in Figure 3-6 is essentially linear with a steady state of shrink-age with time. The temperature of the specimen was recorded to ensure that during thesintering experiment, the preset sintering temperature was maintained.The beginning of region II as indicated by the change from measurable expansion tocontraction was used as the common point of reference from which the shrinkage study wasanalyzed. As shown in Figure 3-6 this time was designated to with a corresponding speci-men height of Lo. Temperature equilibrium within the samples was assumed to haveoccurred sometime after the beginning of shrinkage within region II.The shrinkage curves were digitized for analysis. The curve starting at to was normal-ized with respect to the specimens initial length L.. The strain or normalized shrinkage wasback calculated knowing the final sintering length Lf, the total shrinkage (AL), and usingEquation 3-1.AL  AL E- L. Lf + AL(3-1)Figure 3-7 shows a typical normalized plot of contraction (E) versus time. Only thesecond and the third regions are shown, as region I was intentionally omitted. Theroughness of the plot was not observed on the chart recorder and was attributed todigitization and normalization of the data. All experiments were repeated several times.The deviation between runs was calculated to be AE — ± 5.0x10 -3 . Error bars of thismagnitude are presented in Figure 3-7.54LTemperature Curve:i,I IShrinkage CurveRegion III1^ I200 24040^80^120 160Time (minutes)0.0180.0161 0.014..=c-)  0 012=^...,t, 0.010g 0.008x- 5 0' 006... c4 0.0040e 0.002ct-= 0.000c...)^'73 -0.002=.2 -0.0045 -0.006A 0.008-0.010-0.012012001100-1000900 R.0-800 E.'700 la600 E500 =g400g5a )300acn20010000.130.12 -0.11 -0.1 -0.09 -= 0.08 -0T..3 0.07 -:13 0.06 -0U 0.05 -0.04 -0.03 -0.02 -0.01 -00^40^80^120^160^200Time (minutes)240Figure 3-6. Typical contraction and temperature curve obtained from the start of theshrinkage experiment.Figure 3-7. Typical normalized contraction curve of PbNb 2O6 . Only regions 11 andIII are shown.553.2.2 General ObservationsThe listing of all isothermal sintering experiments performed at various temperatures,dopants, and dopant concentrations was presented in Table 2-2 of Section 2.2. The phasediagram (Figure 1-13) shows the melting point of PbNb 2O6 to be 1343 °C. Forcompositions with a slightly higher PbO content a liquid phase is present at 1233 °C. Inorder to limit grain growth and the presence of liquid unless desired, 1200 °C was chosen asthe highest temperature for the sintering experiments. Consolidation of the compact at1200 °C was slow enough that the end point density was not immediately achieved whiletemperatures lower than 1050 °C were not used as the amount of contraction observed wasvery small.For dopants known to have a low melting point (Bi203 , LiNbO3 , and KNaNb2O6) itwas expected that they would facilitate sintering above their liquidus temperature. Ternaryphase diagrams of PbO, Nb205, and the dopants used are not available so it is not known ifthere is a eutectic at a temperature lower than that of the melting points of the constituentpowders. Table 3-1 lists the various melting points of PbNb 2O6 and the dopants used.These are slightly higher than those temperatures where reactions occurred as indicated bythe DTA.Table 3-1. Melting point of constituent powders.Compound Melting point (°C)SrO 2430Bi203 860LiNbO3 1200KNaNb206 1 100PbNb2O6 1343563.2.3 DensityOver the temperature range that the isothermal compaction experiments wereconducted, final specimen densities between 60 and 95 % of theoretical were obtained.After heat treatment at 1250 °C the dimensions and weight of the pellets were taken again,yielding relative densities between 86 and 95 %.Table 3-2 lists the relative densities of specimens sintered at the various temperatures,dopants, dopant concentrations, and milling procedures. These density values are ofspecimens prior to the heat treatment at 1250 °C. This relative density table can be used inconjunction with the contraction curves for analysis of the data.As expected, with increasing temperature there is greater densification. Thistemperature effect masks the contribution of the dopants at the higher temperature. In orderto study the effect of the dopants on densification their effect at lower temperatures shouldbe considered. Because the powders with the dopants have been milled, they should becompared with the 'milled' powder. The effectiveness of the lower melting point dopantssuch as Bi203 , LiNbO3 , and KNaNb2O6 in densification can be readily seen. There aresome indications that higher dopant concentrations of these dopants would aiddensification. Unlike the low melting point dopants, the addition of Sr() was found toinhibit densification. Table 3-2 also shows the significant effect of milling ondensification.57Table 3-2. Relative density (%) of PbNb 2O6 specimens after sintering with various dopant,dopant concentrations, temperatures and milling procedures.Dopant wt.% 1200 °C 1150 °C 1100 °C 1050 °CSrO 1 89 83 64 59Bi203 1 88 93 86 71LiNbO3 1 92 90 87 72KNaNb2O6 1 90 91 78 63._Sr0 0.5 89 90 66 61Bi203 0.5 92 92 87 64LiNbO3 0.5 92 92 88 69KNaNb206 0.5 92 92 82 63SrO 0.25 90 90 66 62Bi203 0.25 91 93 82 65LiNbO3 0.25 91 90 88 65K4aNb206 0.25 92 91 85 69'Milled * 91 91 94 79'As-rec.' 92 87 76 60* No dopant.3.2.4 Milling DependenceMilling was found to have an appreciable effect on the shrinkage characteristics ofthe compacts as shown by the comparison of shrinkage plots of the 'as-received' powderwith that of the 'milled' powder (Figure 3-8). The undoped milled powder was studied inorder to separate the effect of milling from that of the dopant additions.In general, milling PbNb2O6 was found to facilitate the shrinkage of the compacts.When comparing the 'as-received' contraction curve with the 'milled' contraction curve at1050 °C it can be seen that milling facilitates contraction. The low strength of PbNb 2O6(Table 1-1) and vibratory ball milling contributed to the formation of smaller particles.This was confirmed by particle size analysis, which showed that milling shifted the medianparticle size and mode to smaller values (Section 3.1). The greater contraction rate of the'milled' powder may be explained by the increased number of contact points which resultsin a higher shrinkage rate. For a given mass of powder, smaller particles have a largersurface area and therefore a larger surface energy which aids in sintering.When comparing the contraction curves for the 'milled' and 'as-received' samples at1200 °C it can be seen that the 'milled' powder curve reached its final compaction level farsooner and more abruptly. This shows that milling assists the specimens to reach their endpoint density sooner. Even though the contraction for the 'as-received' sample was largerthan for the 'milled' specimen as indicated in Figure 3-8, Table 3-2 shows essentially thesame relative density after sintering of 92% and 91%, respectively. A possible explanationfor this is that the greater initial density of the 'milled' compact resulted in less contraction.58,I^----^ ----------------------------------------------------------------------/ ---- milled 1200 °Ci^1^I^I^I^I^i^1^1^140 80 120 160 200 240Time (minutes)0.160.15 -0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 --0.01 —00i59Figure 3-8. Contraction curves for 'as-received' and 'milled' PbNb2O6 powdersamples sintered in air at 1050 and 1200 °C.3.2.5 Dopant DependenceThe effect of the various dopants on densification can best be seen at the higherconcentrations and lower temperatures. Figure 3-9 shows the 1100 °C isothermalcontraction curves for specimens having 1 wt. % of the various dopants. Because of thelow melting temperature of Bi203 (860 °C), its presence as a liquid phase aids in thesintering of PbNb2O6 compared to the other dopants. SrO's high melting point meant that itwas a solid and the data suggest that its addition impedes sintering. The effectiveness ofKNaNb2O6 and LiNbO3 towards contraction is intermediate between that of Bi203 and SrO.A clear distinction as to which of the two dopants (KNaNb 2O6 or LiNbO 3) is more effectivecould not be made with certainty.60In Figure 3-9 it can be seen that the addition of any dopants hinders sintering when itis compared to the undoped milled powder. This appears plausible for SrO doped PbNb 2O6as it possesses a high melting temperature. However, this explanation could not be usedwith those dopants that form a liquid over the temperatures studied. A possible explanationis that self diffusion is greater than diffusion in a liquid. It could be that PbNb 2O6 isinsoluble in the liquid formed, thereby impeding diffusion and sintering. Anotherexplanation is that the small amount of liquid dissolves the small particles leaving only thelarger particles that sinter slower.Figure 3-10 compares the effectiveness of the various dopants at 1200 °C. Thediscrepancies in the final contractions for the various dopants did not mean that there was asignificant difference in final density as indicated by Table 3-2. The abrupt change in allcurves from nearly vertical to horizontal over the same period of rime is evident. At thistemperature the effectiveness of the dopant is masked by the sintering temperature. Thechoice of dopant was not found to be particularly important for sintering at 1200 °C.3.2.6 Temperature DependenceA series of contraction experiments were conducted at various temperatures while thedopant concentration was held constant. Figure 3-11 shows contraction curves at differenttemperatures for PbNb2O6 with 1 wt. % SrO. In comparison to the SrO, which is a solidover the experimental temperatures, Bi203 is a liquid throughout this range. Figure 3-12shows contraction curves for PbNb2O6 with 1 wt. % Bi203 at various temperatures. Bothfigures show that with increasing temperature the rate of sintering is also greater asindicated by the steeper initial slope. In general the total contraction increases at highersintering temperature for both dopants.0.12 ^0.11 -0.1 -0.09 -0.08 -0a^0.07 -• 0.06 -0• 0.05 -0.04 -0.03 -0.02 -0.01 -0 I^I^I^I^I0^40 80 120Time (minutes)Figure 3-9. Contraction curves for PbNb 206 samples doped with various dopants at1 wt. % sintered in air at 1100 °C.f-Bi20,_^ ...____,.....^.........•— — KNaNb206160 200SrO240610.160.15 -:••••••• • Y.. •■■•0.14 -0.13 - --------------------------------------0.12 -0.11 -0.1 -0.09 -ct9 0.08 -0.07 - 10.06 -0.05 - Sr0—0.04 - - - Bi 2030.03  ^LiNb00.02 30.01 KNaNb2°60 I I I^ I I0^80^12040Time (minutes)Figure 3-10. Contraction curves for PbNb 206 samples doped with various dopants at1 wt. % sintered in air at 1200 °C.160^200^2401150 °C1100°C .^............................ ....... . ................................. 