The Pennsylvania State University. The Graduate School. Department of Materials Science and Engineering

Transcription

1 The Pennsylvania State University The Graduate School Department of Materials Science and Engineering GRAIN BOUNDARY SEGREGATION AND BLOCKING EFFECT IN DOPED TITANIUM OXIDE AT HIGH TEMPERATURE EQUILIBRIUM A Dissertation in Materials Science and Engineering by Qinglei Wang 2008 Qinglei Wang Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2008

2 The dissertation of Qinglei Wang was reviewed and approved* by the following: Dr. Elizabeth C. Dickey Professor of Materials Science and Engineering Dissertation Advisor Chair of Committee Dr. Clive Randall Professor of Materials Science and Engineering Dr. Long-qing Chen Professor of Materials Science and Engineering Dr. Theresa Mayer Professor of Electrical Engineering Dr. Joan Redwing Professor of Materials Science and Engineering Chair of the Graduate Program *Signatures are on file in the Graduate School

3 ABSTRACT iii Titanium dioxide was studied as a model material to investigate the grain boundary segregation of yttrium and niobium as a function of dopant concentration. Local grain boundary defect chemistry was quantitatively determined by analytical transmission electron microscopy. Electrical properties associated with both bulk and the grain boundaries were obtained using impedance spectroscopy. Experimental findings were compared to the high-temperature equilibrium thermodynamic space-charge models that incorporated both electrostatic and elastic driving forces for solute segregation. For yttrium, solute excess density at the grain boundaries was found to increase with the dopant concentration until the solid state solubility was reached. The activation energy for electrical conduction increased with dopant concentration, which implied a blocking barrier at the grain boundaries introduced by the solute interfacial excess. Local grain boundary defect chemistry, including core and space-charge layers, was employed to understand the relationship between the specific grain boundary resistivity and solute excess. The thermodynamic models showed that the elastic driving force was dominant in this system and electrostatic driving force only contributed a 10-20% correction to the total solute segregation. In comparison to the experimental results, the models provided good predictions for several grain boundary parameters including local stoichiometry, solute interfacial excess, barrier height and space charge layer thickness.

4 TABLE OF CONTENTS iv LIST OF FIGURES...vii LIST OF TABLES...xi ACKNOWLEDGEMENTS...xii Chapter 1 Introduction Grain Boundary Chemistry and Properties in Ionic Materials Objective and Methodologies...4 Chapter 2 Background Physical Properties of TiO Theoretical Background Surface charge concept and space charge models Lehovec model Kliewer and Koehler model Poeppel and Blakely model McLean model Yan and Cannon model Ikeda and Chiang model Maier model Defect chemistry of TiO Interfaces and Interfacial Segregation Degrees of freedom associated with an interface Point defect segregation: space charge concept and defect formation energy Solute segregation: segregation driving forces Numerical solutions of grain boundary segregation Intrinsic defect concentration and Boltzmann distributions Extrinsic defect concentrations Poisson s equation and numerical solutions Improved space charge model with core segregation Experimental Background Energy dispersive X-ray spectroscopy General knowledge about EDXS Quantitative X-ray microanalysis by Cliff-Lorimer technique Electron energy loss spectroscopy General knowledge about EELS Quantitative microanalysis by EELS Impedance spectroscopy...56

5 General knowledge about IS Electrical equivalent circuits (EEC)...58 v Chapter 3 Experimental Procedures Specimen Preparations Starting materials Solute doping and pellet fabrication TEM sample preparations IS sample preparations Density and Grain Size Measurements Scanning Transmission Electron Microscopy Based Chemical Analysis Energy dispersive X-ray spectroscopy (EDXS) TEM and EDX data acquisition EDX data analysis K-factor determination Electron energy loss spectroscopy (EELS) EELS data collection EELS data analysis Impedance Spectroscopy Equipment and data collection Data fitting and analysis...81 Chapter 4 Results and Discussions Densification and Grain Growth Density profile of Y-doped TiO Grain growth of Y-doped TiO TEM Results Interface characterization EDXS results EDS spectra and profiles EDS quantifications EELS results EEL spectra and line profile of Y-doped TiO EEL spectra from TiO 2, Ti 2 O 3 and TiO Impedance Results Rutile TiO 2 single-crystal and pure polycrystalline TiO Acceptor (Y)-doped polycrystalline TiO Activation energy of electronic conduction Barrier height and space charge model Ionic conduction in TiO Simulation Results Pure and stoichiometric TiO

6 4.4.2 Nb-doped TiO 2 (pure electrostatic driving force) Y-doped TiO 2 (electrostatic and elastic driving forces) Improved space charge models with core segregation Questions raised from current models Initial attempt to establish the space charge model with core segregation Outline for Future Work Complete the impedance measurement of bias study Further investigation on Y+Nb co-doped TiO 2 system Further improvements on the core segregation model Chapter 5 Conclusions Bibliography Appendix A Matlab Code for Theoretical Calculations A.1 Frenkel Defect Calculation A.2 Schottky Defect Calculation A.3 Reduced Pure TiO 2 Calculation A.4 Nb Doped TiO 2 Calculation A.5 Y Doped TiO 2 calculation (electrostatic driving force only) A.6 Y Doped TiO 2 Calculation (electrostatic and elastic driving forces) A.7 Y+Nb Co-doped TiO 2 Calculation (electrostatic + elastic driving forces) Appendix B Glossary vi

7 LIST OF FIGURES vii Figure 1-1: Protocols for investigating segregation behaviors in titanium oxide...6 Figure 2-1: Crystal structure of rutile TiO Figure 2-2: Energy diagram of formation energies as referred to the lattice binding energy [55]...14 Figure 2-3: Equilibrium potential diagram assuming that the standard chemical potential is a step function across the core and space charge region...22 Figure 2-4: Sketch of the Kroger-Vink diagram of equilibrium electrical conductivity of updoped TiO 2 single crystal as a function of oxygen partial pressure (T 1 <T 2 <T 3 ) [59]...26 Figure 2-5: (a) Two parts of a crystal with the same coordinate system defining three DOFs: rotation axis (n1, n2) and rotation angle θ (n3). (b) The interface between the two parts of the crystal defined by another two degrees of freedom (n4, n5) Figure 2-6: Typical space charge distribution Figure 2-7: Illustration of EDXS and EELS...45 Figure 2-8: Illustration of integration window and peak intensity Figure 2-9: (a) Brick-layer model for polycrystalline materials where d is the average grain size and δ is the GB width. (b) Electrical equivalent circuit (EEC) with RC branches in series. (c) The spectra corresponding to the EEC shown in (b) Figure 3-1: X-ray diffraction spectrum of TiO 2 starting powder where A presents anatase phase peak and R presents rutile phase peak Figure 3-2: XRD scan of a sintered pellet displaying a single phase of rutile...62 Figure 3-3: A sample PIXIE analysis result of 243ppm Y-doped TiO 2 powder Figure 3-4: Schematics of EDX data collection regions (not to scale) [23] Figure 3-5: Detector input parameters for the DTSA program...75

8 Figure 3-6: The EDXS sensitivity factor, k YTi, measured from an Y 2 Ti 2 O 7 standard, was found to vary linearly with average EDXS count rate which is proportional to the specimen thickness...77 Figure 4-1: Density profile for sintered pellets of pure and 0.025, 0.1 and 0.3 mol% yttrium doped specimens...86 Figure 4-2: Microstructure of 10-hour-sintered pellets shows normal grain geometry and low porosity Figure 4-3: Grain size for sintered pellets of pure and 0.025, 0.1 and 0.3 mol% yttrium doped specimens Figure 4-4: High-resolution TEM image shows clean grain boundary without glassy phase in yttrium-doped TiO Figure 4-5: Z-Contrast image of yttrium-doped TiO 2 shows solute segregation at the grain boundary Figure 4-6: Typical dark-field STEM image of grain and boundary area with rectangular scanning regions (not to scale) identified for EDXS data collection...92 Figure 4-7: Typical EDXS spectra from both grain boundary (grey like) and interior (black like) show yttrium segregation at the boundary while no yttrium was detected in the bulk...93 Figure 4-8: Typical EDX profile across grain boundary in 0.1 mol% yttriumdoped TiO Figure 4-9: Typical EDX profile across grain boundary in 1.0 mol% niobiumdoped TiO Figure 4-10: EEL spectra indicate that the majority of Ti ions remain in the 4+ oxidation state at the grain boundary Figure 4-11: Typical EELS profile across Y-doped TiO 2 grain boundary shows no spatial change in the Ti:O atomic ratio Figure 4-12: Typical EEL spectra obtained from crystalline TiO 2 and amorphous Ti 2 O 3 and TiO, with inset of TEM image showing beam damage effect Figure 4-13: Typical EEL spectra obtained from crystalline TiO 2, Ti 2 O 3 and TiO nanoparticles Figure 4-14: Single-crystal rutile TiO 2 with Cr/Pt electrode at 133 C in air viii

9 Figure 4-15: Polycrystalline pure TiO 2 with Cr/Pt electrode measured within 499~642 C in air Figure 4-16: mol% yttrium-doped polycrystalline TiO 2 with Cr/Pt electrode measured within 450~736 C in air. Inset shows the magnified spectra from 546~736 C Figure 4-17: 0.1 mol% yttrium-doped polycrystalline TiO 2 with Cr/Pt electrode measured within 450~736 C in air. Inset shows the magnified spectra from 642~736 C Figure 4-18: Bulk permittivity calculated for undoped, and 0.1 mol% yttrium doped polycrystalline TiO 2 within the experimental temperature range. Bulk permittivity along c- and a-axis of polycrystalline rutile from literature, shown in dotted curves, are overlapped in the same graph for comparison Figure 4-19: Impedance spectra collected from a 0.025% sample held at 642 C for different time Figure 4-20: Bulk and grain boundary resistivity of pure, and 0.1 mol% yttrium-doped TiO 2 shows a difference in the activation energy between bulk and GB Figure 4-21: Calculated barrier height versus temperature for and 0.1 mol% yttrium-doped TiO Figure 4-22: (a) Electrostatic potential and (b) defect concentrations for pure and stoichiometric TiO 2 at 1300ºC in air Figure 4-23: (a) Electrostatic potential and (b) defect concentrations for Nb-doped TiO 2 at 1300ºC in air Figure 4-24: Comparison of experimental and simulation results at two values of defect formation energy for titanium vacancies in Nb-doped TiO 2 at 1300ºC in air Figure 4-25: (a) Electrostatic potential and (b) defect concentrations for 0.1 mol% Y-doped TiO 2 at 1300ºC in air Figure 4-26: Grain boundary segregation as a function of dopant concentrations at 1300ºC in the air. The vertical dotted line indicates the onset of Y 2 Ti 2 O 7 precipitation Figure 4-27: Space charge model incorporated core region [18] ix

10 Figure 4-28: Improved model with core segregation: (a) Electrostatic potential and (b) defect concentrations for 0.1 mol% Y-doped TiO 2 at 1300ºC in air x

11 LIST OF TABLES xi Table 2-1: Thermal and physical properties of polycrystalline TiO 2 [50]...8 Table 2-2: Mechanical physical properties of polycrystalline rutile at room temperature [50]...9 Table 3-1: Starting material details and dopant concentrations...67 Table 3-2: Geometries of specimens used for impedance measurements Table 4-1: Average quantified yttrium interfacial excess density Table 4-2: Quantified solute excess density from all the grain boundaries analyzed Table 4-3: Comparison of suppression semi-angle, activation energy of conduction and GB solute excess density at different doping concentrations Table 4-4: Comparison of the grain boundary potential and solute excess density between simulation and experimental results

12 ACKNOWLEDGEMENTS xii I would like to give the most cordial thanks to my beloved wife, Hongqi Deng, for her constant support throughout the years of this Ph.D. study. I also want to thank my advisor, Prof. Elizabeth Dickey, and all my committee members for their great guidance and enormous patience during this thesis work. The generous help from Dr. Oomman Varghese and Dr. Gaiying Yang on the analysis of impedance experiments is highly appreciated as well. This work was supported by the Division of Materials Research at the National Science Foundation under grant No. DMR and DMR

13 1 Chapter 1 Introduction 1.1 Grain Boundary Chemistry and Properties in Ionic Materials Point defects and impurities are known to influence various physical properties of materials including electrical conductivity [1], mechanical property [2] and diffusional behavior [3]. Quite often, it is not only the effects that dopants have on the bulk crystalline lattice, but the effects that they have on grain boundary properties that lead to enhanced properties. For example, rare earth elements such as Y and Zr added to polycrystalline Al 2 O 3 at the thousand ppm level significantly decrease grain boundary diffusivity, thus improving the creep resistance of the oxide [4]. Addition of divalent cations (Ba, Sr or Ca) to TiO 2 considerably increases the grain boundary resistivity of the material and allows for varistor behavior. In fact, properties of many electroceramics strongly rely on the grain boundary or interfacial properties while some solid-state devices derive their unique functionalities directly from properties of the interfaces [5]. The most conceptually simple driving force for grain boundary segregation is that due to strain energy minimization when dopant ions are larger than those in the matrix. Grain boundaries can often accommodate larger dopant ions much more easily because of their more open structures. A spontaneous relaxation process takes place to minimize the system free energy when the large ions move to the grain boundary regions where the strain energy can be released to the greatest extent. At equilibrium, a net solute excess at