1050 °C •••••••••■- -..........1 100°C0.160.15 -0.14 -0.13 -0.12 -0.11 -0.1 -0.09 -7:50ct^0.08 -Ud" 0.07 -0.06 -0.05 -0.04 - I^I0 80'^'120^160^200^240Time (minutes)Figure 3-11. Contraction curves for PbNb 206 - 1 wt. % SrO samples sintered in airat temperatures indicated.1200 °C........................... • ....... ......................1050 °C620.160.15 -0.14 -0.13 -0.12 -0.11 -0.1O• 0.09 -0.08 -0  0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01I^I^I^I^I0^40 80 120Time (minutes)Figure 3-12. Contraction curves for PbNb 2O6 - 1 wt. % Bi203 samples sintered in airat temperatures indicated.0 I^I160^200^2403.2.7 Concentration DependenceAt the lower sintering temperatures (1050 and 1100 °C), the effect of dopantconcentration can be most easily seen whereas, at higher temperatures (1150 and 1200 °C),the effect of the dopant concentration cannot be distinguished as the temperature effectpredominates.Figure 3-13 shows contraction curves for SrO doped PbNb 2O6 sintered in air at1100 °C. The 'milled' curve is included as a comparison since it represents the effect ofmilling but with no dopants. The effectiveness of SrO in hindering compaction can beclearly seen. Assuming a comparable amount of milling was done for undoped 'milled'PbNb2O6 and the SrO doped PbNb 2O6 , the SrO addition has a marked effect. The slightdifference between curves of various concentrations in Figure 3-13 does not allow one tosay with certainty the effect of SrO concentration on contraction. Figure 3-14 showscontraction curves for Bi 203 doped PbNb2O6 sintered in air at 1100 °C with the 'milled'contraction curve included. The curves for the Bi 203 doped PbNb2O6 were also found to belie below curves for the 'milled' PbNb 2O6 . This indicates that Bi 203 also hinders sintering,although to a lesser degree than SrO. There are some indication in Figure 3-14 that overthe concentrations studied (0.25-1 wt. %), increasing Bi 203 additions facilitate contraction.Because of the small variation in dopant concentration studied, a comparison ofdopant concentration effectiveness on contraction is not certain. Other factors also madethe concentration effect less distinct. For example, the effectiveness of milling ascompared to the unmilled powder sample can be clearly seen (Figure 3-8). This suggeststhat milling has a marked effect and therefore differences in milling time can have an effecton shrinkage. Slight differences in temperature can effect sintering as well as mask theeffect of the dopants.63- 1.0 wt. %- - - 0.5 wt. %  0.25 wt. %milled..... .. .. .. . . ............. . ..............0.120.11 -0.1 -64^0.09 -^./../.^0.08 -^/oc^0.07-.,,c., /alvs^0.06 -^/." .Uo^0.05 -^i0.04 - i0.03 -0.02 ,0.01 -00^40^80^120^160^ 200I^I^ITime (minutes)Figure 3-13. Contraction curves for SrO doped PbNb2O6 samples sintered in air at1100 °C for concentrations indicated. Milled curved is used as reference.0.12240•••••-.. ................ •0.11 -0.1 -0.09 -0.08 -0 0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 -- 1.0 wt. %---  0.5 wt. %  0.25 wt. %milled0 1^1^I^1^I^1^I0^40 80 120Time (minutes)Figure 3-14. Contraction curves for Bi 203 doped PbNb2O6 samples sintered in air at1100 °C for concentrations indicated. Milled curved is used as reference.I160^200^240653.2.8 Microstructure of Sintered SamplesOne of the primary objectives was to identify or select a dopant which will reducegrain growth significantly and enhance sintering without affecting the dielectric propertiesof PbNb2O6. Densification (sintering) leads to extensive grain growth within the systemwhich results in cracking of the specimens having low mechanical properties. For thisreason, it has not been possible to sinter high density strong specimens of PbNb2O6. Inview of this problem associated with sintering, a study of grain size behavior with dopantsand temperature was carried out.The following micrographs show selected fracture surfaces of specimens with variousdopants and concentrations sintered at different temperatures. All micrographs presentedshow the PbNb2O6 specimens after they have been sintered and have undergone a heattreatment at 1250 °C for 1 hour after the compaction test, unless otherwise stated. Heattreatment was necessary in order to yield the ferroelectric phase required for dielectrictesting. It is known that the heat treatment at 1250 °C and the firing of the Ag electrodes at600 °C would have an effect on the final grain size but since all specimens have undergonethe same treatment it is hoped all would be affected equally by this treatment. Because ofthe post sintering heat treatment, it was not been possible to take into account the effect ofgrain growth on the sintering kinetics but the relative effect of dopant concentration andtemperature on grain size can be qualitatively estimated.After the heat treatment at 1250 °C, the samples possessed a density between 86 and95 % of the theoretical. Intergranular fracture surfaces were chosen to highlight the grains.Sintering and heat treating were found to bring about the formation of acicular PbNb2O6grains with an aspect ratio of —3:1 which is unlike the starting powder shown in Figure 3-3.66Figures 3-15(a-b) show the porosity in the sintered specimens. For large grainedspecimens, intergranular fracture surfaces could be used to observe the porosity while forsmall grained specimens (<10 gm) intragranular fracture surfaces were observed tohighlight the porosity. Figure 3-15a shows the presence of pores on the surfaces of largegrains of the sintered 'milled' powder. In contrast Figure 3-15b shows porosity on theintragranular fracture surfaces of small grained specimens.Figures 3-16(a-d) show the 'milled' and 'as-received' powders sintered at 1050 and1200 °C. The effect of sintering temperature can be observed by moving from left to right,while the effect of milling on microstructure can be seen by moving down the page. Theeffect of sintering temperature on grain size can be seen in Figure 3-16a with its grain sizeof 2.1 gm compared to —3.7 gm for Figure 3-16b The effect of temperature was morepronounced for specimens of the 'milled' powder. Figure 3-16c shows the bimodaldistribution of the 'milled' powder sintered at 1050 °C. The average size of the largeparticles was determined to be —70 gm compared to —3.2 gm for the small particlesyielding; an overall average of 8.7 gm. Figure 3-16d shows the 'milled' specimen sinteredat 1200 °C with an average grain size —120 1.1M assuming that each particle is a grain. Itcan be seen that milling has a pronounced effect on grain growth particularly at the1200 °C.b)Figure 3-15. SEM micrographs highlighting the porosity on a) intergranular fracturesurfaces of 'milled' PbNb2O6 sintered at 1200 °C with its large grainsb) intragranular fracture surfaces of typical small grained specimens (1 wt. % SrO)sintered at 1200 °C.67681050 °C^ 1200 °Ca) b)c)^ d)Figure 3-16. SEM micrographs of intergranular fracture surfaces of PbNb 2O6 usinga) 'as-received' powder sintered at 1050 °C b) 'as-received' powder sintered1200 °C c) 'milled' powder sintered at 1050 °C d) 'milled' powder sintered at1200 °C.69The concentration effect of SrO on grain growth could not be distinguished forspecimens sintered at 1050 °C as can be seen from Figures 3-17a, c, and e. This is unlikethe micrographs of specimens sintered at 1200 °C shown in Figures 3-17b, d, and f whichindicate that increasing SrO concentration inhibits grain growth to a small extent. Movingleft to right, the increase in grain size with temperature can be seen, while moving down thepage the effect of increasing SrO concentration can be observed. The grain size ofPbNb2O6 sintered at 1200 °C and doped with 0.25, 0.5, and 1.0 wt. % SrO was 3.8, 3.7 and3.3 l_trn respectively. Figures 3-17(a-f) show that for all SrO concentrations increasingsintering temperatures yielded larger grains.Figures 3-18(a-f) show the effect of sintering temperature and Bi 203 concentration ongrain size. Figures 3-18a, c, and e all show a grain size of —2.7^for all dopantconcentrations, indicating that Bi 203 concentration has no effect on the grain growth at this1050 °C. Unlike the relatively constant grain size of those specimens sintered at 1050 °C,micrographs of those specimens sintered at 1200 °C did show an effect of Bi 203concentration on grain growth. With increasing Bi 203 concentration, a larger average grainsize was noticed. From the 1200 °C micrographs (Figures 3-18b, d, and f) it wasdetelmined that the grain size for specimens with 0.25, 0.5, and 1.0 wt. % Bi 203 were 3.3,3.0, and 5.7^respectively. This increase in grain size with increasing Bi 203concentration can be explained by the presence of larger quantities of liquid which aidssintering and grain growth.....'incp..Iqa)^ b)d)e)^ f)Figure 3-17. SEM micrographs of intergranular fracture surfaces of PbNb 206 witha) 0.25 wt. % SrO sintered at 1050 °C b) 0.25 wt. % SrO sint. at 1200 °Cc) 0.5 wt. % SrO sint. at 1050 °C d) 0.5 wt. % SrO sint. at 1200 °Ce) 1.0 wt. % SrO sint. at 1050 °C f) 1.0 wt. % SrO sint. at 1200 °C.1050 °C1200 °C 71c) d)a)pokvirkwirviriff 117'r310A^ /■4: ‘1 410Figure 3-18. SEM micrographs of intergranular fracture surfaces of PbNb 206 witha) 0.25 wt. % Bi203 sintered at 1050 °C b) 0.25 wt. % Bi 203 sint. at 1200 °Cc) 0.5 wt. % Bi 203 sint. at 1050 °C d) 0.5 wt. % Bi 203 sint. at 1200 °C e) 1.0 wt.% Bi203 sint. at 1050 °C f) 1.0 wt. % Bi 203 sint. at 1200 °C.72Increasing KNaNb2O6 concentration was found to hinder grain growth at 1050 °C.The effectiveness of KNaNb2O6 in preventing grain growth at 1050 °C may be explained bythe DTA results and/or its melting temperature. At this temperature KNaNb 2O6 remains asolid which may hinder sintering. Figures 3-19a and c show the PbNb 2O6 doped with 0.25and 1.0 wt. % KNaNb2O6 sintered at 1050 °C exhibiting a grain size of —2.8 gm and2.1 gm, respectively. This is in contrast to the KNaNb 2O6 doped PbNb2O6 sintered at1200 °C, which showed no effect of KNaNb 2O6 concentration on grain size (Figure 3-19band d).With increasing additions of LiNbO3 grain growth was found to be hindered forspecimens sintered at 1050 °C. Like KNaNb2O6 this may be explained by the fact thatLiNbO3 remains a solid until —1200 °C. This effect can be seen in Figures 3-20a and cwhich exhibited a grain size of 2.9 and 2.4 gm, respectively. Micrographs of LiNb0 3doped PbNb2O6 sintered at 1200 °C shown in Figures 3-20b and d did not show any effecton grain size with increasing LiNbO 3 concentration.731050 °C^ 1200 °Cc .a)^ b)c)^ d)Figure 3-19. SEM micrographs of intergranular fracture surfaces of PbNb 2O6 witha) 0.25 wt. % KNaNb 2O6 sintered at 1050 °C b) 0.25 wt. % KNaNb 2O6 sintered at1200 °C c) 1.0 wt. % KNaNb 2O6 sintered at 1050 °C d) 1.0 wt. % KNaNb 206sintered at 1200 °C.741050 °C^ 1200 °COa)^ b)c)^ d)Figure 3-20. SEM micrographs of intergranular fracture surfaces of PbNb 206 witha) 0.25 wt. % LiNbO 3 sintered at 1050 °C b) 0.25 wt. % LiNbO 3 sintered at1200 °C c) 1.0 wt. % LiNbO3 sintered at 1050 °C d) 1.0 wt. % LiNbO 3 sintered at1200 °C. Summary of Grain Size DataAs previously mentioned all the doped powders have been milled in order to mix thedopants with the PbNb2O6 . The grain size of the doped specimens should therefore becompared with the undoped 'milled' specimens. It was found that the addition of anydopant at 1050 °C resulted in a more uniform grain size distribution unlike the bimodalone obtained for the 'milled' powder sintered at 1050 °C as shown in Figure 3-16c. It wasnoticed that all the doped PbNb 2O6 specimens sintered at 1200 °C had a far smaller grainsize than that of the undoped 'milled' powder, suggesting that all of the dopants act asgrain growth inhibitors. Table 3-3 summarizes the average grain size obtained from theSEM micrographs and the median particle size from the Horiba CAPA-700 of the startingpowders. Table 3-4 summarizes the average grain size of the specimens after sinteringand heat treating from the SEM micrographs.Table 3-3. Dimensions of starting PbNb 2O6 powder.Powder Average grain size from SEMmicrographs (gm)Median particle size from HoribaCAPA-700 (p.m)'milled' —0.5 1.5'as-received' —1.0 1.975Table 3-4. Grain size of selected PbNb2O6 specimens with various dopant andconcentration sintered at various temperatures.Dopant Conc.wt. %1050 °Cj.tm1200 °'milled' 3.5^70* 120'as-rec' 2.1 3.70.25 2.3 3.8SrOt 0.5 2.3 3.71.0 2.1 3.30.25 2.7 3.3Bi203 1. 