14 2 the grain boundaries is often observed at the grain boundary regions when the solutes are oversized. In ionic materials, there is another significant driving force for segregation called space-charge segregation which can even exist intrinsically in pure polycrystalline materials due to the interaction between the grain boundaries and intrinsic Frenkel or Schottky point defects. Because the energies required to produce various intrinsic defects are not identical, some have greater tendency to stay in the boundary region than others. The discrepancy in defect concentrations results in a net charge in the grain boundary core and compensation space-charge regions on both sides of the grain boundaries. When aliovalent dopant (accepter of donor) ions are placed into a lattice, they interact with the local grain boundary electrostatic field and can be either accumulated or depleted in the space charge region. The segregation phenomenon has been intensively studied from theoretical aspects since Frenkel [6] first proposed a surface charge theory. Notable advancements have been made by Lehovec [7] and Kliewer and Koehler [8] in the theory of space charge segregation to surface and grain boundary, which was later explored experimentally. McLean developed an isotherm model [9] that derived the solute concentration at grain boundaries based on the elastic strain energy. His theory was later applied to doped Al 2 O 3 system [10-12] to predict the segregation level. Using analytical electron microscopy (AEM) techniques, Brown et al. [13] explored the surface segregation of YAG, and Ikeda and Chiang [14] investigated the GB segregation of TiO 2, which were both explained theoretically with Kliewer s space charge model. Yan [15] first developed a theoretical model for segregation when multiple driving forces were

15 3 present. Some of his concepts were afterwards used by Terwilliger [16] and Wang et al. [17] to predict the solute segregation in Ca and Y doped TiO 2 systems. However, this model was over-simplified by ignoring the solute that segregates to the grain boundary core. Recently, more complex models containing specific grain boundary core region proposed by Maier provided better understanding of the core segregation as different values of chemical potential for the core and bulk region were employed [18-21]. The formation of space-charge regions by the segregation of charged solutes and point defects to the grain boundaries is a major characteristic differentiating interfaces in ionic materials from those in metals where no band bending exists. Depending on the solute type and concentration, the potential difference between the grain boundaries and the bulk can be positive or negative, resulting in conducting or blocking boundaries respectively. In some cases, blocking double-schottky type barriers [22-24] can exist at the GBs, characterized by energy barrier height that is usually no more than a few ev and depletion width w d on the order of a few tens of nanometers. The concentrations and distributions of point defects and dopants added to the materials not only influence the bulk electric/dielectric properties but also significantly modify the barrier height at the inter-granular regions by altering the grain boundary core charge. The interfacial accumulation or depletion of the charged carriers depends on the sign of core charge and consequently changes the electrical properties and defect chemistry associated with the grain boundaries. Thus electric measurement can be an effective way to study the defect chemistry and segregation phenomenon and has been applied to several popular oxide systems including ZrO 2 [22-25], CeO 2 [26-28], SrTiO 3 [29-33] and BaTiO 3 [34-41].

16 4 The ultimate goal of studying grain boundary segregation phenomenon is to correlate the grain boundary defect chemistry with the electrical properties of electroceramics from a thermodynamic point of view and through a systematic research composed of experimental approaches and theoretical modeling. It is aimed at providing fundamental understandings of grain boundary defect chemistry and electrical effects in ceramic materials. 1.2 Objective and Methodologies This thesis serves as an essential part of a much larger agenda to understand the segregation and defect formation in metal oxides. Titanium dioxide is used as a model material because 1) it is a relatively simple binary system which could be modeled by the Density Function Theory; 2) it has relatively well understood defect chemistry and been used as a model system [14, 16, 17, 42, 43], which could potentially provide general information that could be applicable to abroad range of oxides; 3) prior work on segregation has been carried out with only one driving force considered. Fig. 1-1 outlines the protocols for understanding the segregation behavior in titanium oxide. Three are three major parts of this study: microstructure analysis, electrical property characterization and theoretical modeling, all based on relevant material information. Solute segregation and associated information such as grain

17 5 boundary stoichiometry will be experimentally determined by quantitative microanalysis, while the effect of segregation on electrical properties such as grain boundary resistance and capacitance are measured by means of electrical measurements. The theoretic models provide predictions for both solute excess and electrical properties based on defect chemistry, thermodynamics and quantum chemistry. The organization of the thesis is designed to first provide background material in Chapter 2 which establishes the context of this thesis research. This chapter will describe the space charge segregation based on defect chemistry. The background material will also cover the basic theory of the techniques used in this study. Chapter 3 will outline the details of experimental procedures and theoretical approaches. Results will be presented in Chapter 4 along with a discussion on the significance and implications of the findings. Chapter 5 will summarize the major conclusions of this thesis research.

18 6 Relevant Material Information Crystallographic information Phase information Physical and mechanical properties Defect chemistry Dielectrical information Microstuctural Analysis Solutes and point defect segregation at grain boundary Grain boundary chemistry and stoichiometry Grain boundary structure Grain Boundary Segregation Electrical Measurements Dielectric behavior Resistance and capacitance Grain boundary blocking effect Theoretic Predictions Defect formation Thermodynamics and segregation driving forces Space charge model Space charge model with elastic interaction Core segregation Figure 1-1: Protocols for investigating segregation behaviors in titanium oxide.

19 Chapter 2 7 Background 2.1 Physical Properties of TiO 2 Titanium dioxide (TiO 2 ) or titania is a ceramic material of technological importance that has been extensively used in various applications. It has three polymorphs: anatase and brookite stable at low temperatures (less than around 1000 C) and rutile stable at all temperatures. Anatase and brookite phases irreversibly transform to rutile at temperatures from 400 to 1000 C, varying with the amount of impurities in the material. The anatase phase demonstrates advantages in applications such as membranes in photovoltaic cells [44] or photocatalysts for photodecomposition because of its high photoactivity. Thermal and physical properties of these three phases of TiO 2 are summarized in Table 2-1. The rutile phase is well known for its extensive usage as white pigment materials because its superior scattering properties provide protection against ultraviolet light [45]. Important mechanical properties of rutile phase are summarized in Table 2-2. Titanium oxide is a dielectric material at room temperature. Because of the structural anisotropy, the low-frequency dielectric constant of rutile phase varies along different crystallographic axis, which is 86 parallel to the c-axis and 170 parallel to the a-axis [46]. The average dielectric constant of polycrystalline rutile is at room temperature with a small negative temperature coefficient of the dielectric constant ( ppm/ C). The average dielectric constant of anatase and brookite is about 80 and 40 respectively while that of the amorphous thin film TiO 2 is about 30. Although showing

20 8 very low electrical conductivity (~10-7 Ω cm at 500 C) at high-purity level due to the large intrinsic band gap (~3.5 ev at room temperature), rutile presents n-type semiconducting behaviors when reduced in low oxygen partial pressure at high temperatures or doped with pentavalent cation (D 5+ ). Since the valence band of TiO 2 is made of filled O 2d states, which are not hospitable to holes, p-type semiconducting behavior is seldom observed in trivalent cation (A 3+ ) doped titania. On the contrary, lightly acceptor-doped TiO 2 may present n-type electrical conduction in low oxygen partial pressure at high temperatures because of the electrons generated in the reducing atmosphere. Because of its interesting electrical properties, TiO 2 has gained applications as high-temperature oxygen sensors [47] and varistors [48, 49]. Table 2-1: Thermal and physical properties of polycrystalline TiO 2 [50]. Property Rutile Anatase Brookite Density (g/cm 3 ) Melting point ( C) 1855 Transformed to rutile Coefficient of thermal expansion (10-6 / C) Reflectance, % At 400 C At 500 C Absorption of UV, %, at 360 um 90 67

21 Table 2-2: Mechanical physical properties of polycrystalline rutile at room temperature [50]. Young s modulus (GPa) 282 Shear modulus (GPa) 111 Bulk modulus (Gpa) 206 Poisson s ratio TiO 2 is a chemically stable material, even under strong acidic or basic environments. As the cation (Ti 4+ ) is already at its highest valence state, TiO 2 cannot be further oxidized. However, when annealed in a reducing atmosphere or in contact with metals at high temperatures, TiO 2 can be easily reduced to lower valence statuses, Ti 2 O 3, TiO, or magneli phases Ti n O 2n-1 (n = 4-10) [51]. Rutile TiO 2 has a tetragonal crystallographic structure as shown in Fig. 2-1 with point group P4 2 /mnm, in which a titanium cation is octahedrally coordinated by six nearby oxygen anions. It has a theoretical density of 4.25 g/cm 2 and lattice parameters of nm along the a-axis and nm along the c-axis [46]. This Ph.D. study uses the rutile phase, the only high-temperature stable phase, of TiO 2 as a model electromaterial because of its relatively simple structure and well understood defect chemistry.

22 10 Figure 2-1: Crystal structure of rutile TiO Theoretical Background Surface charge concept and space charge models Surface charge is a unique concept in ionic crystals. The energy that is necessary to form different lattice defect species is usually not identical. This energy discrepancy results in different amount of charged point defects on the surface and leads to a net surface charge which is compensated by the nearby space-charge layers. Since Frenkel [6] first proposed a surface charge theory, there have been significant advances in theoretical space charge models on the distributions of point defects near the surfaces or interfaces in ionic crystals.

23 Lehovec model 11 Lehovec [7] first explored the electric potential existing between the surface and the bulk region of an ionic crystal. He suggested that the difference in the concentrations of point defects between the surface and the bulk should lead to a surface conduction. He formulated the general numerical approach to calculate the magnitude of the space charge potential and the thickness of the space charge layer near the surface. Based on the values of heat of reaction for NaCl calculated by Mott and Littleton [52], he established the relationship, Eqs. 2.1 and 2.2, between the intrinsic point defect concentrations and the electrical potential, [( h ± qφ) kt ] N F, ± = 2 Z exp F, ± 2.1 [( h m qφ) kt ] N S, ± = 2 Z exp S, ± 2.2 where N i, ± was the concentrations of cation (+) and anion (-) Frenkel (F) and Schottky (S) defects, h was defined as the heat of reaction without the electrical term, and φ was the electrical potential whose zero point was defined at the surface. The magnitude of the space charge was expressed in terms of N i and the Poisson s equation was analytically solved. This method was applied to the NaCl crystal and obtained a V potential difference between surface and bulk for Schottky defects. He also discussed the case when a sufficiently high impurity concentration D existed in the bulk. The space charge density near the surface was approximated by ρ = De thus the potential between the surface and bulk was a function of the impurity concentration expressed as Eq. 2.3.

24 ( D Z ) φ q kt q ln 2.3 = h S, Lehovec s model derived the intrinsic defect concentration and analytically solved the electric potential as a function of distance to the surface. He applied the method to NaCl system and discussed the impact of impurities on the space charge potential to the first order. The accuracy of his model relies on the h values of the intrinsic defects Kliewer and Koehler model Kliewer and Koehler proposed the first comprehensive space-charge model [8], which was applied to ionic crystals with Schottky type disorder M + X. In this model, the equilibrium Helmholtz free energy F was related to the internal energy U I, temperature T and the entropy S by Eq F = U TS 2.4 I The free energy per unit area of half of the disordered crystal could be expressed by Eq. 2.5 F = L dx n + + ( x) g + n ( x) g + n ( x) ( g + g B) + ( x) Φ( x) 1 ρ B TS c 0 2 where n + ( x), n ( x) and ( x) n + were the densities of cation vacancies, anion vacancies and bounded vacancy pairs at x, and S c was the configurational entropy. + g and g were the formation energies of the cation and anion vacancies that included the change in vibrational entropy of the crystal because of the presence of the defects. B was the

25 13 binding energy of the defects. They assumed that the energy terms were independent of distance from the surface but dependent on the temperature. The actual values they used in their model were obtained from previous experimental work by Dreyfus and Nowick [53] and theoretical calculations by Fumi and Tosi [54]. They also gave the formulas to calculate the space-charge profile in the crystal containing divalent cationic impurities, taking into account the impurity-vacancy association. The concentrations of intrinsic Schottky defects, unassociated impurities and associated impurity-vacancy pairs were expressed in terms of the formation energies, electrical potential and binding energy, which only existed at middle-low temperature range Poeppel and Blakely model After Lehovec and Kliewer, Poeppel and Blakely explored the origin of the equilibrium space charge potentials in ionic crystals. Different from previous methods, this model explicitly considered the binding states of ions on surfaces and the surface density of such states. They argued that the individual defect formation energies involved in creating two components of a defect pair were not a unique property of the bulk crystal but depended on the properties of the source. Their model was based on the existence of a finite number of ionic surface sites whose binding energy was different from that of the bulk sites. In their model, the energy to remove an atom from the crystal to vacuum level (infinite) and create a vacancy was defined as EV in Figure 2-2. E L was the binding energy per atom of the perfect crystal, E I was the energy necessary to introduce an atom