0.5 2.7 3.01.0 2.7 5.7KNaNb206t 0.25 2.8 3.11.0 2.1 2.9LiNb03 1. 0.25 2.9 3.11.0 2.4 2.9* bimodal particle size distributionf mixing of dopant and PbNb2O6 accomplished by milling7677Energy dispersive and wavelength dispersive x-ray (EDX and WDX respectively)analyses on the SrO and Bi203 doped specimens were found not to be useful.Intergranular fracture surfaces allowed for individual grains to be clearly distinguished butthe relief hindered element detection. From these x-ray maps of the dopants no highlyconcentrated regions of dopants were observed. Elemental x-ray maps of the polisheddoped PbNb206 were similarly taken but these also could not delineate regions of highdopant concentration. Analysis suggests that the dopants are evenly distributedthroughout the sample or in concentrations lower than the detection limit.The data show significant differences in sintering as a result of milling thereforeslight differences in milling times could have an effect on the sintering characteristics andfinal grain size of the specimens. The effects observed in the sintered specimens couldtherefore be due to the particle size and not the dopant. In order to make certain that theeffects observed were correct, specimens made from powders having the same sizedistribution were tested. This was done in order to confirm and distinguish the effect ofparticle size from the dopant addition.A set of experiments were conducted with undoped 'milled' powders as the startingmaterial. Using this as the starting powder, 2 specimens were made. Some of the powderwas kept aside to which nothing was added. To the remaining powder was added 1wt. % Bi203 . In order to homogenize, the Bi203 doped powder was lightly mixed withmortar and pestle. Mixing of the dopant with powder was not done with the vibratory ballmill as it could have significantly changed the particle size distribution. The undopedmilled powder was similarly mixed with mortar and pestle. The powders were pressedand each specimen was sintered at 1200 °C. These specimens were fractured andexamined without any further heat treatment.78It was noticed that the undoped 'milled' specimen exhibited very large grain growthunlike the Bi 203 doped specimen. This behavior was similar to that previously observed.This confirmed that the difference in grain size was due to the effect of the dopant and notdue to a difference in the initial particle size distribution. Presented in Figure 3-21a and bare SEM micrographs of these specimens. Both specimens started with same particle sizedistribution with the only difference being the Bi 203 addition.The density of the sintered undoped milled pellet (Figure 3-21a) was —90 % of theo-retical while the sintered Bi203 doped specimen (Figure 3-21b) was —79 % PreviousBi203 doped specimens attained a density of —90 % of theoretical. A possible reason forthe previously recorded higher density, could be with the mixing of the Bi 203 with thePbNb2O6 . Both sets of powders have been milled but with the lower density specimen, theBi203 was only lightly mixed with the PbNb 2O6. In contrast the previous high densityspecimens had the dopant and PbNb2O6 vibratory ball milled together allowing for morethorough mixing.Figure 3-21a of the 'milled' unheat treated specimen exhibits the same structure asthe 'milled' heat treated specimen in Figure 3-16d with its large grains and porosity on thegrain boundaries. The higher magnification of Figure 3-21a also shows the presence ofthin film around the pores which indicates that there might be some liquid phase present inthe system. The source of the liquid could be from the off-stoichiometry of the PbNb 2O6or from the contaminants incorporated into the powder during milling. Without heat treat-ment the Bi203 doped specimen sintered at 1200 °C (Figure 3-21b) had a grain size of4.6 p.m compared to 5.7 p.m for the heat treated specimen (Figure 3-180, confirming thatheat treatment does have an effect on grain size.3.3 Electrical PropertiesA table of calculated dielectric constants for the various specimens is shown in theAppendix I and is presented solely for completeness. Analysis of the dielectric data will notbe discussed as it is not within the scope of this thesis. The equation provided by B.M.Hi-Tech Inc. in order to calculate the dielectric strengths from measured values wasCtE 0.08854Awhere C is the capacitance (pF), A is the area of the specimen (cm2), and t is thickness (cm)of the cylindrical specimens.79a)80b)Figure 3-21. SEM micrographs of intergranular fracture surfaces of PbNb 2O6 with a)'milled' PbNb2O6 sintered at 1200 °C, no heat treatment b) 'milled' PbNb 2O6 with1.0 wt. % Bi203 sintered at 1200 °C, no heat treatment.4 DISCUSSION4.1 Sintering TheoriesDepending on how much the specimens have sintered, various sintering theories willhave to be used to analyze the data. For specimens that have densified by less than —7 % oftheir theoretical density, initial stage sintering equations were applied. If densification wasin excess of this, intermediate / final stage sintering equations were used. The similaritybetween the intermediate and final stage sintering equations allowed the data to be analyzedfor both stages simultaneously. In order to determine whether initial or intermediate / finalstage sintering equations should be employed, relative density plots were used.From the shrinkage curves one can derive relative density versus time plots. This canbe accomplished by using the relationship:(4-1)where p is the relative density, Lf is the final height of the specimen, L is the instantaneousheight, and pf is the final relative density. The equation assumes that the pellet shrinksisotropically in all directions which implies that it is homogeneous. It is known however,that specimens are not very homogeneous as there exist density gradients from pressingresulting in a lower centre density. Another important sample inhomogeneity is particlealignment as elongated particles become parallel to one another during pressing. This leadsto a large number of contacts along the axial direction from which shrinkage can occur.Equation 4-1 was used to obtain p - t plots but the limitations of this equation should benoted.814.1.1 Analysis Using Establish Sintering TheoriesFrom the p - t plots presented in Figure 4-1 it can be seen that over the temperaturerange studied, density change varied from —3 - 30 % from an initial density of —60 % of thetheoretical. Because of the wide range in density change the initial, intermediate, final, or acombination of these stages of sintering were recorded. Established sintering modelsdiscussed in Section 1.3 will be used to analyze the contraction data. The stages ofsintering observed were largely dependent on the sintering temperature. In general,specimens sintered at 1050 °C reached only the initial stage of sintering while thosesintered at 1150 and 1200 °C reached the intermediate and final stages.From Figure 4-1 it can be seen that initial stage sintering models can only be usedwith the specimen sintered at 1050 °C as the relative density change was less than 7 %.From the previously mentioned stage 1 models of sintering, plots of In e versus In t shouldyield straight lines whose slopes are indicative of the sintering mechanism. Plots of In E -In t for 1050 °C specimens are presented in Figure 4-2. Lines with slopes of 0.8 and 0.4 areincluded to represent the theoretical models for bulk diffusion and neck bulk diffusion,respectively. The initial portion of the curves fits a line of slope 0.8 then fits a line of slope0.4. It is believed that the initial fit to the bulk diffusion model can be explained as thesintering of small particles which allows for mass transport to occur over the entire particle.After the smaller particles have sintered and the necks reached a stable configuration onlylarger particles remain, limiting diffusion to the neck region.It should be noted that when plotting In e - In t, the error is exaggerated for smallcontractions and times. In addition, the curve's lack of fit to the theoretical lines for smallvalues of time (<10 minutes) can be attributed to factors such as particle rearrangement,8283uneven heating, nonuniform particle size, and the possibility of multiple sinteringmechanisms operating simultaneously. These factors make the initial portion of thecontraction curves more prone to error.The bulk diffusion equations 1-19 and 1-20 for both intermediate and final stages ofsintering can be written in the general form:dP ND yi2^ (4-2)dt KT 1 3where N is dependent on the sintering stage. However, this equation cannot be usedwithout knowing the variation of 1 with time - where / is related to grain size. FromHerring's scaling law62 it is known that for bulk diffusion, grain growth is proportional totime to the 1/3 power, therefore 1 3 = Bt where B is a constant.r o dP dt _ r `iND yildtdt^Je. KT B tP^KTB int Itoc, ND-A.1^gyro (4-3)Thus when density (p= 1 - P) is plotted as a function of In t, 2 linear portions should beobtained, 1 for the intermediate stage and 1 for the final stage of sintering.Because of the greater amount of sintering observed at 1150 and 1200 °C (Figure4-1), equations applicable to intermediate and final stage sintering theories were applied tothose two temperatures. Figure 4-3 shows p - In t plots, where the linearity of the curvesfits intermediate / final stage sintering theories. The 1200 °C curve can be fitted to twolinear portions associated with intermediate and final stages of sintering as drawn in thediagram. The 1150 °C curve can be fitted to a single linear portion suggesting that only theintermediate stage was reached, which is supported by its final relative density of --78 %.Although no grain size versus time data was collected in this study, the validity of bulk0.950.9 -0.85 -0.8 -F, 0.75 --00>.o 0.7 -7Jr:40.65 -0.6 -0.55 -0.5  0I^I^I^I^I^I^I^1^1^1^1^140 80 120 160 200 240Time (minutes)diffusion as the sintering mechanism for PbNb2O6 is justified by the linearity of the plots.According to Coble, 54.55 the significance of the slopes of the p - In t curves is not known.The property to be assessed to determine whether the diffusion models are supported issolely the linearity of curves which is supported by the plots.Figure 4-1. Relative density curves for 'as-received' PbNb 2O6 samples sintered in airat temperatures indicated.8464-4^-2^0^2In Time (In minutes)5-1^1^3In Time (ln minutes)Figure 4-3. Relative density - In t curves for as-recieved PbNb2O6 samples sinteredin air at temperatures indicated.Figure 4-2. In c - In t curves for 'as-received' PbNb2O6 samples sintered in air at1050 °C.85-2-3-4-5-6-7-80.950.90.850.80.650.60.550.5///,//i1200 .!C.....:- ---86Figure 4-4 shows p - t plots of the milled powders. It can be seen that at alltemperatures the density change is greater than that valid for the initial stage sinteringmodels. Only intermediate / final stage sintering analysis is therefore applicable with the'milled' data. Figure 4-5 shows p - In t plots of the 'milled' powders. Lines are drawn into emphasize the linearity of the curves which fit intermediate / final stage sinteringtheories.Analysis using established sintering models will also be conducted on Bi 203 and SrOdoped specimens to highlight any differences in the way they interacted with PbNb2O6during sintering. Bi203 's low melting temperature meant that it was liquid at alltemperatures studied, unlike SrO which was solid. Differential thermal analysis confirmedthat SrO did not form a liquid with PbNb 2O6 over the temperature range studied, unlike theBi203 doped PbNb2O6 .Presented in Figures 4-6 to 4-8 are p - t plots of SrO doped PbNb 2O6 specimenssintered at various temperatures. From these figures it can be seen that a density change of<7 % was realized for those specimens sintered at 1050 and 1100 °C, and therefore they canbe used with initial stage sintering models. Shrinkage curves at 1150 and 1200 °Cexhibited a far greater density change indicative of intermediate / final stage sintering somodels applicable to those stages will be used for those temperatures.