26 14 from vacuum level and form and interstitial defect, g V was the average free energy change of the crystal on forming a vacancy and replacing the displaced atom on the surface, and g I was the corresponding free energy change on forming an interstitial. 0 E I G F g I g V E µ L atoms E V Surface Crystal Interior Figure 2-2: Energy diagram of formation energies as referred to the lattice binding energy [55] The total system free energy G in the defect state was expressed by the defect concentration of vacancies (n v ) and interstitials (n i ), the corresponding formation energies g V, g I, and the configurational entropy terms. Thus the chemical potential of the atoms was obtained as Eq. 2.6 G g = V g I µ atoms = EL kt exp αkt exp 2.6 N kt kt T, P, nv, ni where α was the number of interstitial sites per lattice site. In this model, the chemical potential, which was a constant between surface and the crystal interior, was a function of the binding energy and individual formation energies. Although the sum of

27 15 g V + g = G was available, the magnitude of the individual formation energies I F depended on the level of the binding energy. Following the similar mathematic procedure as Kliewer and Koehler, the bulk potential was calculated by Eq. 2.7, choosing the surface under consideration (source) as the zero point of potential. 1 φ ( ) = ( gv g I + kt ln 2) 2.7 2e McLean model McLean established a model for the solute segregation to the grain boundaries in metals [9]. He suggested that the solute concentration at the grain boundaries, related to the bulk concentration, n b, by Eq. 2.8 n gb, was n gb n = 1 n b b exp + n b ( Fa kt ) exp( F kt ) a 2.8 where F a was the free energy of absorption. Because his model was based on metals, where varying electrical potentials did not exist, the solute concentration at the grain boundaries depended on the bulk doping level and the ionic misfit. He suggested that when there was a size mismatch between the solute and matrix cations, solute segregating to the inherently open grain boundary regions provided a partial relaxation of the elastic strain energy due to the size mismatch. This elastic strain energy provided the root of the free energy of solute absorption at the grain boundaries. Further more, he argued that the absorption energy had a spatial

28 16 dependence because the lattice distortion was a function of the distance from the grain boundaries. The detail of this spatial dependence was related to the structure of grain boundaries Yan and Cannon model Based on the space-charge model of Kliewer&Koehler and the elastic model of McLean, Yan and Cannon proposed a model that combined the space charge, elastic field and dipole interaction [15] on the M + X type solid state crystal systems. They categorized the interactions between solute ions and grain boundaries in ionic crystals into (1) the electrostatic interaction between the charged solutes and grain boundaries, (2) the elastic energy due to the size misfit of solutes in the matrix, and (3) the dipole interaction between the solute-vacancy dipoles and the electrical field in the grain boundary region. Their numerical calculations showed that these interactions, either acting individually or coupled with each other, led to a nonuniform solute distribution near the grain boundary. They showed that the both elastic and dipole interactions could significantly modify the electrostatic potential near the boundary. Unlike the Kliewer&Koehler model (Eq. 2.5), where the elastic interaction was ignored, the total free energy in Yan s model was expressed in terms of the defect + formation energies of cation ( g ) and anion vacancies ( g ); elastic interaction energies between the grain boundary and unassociated vacancies ( U and + s U s ), vacancy complexes ( U ), solute-vacancy complexes ( U ) and unassociated solutes ( U ); B s b s f s

29 binding energy of vacancy-interstitial pair ( B + ) and vacancy-vacancy pair (B); electrostatic interaction energy ρ ( x) φ( x) ; grain boundary surface energy σφ ( 0) configurational entropy term S c as in Eq ; and the F = i= 1 6 j= 1 n m 0 j i L dx{ n + + ( g + U ) + n ( g + U ) + b ( x)[ g + B p E / 2 + U ] + B ( x)[ g + g + B q E / 2 + U ] ρ c ( x) φ( x) } TS σφ( 0) + s i i s s s + n U f f s 2.9 In their model, the defect formation energies + g and g were defined as the excess free energies to form a vacancy in the bulk by moving the appropriate ion from the lattice to the boundary core. The elastic interaction energy U s terms were the excess free energy changes resulting from moving a defect from the bulk to a position near the boundary. All the energy terms utilized the grain boundary as the reference point. After the free energy minimization [15], concentrations of the major defect species were expressed by Eqs. 2.10, 2.11 and [ ( g + U e ) kt ] n+ N = exp s φ [ ( g + U + e ) kt ] n N = exp s φ 2.11 n f N + ( p ) C exp[ ( eφ e U ) kt ] = 1 φ s

30 18 where p was the probability of association between the interstitial-vacancy complex in the bulk, C was the total site fraction of interstitial in the bulk, and N was the number of cation or anion sites per unit volume. In the numerical calculations, they assumed that the elastic energy terms of the charged vacancies did not have any spatial dependence, namely U + ( x) = U ( x ) = 0 s s. Furthermore, they suggested that the elastic term of the unassociated solutes should be a function, Eq. 2.13, of distance from the boundary U f s = n x 2a 0 x U 1 a x > 2a where U 0 was the maximum value of the elastic interaction energy, n was an index that affected the details of the elastic term and a was the lattice parameter. The values of U 0 depended on the size misfit of solutes in the lattice and could be a negative value. When U < 0 0, the elastic interaction tended to attract solutes to the grain boundary which counteracted the electrostatic effect when U < eφ 0. When U << eφ 0, solutes 0 < 0 < became significantly segregated at the grain boundary because of the elastic interaction. When U 0 0 > > eφ, the elastic interaction repelled solutes from the grain boundary region and decreased the electrostatic potential. Yan s model was the first model that incorporated electrostatic, elastic and dipole interactions on the M + X type solid state system with aliovalent impurities. It predicted

31 that the grain boundary segregation behavior would change according to the relative magnitude of the elastic and electrostatic energy terms Ikeda and Chiang model In the study of lattice defect chemistry and space charge potential in TiO 2, Ikeda and Chiang established a pure space charge model [42] based on Kliewer&Koehler s work and predicted the grain boundary segregation of solutes and point defects. Similar to the method described in chapter , the concentrations of point defects were derived using the individual defect formation energy and the electrostatic interaction terms. In their definition, the defect formation energies were not the energy to remove ions to infinity (vacuum level) that were often reported in the defect energy calculations from which the crystal cohesive energy must be subtracted to obtain the formation energy in question. Similar to Yan s definition, the formation energy defined in their model was referenced to individual free surfaces or grain boundaries, which may have distinct values for each particular surface and grain boundary that may vary as much as 1 ev between grain boundaries in a single polycrystalline sample. They started from the pure and stoichiometric crystal and derived the concentration of titanium vacancy, titanium interstitial and oxygen vacancy. The space charge potential, defined as the potential difference between the bulk and boundary, was only related to the individual formation energies of the intrinsic defects. The model then moved to reduced TiO 2, where the compensation mechanism between electrons, titanium interstitial and oxygen vacancy was discussed. An accumulation of electrons to the grain

32 20 boundary was found from the model. In the single donor-doped and the acceptor-doped limiting cases, the spatial distributions of the impurities were calculated to be a function of the bulk doping level and the electrical potential at the grain boundary. Because their model was a pure space-charge model at high temperatures, the elastic and dipole interaction terms discussed in Yan s model were negligible. This model predicted that segregation and grain boundary potential only depended on the net bulk dopant concentration ( D A ) given constant individual defect formation energies and the temperature. At a specific temperature there existed an isoelectric point where the boundary potential equaled to zero. With assumption of the individual defect formation energies, they found this isoelectric point varied with temperature at specific oxygen partial pressure and slightly lied at the donor-doped composition due to the defects from reduction Maier model All previous models only took into account the charge density in the space charge region while treating the grain boundary core as an infinite charge source that provided balance to the net charge in the space charge region. They assumed that the standard chemical potential in the grain boundary core region was equal to that in the bulk. However, it may not be completely true considering that the boundary is a more disordered region comparing to the lattice. Maier proposed a new discrete model that treated the core as a thin region whose standard chemical potential differed from that of the bulk region.

33 The electrochemical potential obtained from chemical potential and electrostatic potential was expressed by Eq [56] where ~ 0 µ j was the electrochemical potential, µ j was the chemical potential, µ j was the standard chemical potential, ~ n n 0 j j µ j = µ j + z j Fφ = µ j + RT ln + z j Fφ = ~ µ 0 j + RT ln 2.14 n n 21 ~0 µ j was the standard electrochemical potential, z j was the charge number, F was the Faraday constant, φ was the electrical potential, number of defects j, and n was the mole number of the available sites. n j was mole Assuming that in equilibrium the electrochemical potential was a constant between the boundary core and the grain interior of the crystal. Since the structure of the core region was significantly different from that of the bulk, Maier s model assume that the standard chemical potential 0 µ j was a step function whose value changed discretely from the core to the bulk. The potential diagram in the core, space charge and the bulk regions was obtained as shown Figure 2-3 [56].

34 22 0 µ j ~0 µ j ~ µ j z j Fφ -s 0 L Figure 2-3: Equilibrium potential diagram assuming that the standard chemical potential is a step function across the core and space charge region. He further argued that the standard chemical potential has a smaller value in the bulk, namely 0 0 j, c = β µ j, µ where β < 1. For example, the β value to move a point defect from a free surface to a kink corner was 2/3. And the β value was expected to be even lower for grain boundaries [57]. In his model, the defect distribution was then divided into two discrete regions: from near the boundary to the bulk ( x 0 ) and in the core ( s < x < 0 region, the profile may be presented as Eq [18] n [ ] ( x T, P) f x; n ( T, P), n ( T P) j ; j0 j, ). In the first = 2.15

35 where the concentration of defect j depended on the distance x from the boundary, concentrations at the boundary n j0 and the bulk n j at specific T and P. The core defect concentration in equilibrium led to Eq a n 0 0 µ jc µ j = exp exp RT ( φ φ ) jc s 0 j0 RT F 2.16 where a jc was the activity of the core defect, which was related to the difference of standard chemical potential and the electrical potential between the core and bulk region Defect chemistry of TiO 2 The defect chemistry of TiO 2 has been extensively studied for decades by means of electrical conductivity measurements at various oxygen partial pressures and temperatures [58]. Denoted in the Kroger-Vink notation, the major intrinsic defect reactions for stoichiometric bulk titania are associated with cation Frenkel, Schottky and electron-hole defects given by Eq. 2.17, Eq and Eq [51], noting that anion Frenkel defect is highly unfavored because large oxygen interstitials are not welcome in such a close packed structure. Ti Ti + V 2.17 Ti + V I... i,,,, Ti TiO2 null V + 2V 2.18,,,, Ti.. O null +,. e h 2.19

36 In reducing atmospheres both oxygen vacancies and titania interstitials are created, which are compensated by free electrons, as shown in Eq and Eq [51]. 24 O O O2 + VO + 2 e, 2.20 Ti... Ti + OO + VI O2 + Tii e, 2.21 Similarly the oxidizing conditions produce the titanium vacancies that are compensated by holes (Eq. 2.22). As discussed previously, the valence band is composed of filled O 2d states which are not hospitable to holes. Thus Eq is not a favored process. O TiO,,,, O V 4 h O + Ti Since electrons or holes are generated in these intrinsic defect reactions at elevated temperatures, the electrical conductivity of rutile is a function of temperature and oxygen partial pressure. The equilibrium defect concentrations are determined by the bulk electron neutrality condition (ENC), Eq :.....,,,, [ Ti ] + 2[ V ] + p = 4[ V ] n i O Ti + which may be simplified to Eq in reducing atmosphere where titanium vacancies and holes are not favored and at much lower concentrations than the other defect species per Eqs. 2.20, 2.21 and [ Ti ] + 2[ V ] n i O =