^0.94 ^0.92 -0.9 -0.88 -0.86 -0.84 ->„^•0.82 -.1=,^0 8 -cnF> 0.78 --cJ0.76 -1=1,1 0.74 -c7-3 •0 72 -40.7 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58  -3 1In Time (In minutes)Figure 4-5. Relative density - In t curves for milled PbNb2O6 samples sintered in airat temperatures indicated.-1 3 50.940.92 -0.9 ----- --1150°C- - ----------^--  -------------------------------------87..........^0.88 -^1200°C ...• • • •..- • •0.86 -0.84 -0.82 - .** 1100 °C0.8 -0.78 -0.76 -0.74 -0.72 -0.7 -0.68 -0.66 -0.64.•••••••.••••••••1050°C0^40^80^120^160^200^240Time (minutes)Figure 4-4. Relative density for milled PbNb 2O6 samples sintered in air attemperatures indicated.0.620.60.58•• •1150 °C1100°C1050 °C0.9 -0.88 -0.86 -0.84 -0.82 -0.8 -0.780.76 -`g 0.74 -0.72 -0.775 0.68 -C4 0.66 -0.64 -0.62 -0.60.580.56 -0.54 ^0I80^120I^I^I160^200 240401200 °C88Time (minutes)Figure 4-6. Density curves for PbNb2O6 - 1 wt. % SrO samples sintered in air attemperatures indicated.^0.92 ^0.9 -0.88 -0.86 -0.84 -0.82 -0.8 -E•- 0.78 --c) 0.76 -> 0.74 -a 0.72 -c4 0.7 -0.68 -0.660.64 -0.62 -0.6 -0.58  0I^I I160 200I^I^I^4 80^120Time (minutes)Figure 4-7. Density curves for PbNb 2O6 - 0.5 wt. % SrO samples sintered in air attemperatures indicated.24000 ■,0^N 00 00 ‹) ct N • 00 ^,0^N ■O 00O 00 00 00 00 c)^• r----^c,^vo tnO c; O cip O c:; O O O O O c:;ki!suap QAp -epliC3■O0U11,-)14.)_ CD^06•8990Plots of In e - In t for 1 wt. % SrO doped PbNb2O6 are presented in Figure 4-9. Lineswith slopes of 0.8 and 0.4 are included to represent the theoretical models for bulk diffusionand bulk neck diffusion, respectively. The 1050 °C contraction curve fits the 0.4 slope lineassociated with bulk neck diffusion while the contraction curve at 1100 °C initially fits the0.8 slope line then the 0.4 slope line. The fit of the 1100 °C curve initially to the 0.8 slopeline may be explained by the greater diffusivity at a higher temperature and the sintering ofsmall particles. With increasing temperatures diffusion increases allowing material fromthe contact point to diffuse farther away from the neck. Curves for 0.5 and 0.25 wt. % SrOdoped PbNb2O6 yielded similar results and are presented in Figures 4-11 to 4-13.When density (p= 1 - P) is plotted as a function of In t, two lines were obtained, onefor the intermediate stage and the second one for the final stage of sintering. Figure 4-10shows a p - In t plot for 1 wt. % SrO doped PbNb 2O6 sintered at various temperatures.Because of the greater amount of sintering observed at 1150 and 1200 °C intermediate andfinal stage sintering theories are applicable. A similar type of fitting to 2 lines can be donewith the 0.5 and 0.25 wt. % SrO doped PbNb2O6 as presented in Figures 4-12 and 4-13.-2-3-47-8.- • . ..• "^.....,0.8 line.- 1 igee91•1-1-9^ I -2^0^2^4^6In Time (In minutes)Figure 4-9. In c - In t curves for PbNb2O6 - 1 wt. % SrO samples sintered in air attemperatures indicated.0.90.850.8E 0.75/u.g 0.71 100 °C0.6-4^-2^0^2^6In Time (ln minutes)Figure 4-10. p - In t curves for PbNb2O6 - 1 wt. % SrO samples sintered in air attemperatures indicated.0.55_______--------'------1050 °C4"640^2In Time (ln minutes)Figure 4-11. In e - In t curves for PbNb2O6 - 0.5 wt. % SrO samples sintered in air attemperatures indicated.4 6-4^-2^0^2In Time (In minutes)92Figure 4-12. p - In t curves for PbNb2O6 - 0.5 wt. % SrO samples sintered in air attemperatures indicated.1150 °C--------- --2-2.5-3-3.5-5.5-6-6.5-70.950.90.850.650.60.55-3-4.0.0-5A=o -6U-5-7-8-9 5 7-3^-1^1^3In Time (ln minutes)..110 °c.•-•'0.4 line931 100°C________-------:1050 -CFigure 4-13. In c - In t curves for PbNb 2O6 - 0.25 wt. % SrO samples sintered in airat temperatures indicated.0.920.90.880.860.840.820.80.780.760.740.720.70.680.660.640.620.60.58 -3^-1^1^3In Time (In minutes)Figure 4- 14. p - In t curves for PbNb2O6 - 0.25 wt. % SrO samples sintered in air attemperatures indicated.1^1594A similar type of analysis was done with Bi203 doped PbNb2O6 . From Figures 4-15to 4-17 of the Bi203 doped PbNb2O6 , it can be seen that initial stage sintering models applyonly to those specimens sintered at 1050 °C. Figure 4-18 shows In e - In t plots of Bi203doped PbNb2O6, which fits well to the theoretical line of 0.8 slope associated with bulkdiffusion. Although Bi203 does form a liquid, sintering data do not indicate that liquidphase sintering models are applicable.The lack of a fit with established liquid phase sintering theories may be explained bythe fact that a very small amount of dopant was added. Assuming no eutectic or otherreactions occurred and that the only source of liquid was from the molten Bi 203 , for 1 wt.% Bi203 there would be —0.74 volume percent liquid. Because of the small amount ofliquid, solution-precipitation associated with liquid phase sintering was not observedotherwise, In e - In t curves should fit lines having a 0.33 slope. The in E - in t curves alsodo not fit lines having a 0.5 slope associated with boundary reaction rate controlled liquidphase sintering. According to Kingery after the completion of these liquid phase sinteringmechanisms the particles form a solid skeleton. The presence of a small amount of theliquid phase meant that the sintering process was controlled primarily by the formation ofthe solid skeleton. This is not to say that Bi203 had no effect on the sintering of thePbNb2O6 . Data show that sintering is facilitated with Bi 203 when compared to the otherdopants. It is believed that the addition of the Bi 203 enhances the diffusion of materialfrom the contact point so that it can spread over the entire particle involved and not only tothe neck, hence sintering is by the bulk diffusion mechanism rather than the bulk neckdiffusion or liquid phase sintering mechanisms.Plots of p - In t for Bi203 doped PbNb2O6 are presented in Figure 4-19 and 4-20.From these plots it can be seen that at 1150 and 1200 °C, there are two distinct linear95portions suggesting that bulk diffusion is the sintering mechanism for intermediate andfinal stages of sintering. Similar results were obtained from p - in t plots of 0.5 wt. % Bi 203doped PbNb2O6 .A similar analysis was also conducted with the LiNbO 3 and KNaNb2O6 dopedPbNb2O6 data but for the sake of brevity the plots are included in Appendix IL The resultsare summarized below. For both doped powders only the 1050 °C data could be analyzedwith initial stage sintering theories. Plots of In E - in t for both powders initially fitted linesof slope 0.8 (bulk diffusion) then fitted lines of slope 0.4 (bulk neck diffusion). Analysis ofdata applicable to the intermediate and final stages of sintering produced linear regionswhen p - In t was plotted confirming that bulk diffusion was method of sintering.Additional plots of p - t, In e - In t, p - In t for the doped PbNb2O6 are presented inAppendix II.Bulk diffusion in general appears to be the mechanism by which PbNb 2O6 sinters.Initial stage sintering data show that sintering is by bulk diffusion and bulk neck diffusion.Data applicable with intermediate and final stage sintering were found to correspond wellwith bulk diffusion.In general the initial stage of sintering for PbNb 2O6 was found to start by bulkdiffusion before it switched to bulk neck diffusion. This can be explained as the sinteringof small particles that diffuse over the entire particle leaving only large particles whosediffusion is limited to the neck region. Depending on the dopant, diffusion can be enhanceor hindered. The addition of low melting dopant such as Bi 203 enhances diffusion allowingfor mass transport farther away from the point of particle contact hence bulk diffusion isencouraged. In contrast the addition of SrO hinders material transport, limiting it to theneck region and facilitating bulk neck diffusion. Increasing the sintering temperature is96another method by which to increase diffusion. This may explain why with increasingtemperature (1100 °C), the SrO doped PbNb 2O6 showed signs of bulk diffusion that waspresent at 1050 °C.For data applicable with intermediate and final stage theories, plots of p - In t yieldedlinear regions proving that bulk diffusion is the mechanism of sintering for these stages.Depending on how far sintering proceeded one or two linear regions were observed.0.950.9 -1150 °C1200 °C,0.851050 °C^...... ,0.65 -0.6 , ...--,0.55^1^l^1^1^1^1^1^1^'^,^,^10 40 80 120 160 200^240Time (minutes)Figure 4-15. Density curves for PbNb 2O6 - 1 wt. % Bi203 samples sintered in air attemperatures indicated....,,,... ■••••••••• ..--- --- --------- . • • • .......1100°C ....•0.8 -0.750.7 -971200 °C- - - - _ -------- ---... •••..• -•, -^1150 °C///- - • • • ...... - • • ........ • . -.• '11.....■•■ -...-• .....-.1050 °C. . -0^40^I^80 I^^12I 0^I^160^I^200^I^I240Time (minutes)Figure 4-16. Density curves for PbNb2O6 - 0.5 wt. % Bi203 samples sintered in air attemperatures indicated.0.940.920.90.880.860.84>.s 0.82•4•=, 0.809 0.780.760.7476 0.72c4^'0.70.680.660.640.620.60.58I1100°C0.650.6 --^0.8 line-2-3-4-5013' -60o _7C-8-9-10-110.95 -0.9 -0.85 -11 50 °C - _•^...... • • -^• .................1 loo1050 °C981200 °C0.55 1^1^1^10 ^40 ^80^120 160 200 ^240Time (minutes)Figure 4-17. Density curves for PbNb206 - 0.25 wt. % Bi203 samples sintered in airat temperatures indicated.-2^0^2^4^6In Time (ln minutes)Figure 4-18. In sE - In t curves for Bi203 doped PbNb2O6 samples of variousconcentrations sintered at 1050 °C.53-1^1In Time (In minutes)1050°C99Figure 4-19. p - In t curves for PbNb2O6 - 1.0 wt. % Bi203 samples sintered in air attemperatures indicated.10.95 1 1 50 °C0.90.85v) • /-c, .804.)-4= 0.751200 °C1 1 00 °CT.)124)0.7••••0.651050 °C0.60.55 I-2 2 4 6In Time (In minutes)Figure 4-20. p - In t curves for PbNb2O6 - 0.25 wt. % Bi203 samples sintered in airat temperatures indicated.4.1.2 Procedures for Calculating Activation EnergyUsing the various sintering theories and equations it is possible to calculate theactivation energy associated with the sintering of PbNb 2O6 with and without dopants.Because of the limited temperatures studied, analysis using these techniques was notpossible for all PbNb2O6 specimens but for completion the procedure is included.Depending on whether analysis is of the initial stage of sintering or the intermediate / finalstage of sintering, different sets of equations are used, although the manipulation of theseequations is similar.From equation 1-18:e _ (6,L j_ ( GD7w )mt.Lo^( a" KT )mL^1 — E= =( L°— 6Lj= _1 GD yo) ei.Lo^Lo^anICT(4-4)Although this equation predicts a straight line for plots of In E versus In t, plots showan initial curved portion followed by a straight section. As previously stated, errors inE (8e) and t (St) affect the initial portion of the in E - In t curve. The choice of Lo and to atthe first signs of contraction was chosen because of its ease as a common reference point.This point does not necessarily indicate the beginning of sintering but rather whencontraction due to sintering exceeds expansion from the apparatus and specimen. Either orboth errors (Se and St) will affect the linearity of the plots but primarily over the initialportion of the curve. These errors were taken into consideration by Johnson and Cutler52'53yielding:1 — E' —( LoL—SL j— 1 — (K ' Dr (t — St)m^(4-5)1 —E' =1 — (K 'Dr (t')m^ (4-6)100with values of SE and St chosen so as to straighten the curve.From Equation 4-6, plots of 1-E' versus (t')m should have a slope of (K'D)m.slope =1.1.' =(K'D)"'^ (4-7)In µ' = m ln(K'D)^D =Be "^(4-8)where A is the activation energy, R is the gas constant, and T is the temperature in Kelvin.AMg' = lnK' +1nB —RT^(4-9)Thus when (In IA' )Im versus 11T is plotted, slopes of -AIR are obtained allowing A tobe calculated.Similarities between the equations used by Johnson and Cutler5153 and Coble54 '55suggested that the above analysis could be extended to Coble's equations. A similarprocedure was followed using the intermediate / final stage sintering equations for bulkdiffusion (Equation 4-3):o ND7C2^IfP IP. KTB int L.P —Po —ND70, (In t — ln to )KTB(4-10)P =K' +K"D In tp = 1—P =1—K' +K"D lnt.