37 Eq indicates that at high-temperature reducing atmospheres, undoped titanium dioxide shows n-type semiconducting behavior. The electrons generated by the reduction of Ti 4+ are compensated by ionic point defects titanium interstitials and oxygen vacancies. There have been debates for a long time on which one of them is the dominant intrinsic ionic defect in such conditions. Some believe that titanium interstitials are preferred over oxygen vacancies in slightly and highly reduced rutile as evidenced by conductivity measurement at varying P O [23, 58], while others have found that oxygen 2 vacancies are dominant at lower temperatures ( 1000ºC) [10, 11].... [ ] n..... Assuming that Ti interstitial is dominant, i.e. [ ] [ ] i V O 25 Ti >>, Eq reduces to 4 Ti i. If the reaction constant of Eq is K R2, the electrical conductivity σ can be derived as a function of oxygen partial pressure [51] (Eq. 2.25), K R2 σ eµ 2.25 = P O2 where e is the charge and µ is the mobility of electrons... Similarly, if the oxygen vacancies are governing, Eq reduces to [ ] n We can get Eq. 2.26, 2. V O where K R1 is the reaction constant for Eq Comparing Eq and Eq. 2.26, it is easy to see change of the logarithm values of σ is related to the oxygen partial pressure by different multiplication factors. If the K R1 σ = eµ PO 2

38 values of σ are plotted as a function of..... region will be -1/5 for [ Ti ] and -1/6 for [ V ] i P O 2 on a logarithm scale, the slope at low O 26 P O 2. From the precise conductivity study of undoped TiO 2 single crystal by Baumar et al. [59], a slope of -1/5 was observed in the range of atm and K, as sketched in Fig Our typical test conditions, T = 1573 K and P O 2 = 0.2 atm, falling into the region where the slope is -1/5, lead to conclusion that titanium interstitial is the dominant ionic defect. Logσ (Ω -1 cm -1 ) Slope = -1/5 T 3 > 1573K T 2 = 1573K T 1 < 1573K 1 P O2 (atm) Figure 2-4: Sketch of the Kroger-Vink diagram of equilibrium electrical conductivity of updoped TiO 2 single crystal as a function of oxygen partial pressure (T 1 <T 2 <T 3 ) [59].... Although the dominant ionic defect is [ ] and low-medium O 2 Ti at temperatures higher than 1000 C... P range, trivalently ionized titanium interstitial [ ] i Ti may exist at regimes of lower temperature (<1000 C) or very low oxygen partial pressure (<10-10 atm) [51], which is associated with the nonstoichiometry of the titania at those conditions. i

39 Trivalent acceptor cations (A 3+ ) with similar sizes to the titanium ion will be incorporated substitutionally on the titanium lattice sites and form the negatively charged A, Ti defects that are compensated by either titanium interstitials or oxygen 27 vacancies. Once again, holes are not favored in this system at low-medium P O 2 conditions thus its concentration is usually negligible. The defect reactions, Eq and Eq. 2.28, are [51], 2 2A O TiO, Ti Ti + Vi 4ATi + Tii + OO 2TiO2,.. A O 2A + V + 3O Ti O O Pentavalent donor cations (D 5+ ) that are incorporated substitutionally may be compensated by either titanium vacancies or electrons, noting that oxygen interstitial is highly unstable in the system. The donor-doped limiting cases of purely ionic or purely electronic compensation are given by Eq and Eq [51], 5TiO2.,,,, 2D O 4D + V + 10O Ti Ti O 2TiO 1 2., D2O5 2DTi + 2e + 4OO + O2 ( g)

40 Interfaces and Interfacial Segregation Degrees of freedom associated with an interface Sufficient understanding of interface or grain boundary structures is essential before we start to investigate segregation behavior at the interfaces or GBs. From a thermodynamic point of view, any interface can be described in terms of state variables. To describe the geometry of an individual interface between two crystals, we need eight geometrical parameters, which are usually divided into five macroscopic degrees of freedom (DOF) and three microscopic degrees of freedom. The five macroscopic degrees of freedom are generally regarded as the minimum requirement to determine a specific crystallographic orientation relationship between the two parts of a bicrystal [60]. Defining the same crystallographic coordinate system or reference, the orientation relationship can be thought of being formed by rotating the top part along with an axis characterized by (n1, n2) by a specific angle θ, as shown in Fig. 2-5 (a) and (b). The interface plane is determined by another two degrees of freedom (n4, n5) that complete the five macroscopic degrees of freedom.

41 29 (a) (b) z x z θ z (n1,n2) y x (n4,n5) z y x y x y Figure 2-5: (a) Two parts of a crystal with the same coordinate system defining three DOFs: rotation axis (n1, n2) and rotation angle θ (n3). (b) The interface between the two parts of the crystal defined by another two degrees of freedom (n4, n5). The microscopic degrees of freedom are a translational operation T= (T x, T y, T z ) along with x, y and z directions respectively. These DOFs are determined by relaxation processes at the interface [60]. The presence of excessive amount of solute at surfaces or grain boundaries has been studied for decades. The interfacial segregation phenomena usually results from the interaction between the interface and point defects or solute ions. From the thermodynamic point of view, the amount of segregation may be quantified by the Gibbsian interfacial excess Γ given for a binary system as Eq [61] Γ i γ = µ i T, P, DOF 2.31

42 30 where γ is the free energy of the interface, µ i is the chemical potential of component i, T is the temperature, P is the pressure, and DOF is the aforementioned five macroscopic degrees of freedom associated with the interface Point defect segregation: space charge concept and defect formation energy As described briefly in previous chapters, intrinsic point defects (interstitials and vacancies) are generated under reducing or oxidizing atmospheres while substitutional solute defects are created when the material is doped. The charged point defects and solute ions may accumulate to or deplete from interfaces or grain boundaries by interacting with the surrounding electrostatic field. The resulting space charge segregation is an important concept in ionic crystals. The formation of such an electrostatic field inherently originates from the difference in the defect formation energies of individual point defects. The defect formation energy for an individual defect i, denoted as g i, is defined as the energy required to move a point defect from its source (surface or interface) to a lattice site. The individual defect formation energy is generally a positive number, which implies the point defect is more stable thermodynamically when staying at the surface of interface. For example, considering a pair of Schottky defects V M and V X in a pure ceramic material MX, individual defect formation energy is numerically equivalent to the energy released when a point defect is moved from a lattice site to the source (surface or interface). If the formation energy of the cation vacancy V M is smaller than that of anion vacancy V X, less energy will be required to move the same amount of M ions to the