^ (4-11)Thus when p versus In t is plotted, slopes of K"D are obtained.^_ A 1^ (4-12)slope =p," =K"D^D =Be "A 1^(4-13)1nµ" = lnK "B —RT101mThus when In p." versus 117' is plotted, slopes of -ii i/R are obtained, allowing theactivation energy A l to be calculated. This meant that some useful information can beextracted from the slope unlike that originally suggested by Coble.It should be noted that when using either equation 4-4 or 4-10 for calculating theactivation energy, it was assumed that the diffusion coefficient 'D' is temperatureAdependent exponentially according to the Arrhenius equation (e ") while the energy termKT in the denominator of both equations, proportionally affected the strain (e) and porosity(P). For this reason neither e nor P was corrected with respect to the temperature (T) in thedenominator. The method used here exactly follows the techniques used by previousworkers for calculating the activation energy.As previously mentioned the limited temperatures studied (1050 °C, 1100 °C,1150 °C, 1200 °C) made this type of analysis difficult. In general only the lowertemperatures (1050 °C, 1100 °C) yielded data valid over the initial stage of sintering. Thismeant that at most 2 temperatures could be used to calculated -AIR when (In µ')/m versus11T is plotted. Because only two temperatures were used to obtain slopes of -AIR, there issome doubt as to its validity. Similarly, use of the intermediate / final stage analysisprocedure was made difficult by the fact that not all the temperatures studied could be used.1024.1.2.1 CalculationsFor completeness, calculations of the activation energy of selected data is provided.From the p - t plots of the 'milled' PbNb2O6 (Figure 4-4) it can be seen that for alltemperatures, the intermediate stage of sintering was attained. Unlike other specimensthis meant that all 4 temperatures could be used. From the analysis above, tangent lines tothe intermediate portion of the p - In t plots for duplicate runs were drawn and the slopescalculated. The activation energy can then be obtained by plotting in (slope) as a functionof reciprocal temperature (Figure 4-21). The activation energy was calculated to be180 ± 18 kJ/mole. No other researchers have reported any activation energy valueassociated with the sintering of PbNb 2O6. Included in Appendix III are some plots for thecalculation the activation energy of SrO doped PbNb2O6. Because of the limited numberof points in those plots the data and analysis will not be discussed any further.103-3.3^1^1^I^1^i^1^I^r^0.00067^0.00069^0.00071^0.00073^0.000751/Temp. (1/K)Figure 4-21. Activation energy plots for milled PbNb2O6 from intermediate stagesintering data.4.2 Viscoelastic ModelAttempts made to correlate the sintering of PbNb 2O6 with known sinteringmechanisms, such as those postulated by Frenkel 47 , Kuczynski48 , ICingery4551 , andJohnson5253 , were only partially satisfactory and required the use of multiple theories. Therange of temperatures studied made it necessary to use initial stage sintering theory for someof the data and intermediate / final stage theories for others. Since initial stage sinteringequations are only valid for shrinkage to a few percent it is of limited use for PbNb 2O6specimens tested that exhibited linear shrinkages in the range of 4 % to 30 %. As waspreviously mentioned, factors such as grain growth, particle rearrangement, and unevenheating are not accounted for in these sintering models. In addition, as shown in Figure 2-1,the starting PbNb 2O6 powders are neither uniform nor spherical.Previously, in this department there have been several attempts to mathematicallymodel the total compaction behavior using viscoelastic elements, assuming the total processis a continuum. Following these analyses, a mechanical translational model has beendeveloped to simulate the pressureless sintering of PbNb 2O6. Unlike the approach ofestablished sintering models which begins with the sintering of individual particles andextends this to compacts, the translational model approach accepts the results from sinteringtests and tries to explain events in terms of the translational elements. Instead of only beingvalid over a small temperature range, the translational model can simulate the behavior of allsintering mechanisms individually or concurrently over the entire sintering period for alltemperatures.Bradbeer66 started by fitting normalized hot pressing data to an equation of the generalform:e = K(1 —ile' —Be -13` — Ce' — .....)^(4-14)104105where K, A, B, C, a, p, and y are experimental coefficients which vary with the curve beingfitted. Bradbeer66 and Penugonda67 used the 2 exponential term form of Equation 4-14 whileGi1168 and Chow69 used 3 exponential terms to fit their respective hot pressing data.dEBradbeer66 and Gi1168 plotted In (7) versus time and assumed non-overlapping terms, fromwhich the slopes and intercepts were obtained allowing the coefficients in Equation 4-14 tobe calculated. Penugonda 67 instead used a curve fitting routine that was able to obtain thecoefficients directly. For the work with PbNb2O6, Penugonda's approach was used.As was previously mentioned all experiments were repeated several times. Figure 4.22shows the magnitude of the maximum deviation between runs under the same experimentalconditions. This error between similar runs does not mask the distinction between curves ofdifferent temperatures, but made it difficult to assess the dopant concentration effect. Themagnitude of this error was calculated to be Ae ± 5.0x10 -3 .^0.1to 0^40^80^120^160Time (minutes)Figure 4-22. Contraction versus time plots highlighting the deviation betweenexperiment conducted under the same experimental conditions.2402000.08 -000.06 -00.04 -0 ^0 40^80^120Time (minutes)I^I^I160 200 240106The use of Equation 4-14 to fit the data allowed for the roughness in the plots to beeliminated. This also allowed for the averaging of contraction curves for repeated runs ofexperiments under the same testing conditions. Thus, more representative contraction curveswere obtained. Figure 4-23 compares the curve fit using 1, 2, and 3 exponential terms withone compaction curve. The curve with 1 exponential term did not fit the data for times>120 minutes unlike the 2 and 3 exponential term curves. Because of the good fit and itsgreater simplicity the 2 term form of Equation 4-14 was used. Due to errors in digitizing theshrinkage plot, there is a small but inherent amount of scatter in the experimental data.Because of the small difference between the digitized and calculated curve, the calculatedcurves were deemed to be a good representation of the contraction data. The 2 exponentialterm form of Equation 4-14 is:e =K —KA Cc" —KB e -13`^ (4-15)Figure 4-23. Comparison of curve fit, using 1, 2, and 3 exponential terms, to data.107Strain (e) is related to stress (a) via the following relationship:(4-16)where x is the transfer function. Although there was no externally applied pressure Equation4-16 can be valid. The stress in pressureless sintering arises from the contact stressesassociated with solid state sintering or the capillary forces during liquid phase sintering 48,51.'°Taking the Laplace transform for e (Equation 4-15) yields:E = £[e] =—K – KA KBs s+a s+0where £ represents the Laplacian operator. Rearrangements leads to:E = K(s + a) (s + (3) – KAs (s +(3) – KBs (s +a)s(s + a)(s +13)E = Ks 2 + K(a + (3)s + Kar• – KAs 2 – KA Ds – KBs 2 –KB ass [s 2 + (a + (3)s + 4]E = K(1 –A –B)s 2 +Ka(1–B)s +10(1 –A)s +Kal3s[s 2 +(a+(3)s +a(3](4-17)By considering the initial conditions Equation 4-17 can be further reduced. At t = 0Equation 4-15 becomes:et . ° ---- K(1 – ACas –Best)0 = K(1 – A –B)B =1–A^A =1–Btherefore Equation 4.17 becomes:E – KA as +KB13s + Kaf3s [s 2 + (a + 13)s + 4]E = Ks(A a +B(3) + Kar3s [s 2 + (a + Ms + oc13] .(4-18)(4-19)(4-20)108Previous work by Bradbeer66 , Chow69 , Gill68 , and Penugonda67 have assumed stress tobe constant for t > 0 as they applied a stress on the system and thus used a step function tomodel the stress. Since these previous studies have dealt with hot pressing, where theapplied pressure is far greater than that arising from the contact or capillary stresses, the useof a step function to represent hot pressing was appropriate. Schematically, the unit stressinput and resultant strain is shown in Figure 4-24. Because the PbNb 2O6 sinteringexperiments have essentially been carried out without any applied pressure, all the stressescome from the contact or capillary stresses.INPUT (stress)..•a = ao u_1 (t)SYSTEMDYNAMICS rE=K —KACca --KBCPOUTPUT (strain)Figure 4-24. Schematic representation of 'system dynamics' relating output (strain) toinput (unit stress).For pressureless sintering a better approximation would be a decaying stress function.Theory suggests that the stress-time relationship is perhaps of the form a =11(e) for n > 0.A problem with using this relationship is the singularity at t = 0. Use of such a function withthe Laplace transform technique also does not allow for an explicit solution. As anapproximation to this type of function an exponential decay stress function was used whichremoves the singularity at t = 0 and allows for an explicit solution. The Laplacian of anexponentially decaying stress function yields:E = f[a] = f[T Cal — T (s +a)(4-21)=TE[s 3 + (a+ (i)s 2 + ails])K(Aa+B(3)s 2 +(Kal3+aKAa+aKB(3)s +Ka13a(4-22)The transfer function in s-space becomes:Ts [s 2 + (a + (:1)s + cci3] X = f[x]= E = (s + a)[K (A a+ B (3)s +Kaf3]X =^[s3 + (a+ (3)s 2+ ar3s] K(A a +B(3)s 2 + KaPs + aK(A a +B f3)s +Kar3a109By taking its inverse Laplace transform (f t ) and with rearrangement Equation 4-22becomes:• K(A a +B(3)cy (Kap + aKA cc + aKB(3)0 KaPaaE+ (cc + (3)E + ar3E =(4-23)where the number of dots above the respective variable refers to the order of the derivativewith respect to time. Equation 4-23 is a third order differential equation which describes thepressureless sintering behavior of PbNb 2O6 with constants T, a, K, A, B, a, and(?).A mechanical translational model based on springs, dashpots, and masses whoseresponse is similar to Equation 4-23 has been developed. The mechanical elements werecorrelated to T, a, K, A, B, a, and 13. A spring is a device which exhibits a stress proportionalto the strain (e), while a dashpot is a device whose stress is proportional to the strain rate (e),and a mass is a device whose stress is proportional to the second derivative of strain (e). Thedetails of the development of the viscoelastic model are given in Appendix IV. An electricalcircuit whose response can be represented by a differential equation similar to Equation 4-23and used to develop the mechanical model is presented in Figure 4-25.The differential equation relating a - e for the viscoelastic model in Figure 4-25b is:((^ (4-24)^4. M^±.^M .,: 1 . .5.4.^+M 1 ,;± - +-M M a.1 12^1-1, ,^111^111112 11^gill^11112110L112m/h//mimmmmmFigure 4-25. Interactive a)electrical circuit and b)viscoelastic models used torepresent sintering behavior of PbNb206.By comparing coefficients between Equation 4-23 and Equation 4-24, one obtainsM—= a+(3112M = af31 K(Aa+B(3)M + –1 Kar3+aKAa+aKB(311112M M Ka(la —+ –P.112^T(4-25)(4-26)(4-27)(4-28)(4-29)Solving for ill, M, 112 from Equations 4-25 to 4-29 yields:T^ (4-30)K(Aa+B(3)(4-31)111 — Kar3+ aKA a+ aKB 0 ( a+RM = gai3M12 —^•a +13111T^11(4-32)(4-33)4.2.1 ProgramAn MTS FORTRAN IV program was written using the available subroutineNL2SNO to obtain the coefficients K, A, B, C, a, 13, and y. NL2SNO is a curve fittingsubroutine particularly suited for equations consisting of exponential terms similar toEquation 4-14 used to describe the sintering behavior of PbNb 2O6. The program uses thesubroutine NL2SNO to fit the data with versions of Equation 4-14 containing 1, 2, and 3exponential terms.Inputs into the program are the raw shrinkage data and the final specimen size Lf.The raw shrinkage data was normalized then used by the curve fitting routine NL2SNO thatdetermined the coefficients. Seed values needed to initiate the starting values of NL2SNOwere also supplied. Depending on the fit desired 2, 4, or 6 seed values were used. Theinitial condition (t = 0) that e = 0 was also incorporated.After curve fitting, the generated numbers were compared with the normalized input.The form of the equation with the 1 exponent term did not fit well and was not analyzedfurther. From Figure 4-23 it can be seen that both of the curves with 2 and 3 exponentterms fit the data well. From the 2 and 3 exponential curve fitting coefficients, themechanical translational coefficients were calculated. Depending on the desiredtranslational model various subroutines can be used to obtain these coefficients. The modelVDETERMINETRANSLATIONALCOEFFECIENTPRINTCOEFF.112ultimately used and discussed in Section 4.2 is incorporated in the subroutine MECH.Presented in Appendix V is the program written to obtain the curve fitting andtranslational coefficients of the model presented in Figure 4-25b and described by equation4-30 to 4-33. A schematic of the program is presented in Figure 4-26.( START )/ READ DATAVNORMALIZE DATAUSE NL2SNOCURVE-FIT ROUTINEPRINTCOEFF.( STOP )Figure 4-26. Schematic of computer program.4.2.2 InterpretationIn Appendix VI, tables of the viscoelastic coefficients, M, µ, 11 1, and /12 for runsconducted at the various temperatures and dopants studied are included. The values in thetables are the average of multiple runs. Some of these final results are presentedgraphically in Figures 4-27 to 4-30, where they are plotted as a function of temperature.The values for the mass or inertia (IA) and viscosity (rh) terms are much larger than those ofthe elastic (M) and viscosity (1 2) terms so the latter two terms are intentionally omitted.Previous viscoelastic studies by Bradbeer66 , Gill68 , Chow69, and Penugonda67 withreactive hot pressing yielded viscoelastic parameters that varied with temperature which inturn correlated well to specific reactions. There was found to be no correlation with anyreaction recorded by the DTA of the doped PbNb2O6. This could be due to the smallamount of dopants, the phase stability of PbNb 2O6, or the fact that experiments wereconducted under different conditions (pressureless versus pressure).Comparison of the viscoelastic terms from the 'as-received' PbNb 2O6 (Figure 4-27)with the 'milled' PbNb2O6 (Figure 4-28) clearly shows the much smaller values ofil and 111for the 'milled' powder. As mentioned previously, milling facilitates sintering. Anexplanation for the sintering behavior of 'milled' PbNb 2O6 in terms of the viscoelasticresults is that there was less inertia and viscous drag that allowed for greater contractionunder the sintering stress. The difference in the values is most noticeable at the lowertemperature of 1050 and 1100 °C. The small difference at the higher temperatures can beexplained by the fact that at the higher temperatures, milling and dopant additions werefound to be masked by the effect of temperature on sintering.Figures 4-29 and 4-30 show the viscoelastic terms 4 and i i , for PbNb2O6 doped with1 wt. % SrO and Bi203 , respectively. The values for the Bi 203 doped PbNb2O6 are aboutone-half of those for the SrO doped PbNb 2O6. From the viscoelastic point of view, the113114smaller values of j.t. and Th meant that its contraction is greater. As mentioned in Section3.2.5 the 'milled' powder contracted the most of any doped or undoped powder. Of all thedopants, Bi203 aided sintering the most and SrO the least. This ranking of relativeeffectiveness for sintering is supported again by the viscoelastic analysis.The graphs clearly indicate that the mass component p. and viscosity term rh decreasewith increasing temperature. Although there may not be any liquid phase during sintering,the viscosity term of viscoelastic model represents all flow behavior in the system. Withincreasing temperature powder moves more during sintering and grain growth, which isreflected in the decrease in viscosity. The mass component II of the model was also foundto decrease with temperature. The physical significance of the "mass" term .t is not clearlyunderstood but its effect on strain (e) is similar to that of mass on displacement. If the massterm is to be interpreted as a real mass, Nelson et al. 34 results show increasing PbNb 2O6 losswith increasing temperature in the order of 2 atomic %. This increasing PbNb 2O6 lossmeans that the mass is less at higher temperatures as shown by all the plots. Because of thesmall PbNb2O6 loss and the dramatic decrease of in the value .t this interpretation ofµ wasfelt to be improper.The need for the mass-like termg may be explained by studying the stress - strainrelationship of a dashpot:a =.f(i)=f l(a)where a is function of time. For pressureless sintering, contact or capillary stresses giverise to compaction. These stresses decrease as the necks between particles grow. Thismeans that stress is function of time and thatate ariasat2 as at115Since i is not zero there is a need for mass-like term p, to relate a and E.The viscoelastic analysis as carried out here is the first of its kind to interpret thepressureless sintering phenomena, although similar studies were made to analyzehot-pressing data. In the present case, the time dependent sintering stress (the driving forcefor sintering) makes the system more complex. It is hoped that with more understanding ofall concurrent sintering mechanisms, such an analysis as developed here may be able toincorporate all sintering behavior into the mechanical elements for interpretation andanalysis.1.05^1.07^1.09^1.11^1.13^1.15Temperature ( °C x 1000)Figure 4-27. p. and II I as function of temperature for 'as-received' PbNb 2O6specimens., 1.6o •2 1.4x1.,=300a2001000116-1001.05^1.07^1.09^1.11^1.13^1.15Temperature ( °C x 1000)Figure 4-28. p, and m as function of temperature for 'milled' PbNb 2O6 specimens.1.191.171171.^1.6c^1.07^1.09^1.11^1.13^1.15^1.17^1.19Temperature ( °C x 1000)Figure 4-29. and Ili as function of temperature for 1 wt. % SrO doped PbNb 206specimens.1.05^1.07^1.09^1.11^1.13^1.15^1.17^1.19Temperature ( °C x 1000)Figure 4-30. p, and rh as function of temperature for 1 wt. % Bi 203 doped PbNb2O6specimens.5 SUMMARY AND CONCLUSIONI) Isothermal sintering experiments were carried out to investigate the sintering kineticsof PbNb206 over the temperature range 1050 - 1200 °C, with various dopants (SrO, Bi 203 ,LiNbO3 , and KNaNb2O6) at concentrations from 0 to 1 weight percent. Since mixing of thedopant and the PbNb2O6 was done by vibratory ball milling, the effect of milling on sinteringkinetics was also investigated.II) Vibratory ball milling resulted in the greatest shrinkage at lower temperatures inaddition to reaching the end point density sooner at higher temperatures. Milling was foundto decrease the average particle size which led to excessive grain growth when sintered at1200 °C and a bimodal grain size when sintered at 1050 °C.III)Additions of any dopant to PbNb 2O6 prior to milling was found to hinder graingrowth and sintering when compared to the 'milled' powder. Results suggest that increasingadditions of SrO, LiNbO3 , and KNaNb2O6 decrease the grain size unlike increasing additionsof Bi203 which led to a larger grain size. Of the 4 dopants, Bi 203 doped PbNb2O6 had largestgrain size and exhibited the greatest contraction. SrO additions resulted in the leastcontraction while the effectiveness of LiNbO 3 and KNaNb2O6 was intermediate between theother two dopants.IV) The sintering mechanisms of PbNb 2O6 during the initial stage were found to be bulkdiffusion and bulk neck diffusion. The sintering mechanism was dependent on the dopantaddition and sintering temperature. The presence of a low melting dopant encouraged bulkdiffusion while the presence of a high melting dopant encouraged bulk neck diffusion. Ingeneral, initial stage sintering of PbNb2O6 was found to begin by bulk diffusion (e cc 19.8)before it switched to bulk neck diffusion (e oc 114). For some dopants this change inmechanism was not noticed at 1050 °C but was noticed at 1100 °C118119V) For intermediate / final stage sintering data the analytical method employed byCoble54'55 was used. Plots of p - In t yielded linear regions which are indicative of bulkdiffusion for these stages. One or two linear regions were observed in the plots depending onwhether the final stage of sintering was reached.VI) Analysis developed by Johnson and Cutler 52'52 to determine the activation energy forinitial stage sintering was extended to the intermediate / final stage sintering equations ofCoble's54 '55 . Because of the limited number of temperatures used in these tests either one orthe other type of analysis on selected data could be used. For 'milled' PbNb2O6 an activationenergy of 180 ± 18 kJ/mole was obtained.VII) Isothermal compaction curves were fitted to an equation of the formc . AL/Lo = K(1 — Ae' —Be -13t), from which a viscoelastic model for the pressurelesssintering of PbNb2O6 was developed. This model uses a decaying stress function to representthe decreasing contact or capillary stress as sintering proceeds. A computer program waswritten to find the solution for the various components. Of the four viscoelastic element onlythe mass and one of the viscosity components were found to be significant andtemperature-sensitive. In general, the contraction behavior could be explained by thedecreasing values of these two terms with temperature.6 RECOMMENDED WORKI) The limited temperature tests carried out in this investigation meant that some datawas applicable to initial stage sintering analysis while others to intermediate / final stagesintering analysis. For this reason the limited amount of data for either type of analysis madedetermination of the activation energy difficult. More sintering data at other temperatures areneeded to determine the activation energy of these stages. It would be interesting to see if theactivation energy associated with intermediate / final stage sintering yields values similar tothose obtained from Johnson and Cutler's 52 ' 52 technique with inital stage sintering.II)As noted, milling was found to lead to an excessive amount of grain growth. Sincethe addition of dopants generally results in a decrease in the curie point. milling appears to bea way to enhance densification without any dopant addition. A quantitative analysis of initialparticle size distribution and its effect on final grain size and sintering would be worthy offurther investigation. In addition, a controlled rate sintering schedule should be studied toreduce grain growth in the 'milled' powder.III)Because of the small range of dopant concentrations studied, its effect could not bedetermined with certainty. A greater range of concentrations should be investigated todetermine any correlation. It would be of interest to know if and/or when greaterconcentrations of a liquid forming dopant causes the sintering mechanism to change to liquidphase sintering from by bulk diffusion.IV) To determine the applicability of the viscoelastic model developed for thepressureless sintering of PbNb2O6 it should be used with pressureless sintering data of othermaterials.1201217 REFERENCES1. M. K. Murthy, "Advanced Ceramics", Report #3 for the Office of IndustrialInnovation, Government of Canada, Jan. 86.2. E. J. Kubel, "Good Opportunities for Advanced Ceramics", Advanced Materials &Processes, no. 9, 1987, p. 55-60.3. L. M . Sheppard, "Canada Chases the Ceramics Market", Advanced Materials &Processes, no. 3, 1987, p. 11-14.4. L. M. Sheppard, "Ceramics at the Cutting Edge", Advanced Materials & Processes,no. 8, 1987, p. 73-79.5. N. Ichinose, "Electronic Ceramics for Sensors", J. of the Am. Cer. Soc., vol. 64, no. 12,1985, p. 1581-1585.6. R. C. 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Buchanan, "Low-Temperature Densification of LeadZirconate-Titanate with Vanadium Pentoxide Additive", J. of the Am. Cer. Soc., vol.64, no. 8, 1981, p. 485-490.27. R. Lal, N. M. Gokhale, R. Krishnan, "Effect of Sintering Parameters on theMicrostructure and Properties of Strontium Modified PZT Ceramic Prepared UsingSpray-dried Powders", J. of Materials Science, vol. 24, 1989, p. 2911-2916.28. K. W. J. Fickert, K. A. Kappes, S. Bateson, "Ferroelectric Ceramics in the SystemPb0-Nb2O5", J. of the Canadian Ceramic Society, vol. 35, 1967 p. 55-60.29. A. M. Golub, N. C. Wen, F. F. Grigorenko, "Lead Niobates", Translated from ZhurnalNeorganisheskoi Khimii, vol. 14, no. 5, 1969, p. 607-610.30. V. A. Titova, I. F. Cherednichenko, N. G. Kisel, "Preparation of Lead Metaniobate",Translated from Zhurnal Neorganisheskoi Khimii, vol. 12, no. 6, 1967, p. 769-772.12331. V. V. Prisedskii, L. G. Gusakova, V. V. Klimov, "Conditions and Mechanism ofPreparation of Lead Niobates", Translated from Zhurnal Neorganisheskoi Khimii, vol.21, no. 10, 1976, p. 2679-2685.32. D. R. Biswas, "Development of Novel Ceramic Processing", J. of Materials Science,vol. 24, 1989, p. 3791-3798.33. D. Waller, T. Iqbal, A. Safari, "Poling of Lead Zirconate Titanate Ceramics andFlexible Piezoelectric Composites by the Corona Discharge Technique", J. of the Am.Cer. Soc., vol. 72, no. 2, 1989, p. 322-324.34. K. E. Nelson, R. L. Cook, "Ceramic and Dielectric Properties of Lead MetaniobateElectrical Ceramics", J. of the Am. Cer. Soc., vol. 42, no. 3, 1959, p. 138-142.35. M. C. Bhardwaj, "Principles and Methods of Ultrasonic Characterization of Materials",Advanced Ceramic Materials, vol. 1, no. 4, 1986, p. 311-323.36. M. C. Bhardwaj, "Modern Ultrasonic Concepts of NDT", Advanced Materials &Processes, no. 5, 1989, p. 53-60.37. M. H. Francombe, "The Relation between Structure and Ferroelectricity in Lead andBarium Strontium Niobates", Acta Cryst., vol. 13, 1960, p. 131-14038. E. C. Subbaro, "Nonstoichiometric PbNb2O6-Type Solid Solutions", J. of the Am. Cer.Soc., vol. 42, no. 9, 1959, p. 448.39. E. C. Subbaro, J. Hrizo, " Solid Solutions Based on Ferroelectric PbNb 2O6", J. of theAm. Cer. Soc., vol. 45, no. 11, 1962, p. 528-531.40. M. H. Francombe, "Polymorphim in Lead Metaniobate", Acta Cryst., vol. 9, 1956, p.683-684.41. R. S. Roth, "Pb0-Nb 2O5 Phase diagram" , J. of the National Research Bureau ofStandards, vol. 62, no. 1, 1959, p. 34.42. E. C. Subbaro, G. Shirane, "Nonstoichiometry and Ferroelectric Properties ofPbNb2O6-Type Compounds", J. of Chemical Physics, vol. 22, no. 6, 1960, p.1846-1851.43. M. H. Francombe, B. Lewis, "Structural, Dielectric and Optical Properties ofFerroelectric Lead Metaniobates", Acta Cryst., vol. 11, 1958, p. 696.44. W. D. Kingery, H. K. Bowen, D. R. Uhlmann, Introduction to Ceramics,  John Wiley& Sons, New York, New York, 1976.45. W. D. Kingery, "Sintering in the Presence of a Liquid Phase" Kinetics ofHigh-Temperature Process, Part IV, Technology Press, Cambridge, Massachusetts,John Wiley and Sons, New York, 1959.12446. L. E. Murr, C. Stein - Editor, Frontiers in Materials Science, Marcel Dekker Inc., NewYork, New York, 1976, R. J. Charles, Sintering and its Role in Material Processing, p.456-491.47. J. Frenkel, Journal of Applied Physics: USSR, vol. 9, 1945, p. 385.48. G. C. Kuczynski, "Self diffusion in Sintering of Metallic Particles" MetalTransactions, Feb. 1949, p. 169-177.49. R. L. Coble, "Initial Sintering of Alumina and Hematite", J. of the Am. Cer. Soc., vol.41, no. 2, 1958, p. 55-62.50. W. D. Kingery, J. M. Woulbroun, F. R. Charvat, "Effect of Applied Pressure onDensification During Sintering in the Presence of a Liquid Phase", J. of the Am. Cer.Soc., vol. 46, no. 8, 1963, p. 391-395.51. W. D. Kingery, M. Berg, "Study of the Initial Stages of Sintering Solids by ViscousFlow, Evaporation-Condensation, and Self-Diffusion", J. of Applied Physic, vol. 26,no. 10, 1955, p. 1205-1211.52. D. L. Johnson, I. V. Cutler, "Diffusion Sintering: I, Initial Stage Sintering Models andTheir Application to Shrinkage of Powder Compacts", J. of the Am. Cer. Soc., vol. 46,no. 11, 1963, p. 541-545.53. D. L. Johnson, I. V. Cutler, "Diffusion Sintering: II, Initial Sintering Kinetic ofAlumina", J. of the Am. Cer. Soc., vol. 46, no. 11, 1963, p. 545-550.54. R. L. Coble, "Sintering Crystalline Solids. I. Intermediate and Final State DiffusionModels", J. of Applied Physics, vol. 32, no. 5, 1961, p. 787-792.55. R. L. Coble, "Sintering Crystalline Solids. H. Experimental Test of Diffusion Modelsin Powder Compacts", J. of Applied Physics, vol. 32, no. 5, 1961, p. 793-799.56. W. Beere, "The Second Stage Sintering Kinetics of Powder Compacts", ActaCrystallica, vol. 23, no. 1, 1975, p. 139-145.57. G. H. Gessinger, H. F. Fischmeister, H. L. Lukas, "A Model for Second-Stage LiquidPhase Sintering with Partially Wetting Liquid", Acta Metallurgica, vol. 21, 1973, p.715-724.58. W. D. Kingery, E. Niki, M. D. Narasimhan, "Sintering of Oxide and Carbide-MetalComposition in Presence of a Liquid Phase", J. of the Am. Cer. Soc., vol. 44, no. 1,1961, p. 29-35.59. J. K. Hulm, "Low-Temperature Dielectric Properties of Cadmium and Lead Niobates",Physics Review, vol. 92, no. 2, 1953, p. 504-5.60. R. S. Roth, "Unit-cell Data of the Lead Niobate PbNb 206 ", Acta Crystallica, vol. 10,1957, p. 437.12561. P. W. Sunderland, Master's Degree Thesis - Deformation of Compacts of Magnesiumhydroxide during dehydroxylation., Univ. British Columbia, Dept. Metals andMaterials Engineering, Dec. 1987.62. C. Herring, "Effect of Change of (Size) Scale on Sintering Phenomena", J. AppliedPhysics, vol. 21, 1950, p. 301.63. M. F. Ashby, "A First Report on Sintering Diagrams", Acta Metallurgica, vol. 22,Mar. 1974, p. 275-289.64. A. Mohon, N. C. Soni, V. R. Moorthy, "Mechanism of Sintering", Transactions of theIndian Institute of Metals, vol. 33, no. 6, Dec. 1980, p. 423-431.65. F. B. Swinkels, M. F. Ashby, "A Second Report on Sintering Diagrams", ActaMetallurgica, vol. 29, 1981, p. 259-281.66. R. S. Bradbeer, Master's Degree Thesis - Reactive Hot Pressing of Boehmite, Univ.British Columbia, Dept. Metals and Materials Engineering, Jan. 1972.67. M. R. Penugonda, Master's Degree Thesis - Al 203-SiC Composites fromKaolinite-Carbon Precursors by Hot-pressing, Univ. British Columbia, Dept. Metalsand Materials Engineering, Dec. 1987.68. W. W. Gill, Master's Degree Thesis - Material Characteristics Affecting Formcoke,Univ. British Columbia, Dept. Metals and Materials Engineering, June 1979.69. C. Chow, A. C. D. Chaldader, I. H. Warren, W. R. Leeder, "Hot-compaction Behaviorof Char/binder-coal Systems", Fuel, vol. 57, July 1978, p. 387-393.70. W. A. Kaysser, G. Petzow, "Liquid Phase Sintering of Ceramics", Emergent ProcessMethods for High-Technology,  Edited by R. Davis, H. Palmour III, R. Portor, PlenumPress, New York, 1984, p. 225-231.71. D. E. Scott, An Introduction to Circuit Analysis: A System Approach, McGraw-HillBook Company, New York, 1987, p. 141-173.Appendix IDielectric constants of the test specimens were determined by Dr. E. Prasad. Thesevalue are comparable with the known dielectric value (-225) for PbNb2O6. Milling yieldedsignificantly lower dielectric constants than those of the other samples. Francombe andLewis43 found that conversion of PbNb 2O6 to the orthorhombic phase was difficult even afterheating to 1250 °C, particularly if it is present as large crystals. This could the reason for thelow dielectric constants of specimens produced from the 'milled' powder. The SEMmicrographs confirm this observation.Table I-1. Dielectric constants of PbNb2O6 specimens with various dopants, dopantconcentrations, temperatures, and milling procedures.Dopant wt.% 1200 °C 1150 °C 1100 °C 1050 °CSrO 1 537 463 371 495Bi203 1 419 324 371LiNbO3 1 401 371 376 408KNaNb2O6 1 354 328  323 428SrO 0.5 285 330 356 436Bi203 0.5 412 202 626LiNbO3 0.5 264 460 216 882KNaNb206 0.5 383 286 329 593SrO 0.25 361 632 312Bi203 0.25 411 255 325 538LiNbO3 0.25 305 425 293 381KNaNb206 0.25 358 330 319 341'Milled' * 12 45 62 47'As-rec.' 383 358 249 464126No dopant.Appendix II127rn tr.)0000xskz..0O1.••••■I1X.1 TS LIOp anuutag1150 °C1 100 °C... • . . . ................... - - - • ............. • • .....1050 °C0.950.90.85...,'^0.809-00.)  0.75I;) 0.7... • - * •• .......... ......^,,...... ....■ ••■ ...•••• .■■•.....I. .•■••••,. ...•■•••. ••■••• •■•... ........ ■ •■■•■•••128...... - - - - - • '1^i80^120^160^200^240Time (minutes)Figure 11-2. Relative density curves for 1 wt. % KNaNb 2O6 doped PbNb2O6 samplessintered in air at temperatures indicated.0.954^60.9 -0.85 -0.650.60.55-4^2^0^2In Time (In minutes)Figure 11-3. Relative density - In t curves for 1 wt. % KNaNb 2O6 doped PbNb2O6samples sintered in air at temperatures indicated.0.650.60.551290.940.920.90.880.860.84_4,, 0.82E 0.80.780.760.740.720.70.680.660.64.62  40^80^120^160^200^240Time (minutes)Figure 11-4. Relative density curves for 0.25 wt. % KNaNb2O6 doped PbNb2O6 samplessintered in air at temperatures indicated.1200 °C1150 °C1 10 ° c1050 °C0.940.920.90.880.860.840.824.9 0.87:3 0.780.760.740.720.70.680.660.640.621200 °C1150 °C1 10 ° c1 05 °C-3^-1^1^3^5In Time (ln minutes)Figure II-5. Relative density - In t curves for 0.25 wt. % KNaNb2O6 doped PbNb2O6samples sintered in air at temperatures indicated.1050 °CI^I^I^I^I^I^I^I^I^I^I^I0.920.9 -0.88 -0.86 -0.84 -0.82 -, 0.8 -0.78-8 0.76.>".., 0.74rt' 0.7273r:4 0.70.680.660.640.620.6 -0.58  1300^40 80 120 160 200 240Time (minutes)Figure 11-6. Relative density curves for 1 wt. % LiNbO 3 doped PbNb2O6 samplessintered in air at temperatures indicated.0.920.90.880.860.840.82>, 0.8'v., 0.78=-8 0.764.) 0.740.72(24) 0.70.680.660.640.620.60.58-2^0^2^4^6In Time (In minutes)Figure 11-7. Relative density - In t curves for 1 wt. % LiNbO 3 doped PbNb2O6 samplessintered in air at temperatures indicated.I5■^I^I1 3In Time On minutes)1050 °C0.920.90.880.860.840.82▪ 0.8•6,1▪ 0.78-8 0.760..)> 0.74.-0.72p4 0.70.680.660.640.620.60.581310^40^80^120^160^200^240Time (minutes)Figure 11-8. Relative density curves for 0.25 wt. % LiNbO 3 doped PbNb2O6 samplessintered in air at temperatures indicated.0.920.9 -0.88 -0.86 -0.84 -0.82 -)., 0.8 -' 0.78 --8 0.76 -> 0.74 -0.72 -0.7 -0.68 -0.66 -0.64 -0.62 -0.6 -0.58-1Figure 11-9. Relative density - In t curves for 0.25 wt. % LiNbO 3 doped PbNb2O6samples sintered in air at temperatures indicated.1320^2^4^6In Time (in minutes)Figure II-10. In e - In t curves for KNaNb2O6 doped PbNb2O6 samples sintered in air attemperatures indicated.0^2^4^6In Time (In minutes)Figure II-11. In £ - In t curves for LiNbO3 doped PbNb2O6 samples sintered in air attemperatures indicated.Appendix IIIFor completeness, calculations of the activation energy of selected data is provided.Tangent lines to the intermediate portion of the p - In t plots were drawn and the slopes werecalculated. Following the procedure for analyzing intermediate stage sintering data activationenergies of 193 ± 110 kJ/mole for 'as-received' (Figure III-1), 151 ± 26 kJ/mole for SrOdoped PbNb2O (Figure 111-2), 95 ± 19 kJ/mole for Bi203 doped PbNb2O (Figure 111-3),138 ± 32 kJ/mole for KNaNb 2O6 doped PbNb2O (Figure 111-4), 140 ± 26 kJ/mole for SrOdoped PbNb2O (Figure 111-5) were obtained. These values agree agree reasonably well withthe activation energy determined previously for the 'milled' powder.133-1.3-1.4 --1.5 --1.6 --1.7 --1.8 --1.9 --2 --2.1 --2.2 -o -2.3-2.4 --2.5 --2.6 --2.7 --2.8 --2.9 --3 --3.1 --3.2 --3.3 --3.4 ^0.00067 0.00069^0.00071^0.00073^0.000751/Temp (1/K)Figure III-1. Activation energy plot for 'as-recieved' PbNb 2O6 from intermediate stagesintering data.134-2.5 1^1^I^1^1^11110.000678 0.000686 0.000694^0.0007021/Temp (1/K)Figure III-2. Activation energy plot for SrO doped PbNb 2O6 from intermediate stagesintering data.-2.5 --2.6 --2.7 --2.8 1^1^1^1^1^i^1^10.00067^0.00069 0.00071 0.00073 0.000751/Temp (1/K)Figure III-3. Activation energy plot for Bi203 doped PbNb2O6 from intermediate stagesintering data.^-1.9 ^-2 --2.1 --2.2-2.3 --2.4 --2.5 -4)2., -2.6 -c -2.7 --2.8 --2.9 --3-3.1-3.2 --3.3 --3.4 ^0.00067135I^I^I^I^I^I0.00069 0.00071 0.000731/Temp (1/K)Figure III-4. Activation energy plot for KNaNb 2O6 doped PbNb2O6 from intermediatestage sintering data.i0.00075-1.7 ^-1.8 --1.9 --2 --2.1 --2.2 --2.3 -1.-') 2.4 -ra.-2.5 -0tg -2.6 --2.7 --2.8 --2.9 --3 --3.1 --3.2 --3.3 ^0.00067 0.00069^0.00071^0.000731 10.000751/Temp (1/K)Figure III-5. Activation energy plot for LiNbO3 doped PbNb2O6 from intermediatestage sintering data.Appendix IVAn electrical analog to the mechanical model was used in the derivation for ease of useand because of the extensive literature on circuit analysis. At a later stage it can be convertedto the mechanical model using the following Table and Rule:Table IV-1. Mechanical-Electrical Conversion Table. [71]Mechanical^Electricalspring (1/M)^capacitor (C)dashpot (1) resistor (R)mass (p.)^inductor (L)stress (a) voltage (v)strain (e)^charge (q)Rule IV-1.) A mechanical model of an electrical model can be made by replacingcomponents in parallel by those in series and those in series with components inparallel. This is necessary because the matching forms of the relationships between v -qand e-a is in the impedance format Z(s) for v-q compared to the admittance format Y(s)for E-G as presented in Figure IV-1.(4-23)••^• K(Aa+B(3)a (Ka43+aKAa+aKB13)a aKaPae+(a+(3)e+aPe- T^+^T^+ TThe electrical analog of Equation 4 -23 can be converted using Table IV-1 and thedefinition i = q yielding:i + (a+ IV + aPi = K (A a + B 13)v (KaP + aK A a+ aKB (3)v aKaPv whose Laplace transform is:[s 2 + (a+ (3)s + apsj/ =[ K(A a + B (3) s 2 + (KaP+ aKAa+ aKB(3)s +aKafl ywhere /=£(i] and V=114. Upon rearrangement:Z — V  ^T [s 2 + (a + 13)s + I K(Aa+B(3)s 2 +(Kar3+aKAa+aKB(3)s +aKal3(N- 1)136C1Ls 2 1cyLs 2137VQ =—sY(s)^ E = EY(s)^EZ(s)Y (s) Z(s)^Y(s)1Rs111us^usV = QsZ(s)sZ(s)--IMAAAAn RsRVNM))LV 1CC M 1MFigure AIV-1. Defining equations in s-space for a) electrical systems b) mechanicalsystems in impedence Z(s) and admittance Y(s) format.Various combinations and arrangements of resistors, capacitors, and inductors weretried and their transfer functions calculated. The simplest arrangement of resistors,capacitors, and inductors that produces a similar transfer function as in Equation AN-1 ispresented in Figure AN-2a. This electrical analog was converted to its mechanicalequivalent following Rule AN-1 and Table AN-1. The impedance transfer function forelectric circuit is:11sL +1 ) R2sC1R 111- ±^R 1^R2 sL + sCR+ 121R1 + (sL)(sCR2 + 1) + R2[sL(sCR2 + 1) + R2]R 1=(sL)(sCR2 +1)+ R2 + (SCR2 + 1)R1S 2LCR 1R2 + sLR I -I- R iR2 1[s 2LCR2 + s(L +CR 1R2)+ (R2 + R 1 )[and noting that I= sQ2^1^1V ,^S + S a2 + LC— = LQ^c 2( 1^1^1 ) ( 1 ^1S'-' R 4.1 ) ' S ( CR IR2 + 1, + La i LCR2 )(I V -2)(IV-3)1381^sCR2 + 1LC R iR2LCR1R2139Note the similarity in the transfer function between Equation AIV-2 and Equation AIV-1confirming their similar behavior to a given response. The electronic values presented inEquation AIV-3 were converted to mechanical values following Table AIV-1 yielding FigureAIV-2b and Equation AIV-3.— = ^E s 23S F S 2( 71 )± S( 111- )(AIV-3)/ 1 v+I s(2—+-1 )+(-111 + —A111112^1 ^11111^In12 L112Figure AIV-2. Interactive a) electrical circuit and b) viscoelastic model used torepresent sintering behavior of PbNb2O6 .Appendix VC THIS PROGRAM USES THE SUBROUTINE NL2SNO AVAILABLE ON MTS TO FITC SHRINKAGE DATA. THE INPUTS ARE TIME (X) AND SHRINKAGE (Y) ASC WELL AS THE FINAL PELLET THICKNESS (YF). THE MODEL IS ASSUMED TOC BE THE SUM OF EXPONENTIAL FUNCTIONS WITH THE TRIAL COEFFICIENTSC PROVIDED. THIS PROGRAM DOES ALL THE CALCULATIONS (CURVE FITTING ANDC VISCOELASTIC) OF SECOND AND THIRD ORDER EXPONENTIAL TRIAL FUNCTIONSC TO BE FI FIED. BECAUSE OF LIMITED SPACE SUBROUTINE MECHBRAD,C MECHGILL, MECHCHOW, MECHOWI, MECHNLE ARE NOT INCLUDED WITH THISC CONDENSED VERSION OF THE PROGRAM.IMPLICIT REAL*8(A-H 2O-Z)DIMENSION P(6),IV(70),V(2200),YCALCD(3,200),ALNY(3,200)EXTERNAL CALCRCOMMON X(200),Y(200)REAL*8 YF,B(3,8),MU(3,3),ETA(3,3),YCALC(3,200),TEMP(200)INTEGER I,J,K,N,M,NAMCHARACi ER*11 SPECC BEGIN MAIN, GUESS INITIAL VALUES OF COEFFICIENTSC P ARRAYS STORES SEED VALUES FOR CURVE FITTING ROUTINEP(1)=-0.01170D0P(2)=-0.03175D0P(3)=1.0* (-0.05757D0)P(4)=1.0*(-0.00049D0)P(5)=1.0*(-0.00001D0)P(6)=1.0*(-0.00001D0)C B MATRIX IS WHERE CALCULATED VALUES FOR CURVEC FITTING COEFFICIENTS ARE PLACED. SETS B(I,J)=0DO 5 1=1,3DO 4 J=1,8B(I,J)=04 CONTINUE5 CONTINUEC READS SHRINKAGE DATA. FIRST TWO LINES OF DATA FILE CONTAINSC FILE SPECIFICATION AND NAME BEFORE READING IN YFREAD(5,*) SPECREAD(5,*) NAMREAD(5,8) YFN=110 READ(5,11,END=15) X(N),TEMP(N)11 FORMAT(F9.3,1X,F8.6)N=N+1GOTO 1015 CONTINUEC CALCULATES STRAIN (IE NORMALIZE)C SET COUNTER (H) TO ZERON=N-1DO 18 I=1,NY(I)=(TEMP(I))/(YF+TEMP(N))18 CONTINUEC FIND COEFFICIENTS TO FIT DATA USING NL2SNOM=2DO 30 I=1,3IV(1)=0140CINHINI21f11321((9`Z)E1 -1-(17`6$1)/(0`61 .11/1-)=(Z`Z)VI3(9`Z)3*(17`61:1*(Z`Z)flIA1=(I`Z)flIAI((I `Z)VI3/((9`611+(t7`611)+(9`Z)E1*(S`Z)fl^++(t,Z)3*(E`Z)E1+(9`Z)3*(17`Z)11*(1`Z)3)/(01)=(Z`Z)f1IAT((Se Z)E1*(S`Z)11+(17`Z)E1*(E`Z)11)/(0' I)=(I' Z)VIR'SSVIAI au, (z`z)niAl (INV `INVISNOD ONIIMS Ma SI (I `Z)flIAI D`swum, AIISODSIA 3111 auv via •st•rouLvriba dO 12S DNIM0-1103 RHI DAEI URA-10S a{ NVD II N3ILL S'TVIIN3N0dX3 OMI SaSfl "MOW Ma dI DHDRIAI NIDREI D(9)X(9)XX NOISNMAIIGN411 IIRDall\II(8`E)H`(E`E)va,3`(c`c)rArda`33'au`A`(b)v 8*Tvra2I(z-oli-v) 8*TVR8 Iondinil(via`runrE)oisloHDavq gNunougas'via CINV flIAI XIILLVIAT MUNI (13110IS 31IV sllsonidaaop DDIISVTRODSIA asau •SINVISNOD IOcillSVU (INV `SSVIAI `DNIIMS dO DnsamagoD ima-rvionba TVDINVHDMAI 3ILL sanirmo-w3 9N I1011 SIMI. DDI1IV119011c1 NIVIAI UNR DaNgdOIS(. „.V..„)IVIAT2103 01733dS (01,9)R.II2TM(17'0ID'XI`V0ID'XI`VOID'XI`V0ID'XI`V0ID'XI`VOID +` .171ID'XI 4171IYXI`i7lID'XI`VITO'XI`EDIVIAT110.4 6E(E`E)vag(z`z)via`(T`E)vis`(E`E)nw +`(z`z)funr(T`E)ruAr(z`z)vaz`(I`z)via`(z`z)npt (ii`z)aysrvsnix (6V9)31111A1'CaLlIOS GNV NI (Beau ugori SI II 3113BM `3•-n4 V OZ INRS CINV DVIVU '1'1V SSaDMId 01 I1IV21002Id H3IVE1 V HIIM NOIIDNIIINOD NI COM SI Sall Dsmai,'IVIIN3N0dXR E CINV Z 110.4 'adaop 'ODRA T1V 1f10 SINIIld MOTREI NOLLDRS D(vJaniAra)a-usa-DmAi TIV3 D(VIR`flIVEDIMOHDRIAI TIV3 D(vIa`fliAril)MOHDHDRIAI 'YIVD D(VIH'IlIArEOTTIDHDRIAI T1V3 D(via`mArE)UV/IIIHDMAI T1VD D(VIR`flIAril)HDRIAI 'TIV3MINI,LNOD SE3fINIINO3 VE0=(11)111AI0=(1"I)VIR£`I=1 17E OCIE`I=I SE OCE0113Z OZ XDIJATIAI flIAI (INV VIR IRS DMINIINOD OEZ+IAI=IAI3f1NLLNO3 ZZ(f)d=(Z+1`I)EfIAn=f ZZ OCEMlNIINOD OZ( I `I) E1+()Dd-=( 1 ' I)E1VW' I=31 OZ OU(I1II1Vdd'IARIVc111`Y^IIIVdrA'AI`ILD'IV3`cri1rmONSZTIsI TIV3Ti, 'C END MECHCC SUBROUTINE CALCR IS CALLED UPON BY NL2SNO AND PASSES THEC FORM OF THE FUNCTION TO FIT THE DATA.SUBROUTINE CALCR(N,M,P,NF,R,IPARM,RPARM,FPARM)IMPLICIT REAL *8(A-H 2 O-Z)DIMENSION P(M),R(N)COMMON X(200),Y(200)INTEGER I,JC BEGIN CALCRDO 100 I=1,NR(I)=0DO 99 J=1,M,2IF ((P(J+1)*X(I)).GT.165) THENC CONTROL OVER EXPONENTIAL OVERFLOWR(I)=P(J)*5.0D71+R(I)ELSEC INITIAL CONDITION MAINTAINED EPS=0 AT T=0R(I)=P(J)*(DEXP(P(J+1)*X(I))-1)+R(I)ENDIF99 CONTINUER(I)=R(I)-Y(I)100 CONTINUERETURNENDC END CALR142Appendix VITable VI-1. Viscoelastic coefficients (M, 4, T) 1 ,12) of PbNb2O6 specimens with variousdopants, dopant concentrations, sintering temperatures, and milling procedures.DopantConc.wt %Temp.°Cavg. M avg. p. avg.ii, avg.%milled 1200 0.000 44.9 34.8 0.000milled 1150 0.060 61.4 51.4 0.261milled 1100 0.067 158.6 148.9 0.947milled 1050 0.185 523.8 500.4 3.046as-rec'd 1200 0.000 68.5 60.9 0.000as-rec'd 1150 0.001 165.9 156.6 0.013as-rec'd 1100 0.002 1252.0 1244.7 0.459as-rec'd 1050 4.360 2467.7 2143.7 8.302SrO 1 1200 0.092 37.8 28.2 0.330SrO 1 1150 0.072 356.2 343.5 1.143SrO 1 1100 0.049 1232.3 1187.0 0.888SrO 1 1050 1.220 2376.8 2259.8 7.807SrO 0.5 1200 0.023 81.3 72.3 0.176SrO 0.5 1150 0.139 88.1 76.5 0.957SrO 0.5 1100 0.241 1220.7 1181.0 4.781SrO 0.5 1050 1.767 1757.8 1572.8 12.892SrO 0.25 1200 0.000 58.2 48.9 0.001SrO 0.25 1150 0.000 77.4 67.7 -0.000SrO 0.25 1100 0.726 1567.5 1498.0 8.655SrO 0.25 1050 3.034 2147.5 1826.5 15.620Bi203 1 1200 0.074 46.6 36.4 0.289Bi203 1 1150 0.000 120.2 111.6 0.000Bi203 1 1100 0.000 497.7 488.4 0.009Bi203 1 1050 0.004 1139.0 1073.5 0.052Bi203 0.5 1200 0.043 49.3 39.0 0.194Bi203 0.5 1150 0.056 64.1 54.7 0.331Bi203 0.5 1100 0.005 838.8 830.2 0.750Bi203 0.5 1050 0.006 2716.7 2655.3 0.215Bi203 0.25 1200 0.055 53.7 44.4 0.260Bi203 0.25 1150 0.062 65.4 55.5 0.389Bi203 0.25 1100 0.103 620.5 597.0 1.898Bi203 0.25 1050 0.002 2362.0 2325.5 0.112143DopantConc.wt %Temp.°Cavg. M avg. j.t. avg.% avg.r12LiNbO3 1 1200 0.161 34.1 22.6 0.430LiNbO3 1 1150 0.340 90.6 65.8 1.171LiNbO3 1 1100 0.308 209.6 176.3 1.044LiNbO3 1 1050 0.323 1063.5 1020.1 4.064LiNbO3 0.5 1200 0.070 56.9 45.7 0.289LiNbO3 0.5 1150 0.029 88.4 77.8 0.387LiNbO3 0.5 1100 0.194 137.8 126.1 0.375LiNbO3 0.5 1050 0.005 2768.5 2727.0 0.257LiNbO3 0.25 1200 0.013 56.8 44.2 0.045LiNbO3 0.25 1150 0.135 94.2 83.4 1.036LiNbO3 0.25 1100 0.044 509.5 497.8 1.029LiNbO3 0.25 1050 0.094 878.9 848.4 1.065KNaNb2O6 1 1200 0.037 84.2 74.8 0.348KNaNb2O6 1 1150 1.031 69.5 53.7 1.234KNaNb2O6 1 1100 0.042 1113.5 1097.0 2.379KNaNb2O6 1 1050 0.090 4967.0 4872.5 1.288KNaNb2O6 0.5 1200 0.073 50.4 39.2 0.294KNaNb2O6 0.5 1150 0.074 85.5 75.3 0.508KNaNb2O6 0.5 1100 0.059 951.7 939.4 2.188KNaNb206 0.5 1050 0.659 3249.5 3137.0 9.882KNaNb2O6 0.25 1200 0.000 48.8 38.2 0.001KNaNb2O6 0.25 1150 0.079 81.1 71.2 0.441KNaNb2O6 0.25 1100 0.083 422.6 407.3 1.509KNaNb2O6 0.25 1050 0.122 1457.4 1409.6 2.256144


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