43 interface. Thus the interface tends to have more M ions than X ions when the total free energy of the system is minimized. The interface, as a result, becomes nonstoichiometric and bears a net positive charge that introduces a positive electrostatic potential φ. This positive charge has to be electrically balanced by the negative space charges nearby to maintain the global electric neutrality. Thus a electrical filed is formed as the interfacial potential decays to its bulk value with the increasing distance away from the interface, which complies with Poisson s Equation (see chapter for details). The electrostatic field in the space charge region interacts with charged species in adjacent regions that finally depicts the spatial distributions of those point defects. Although the total formation energy of a Frenkel defect ( G = g + g ) or f M i V M Schottky defect ( G = g + g ) could be experimentally derived, individual defect energies ( g, M i gv Mi s and V M V X g V Xi ) are not readily obtained. One of the most important reasons is the energy reference point, where the energy of a defect is defined as zero. In the present study, the absolute values of the individual defect formation energies are all referenced to the surface or interface whose energy is treated as zero. This reference point has been an important issue in the discussion of defect formation energy as explained by Poeppel and Blakely [55]. However, the energy of different types of surfaces or grain boundaries may not be the same, which results in the fact that the energy varies when point defects or solutes segregate to different type of GBs. For example, point defects are less likely to segregate to the low-energy interfaces such as twin boundaries or coincident site lattice (CSL) interfaces. In fact, variation of interfacial excess of point defects with grain boundary geometry has been explored in a variety of systems [62, 63]. Thus this 31

44 32 reference point, set to zero, should be an average value of the energies of all types of GBs in a system. As a result, the individual defect formation energy is an average value with obvious standard deviation as well as the interfacial excess that is determined by the formation energy. Individual defect formation energies must therefore be theoretically calculated. The most commonly cited values from Catlow s calculations [16, 17] used vacuum level as the reference thus they are different from our definition. Ab-initio calculations [64-66] based on density functional theory (DFT) could yield more accurate results. In our present study, ab-initio calculations have been applied to TiO 2 by the group of Prof. Susan Sinnott in the University Florida. A volume of unit cells is established as a simulation module, where an arbitrary point defect is placed inside of the module for the first time while on the most stable surface, <110> for rutile TiO 2, for the second run. The structures are relaxed to find the minimum points of total system energy, the difference of which is defined as the individual defect formation energy of this specific point defect. This method has been employed to provide reasonable segregation energies for the ZrO system [67]. The biggest drawback of ab-initio calculations is very time consuming so that the size of the simulation module has to be limited Solute segregation: segregation driving forces As discussed in the previous chapters, electrostatic fields are established in the near grain boundary regions because of nonstoichiometry at the grain boundaries resulting from the difference in the individual defect formation energies. Charged dopant

45 33 (acceptors or donors) cations, during the interaction with the electrostatic fields, tend to segregate to or deplete from the grain boundaries depending on the charge polarity to minimize the total electrostatic energy. On the other side, if the size of the dopant ions is significantly different from that of the matrix cations on the substitutional site, considerable strain field is created around the alien ions due to the large ionic mismatch. The high strain energy has to be released in some way for the system to stabilize. Before the solid state solubility of the dopant is achieved, those alien ions can be better accommodated in the grain boundaries than in the bulk. From the thermodynamic point of view, the total system free energy reaches its lowest point when the strain energy is released to the greatest extent where the system is the most stable. In such a condition, the impurity ions enter sites at or near the grain boundaries, referred as solute segregation, where the oversized ions can be accommodated much more easily. Solute and point defect segregation to surfaces and interfaces has been an important topic since the study of metals [9]. As described in previous chapters, the segregation phenomenon is mainly driven by two types of energies: 1. Electrostatic energy resulting from the electrostatic interactions between the charged defects and the space charge potential φ; 2. Elastic interaction energy (U s ) that is the consequence of the release of the strain energy due to the elastic misfit of the large solute cations in the host lattice. There are also some other driving forces for segregation such as dipole interaction [15]. The charged point defects at opposite polarities can form defect dipole pairs, which interact with electrostatic field. But their effect can be negligible compared to either of

46 34 the principle driving forces at high temperatures. In order to understand the fundamental principles behind segregation we must analyze the contribution of each effect. The electrostatic energy is relatively easy to define as Z i eφ, where Z i is the effective charge number of a type of defect, e is the electron charge. As an important factor governing defect concentrations, the elastic interaction energy, U s, is another independent parameter, which is given by Eq [15] (x) U s = n x 2a 0 x U 1 a x > 2a where n is a constant describing how quickly the elastic-strain energy converges to its bulk value, x is the distance from grain boundary and a is the lattice parameter. The effective elastic interaction is limited to a small distance, usually in the range of two lattice constants. Assuming that the strain energy is fully relieved upon segregation to the grain boundary, the peak point of the elastic interaction energy, U 0, is related to the ionic misfit r, bulk modulus B and shear modulus µ of the matrix, Eq [9, 15], U 0 3 r 6πr B r = 1+ 3B 4µ Yan and Rhodes gave a temperature-dependent expression for this energy based on TiO 2 single-crystal elastic constants, Eq [68]: U 0 2 r T = (3.45 ) ev 2.34 r 1700

47 35 The amount of contribution to this driving force depends on the relative size of the solute and host ions r, specifically the square of this misfit. Most models agree that the larger ionic misfit implies greater driving force for segregation. The elastic interaction energy may vary considerably from one type of dopant cation to another. For solute cations such as Al 3+ (0.53 Å) or Nb 5+ (0.61 Å), the ionic radii are close to that of the host Ti 4+ cation (0.68 Å). At 1300ºC, the elastic interaction energies for Al and Nb are and ev respectively. For large cations such as Y 3+ (1.02 Å) or Ce 4+ (1.01 Å), U 0 is 0.64 and 0.60 ev respectively. For isovalant dopants such as Ce 4+, the elastic interaction energy is the dominant driving force for solute segregation. This type of segregation can be easily modeled by the McLean isotherm model [9] in which the solute concentration at grain boundary n gb is expressed in terms of the absorption free energy, F a, and the bulk concentration, n b, as in Eq n e = 2.35 Fa / kt b n gb Fa / kt 1 nb + nbe For a system where only elastic driving force exists, the absorption free energy is approximately equivalent in magnitude to the elastic energy defined by Eq. 2.32, Eq and Eq The shortcoming of this model is that it doesn t provide any information about the grain boundary potential or distribution of the intrinsic defects. The existence of a space-charge region is a special characteristic for ionic materials. In the bulk, local electroneutrality condition (ENC) is achieved so as to prevent an infinite electrostatic energy. This ENC is not valid near the surface or interface due to the inequality of the individual defect formation energies. The global ENC is achieved by

48 36 obtaining the equilibrium of core charge with the space charge in the depletion region, as depicted in Fig. 2-6, where the space charge region bears a net positive charge that is balanced by the negative core charge. An energy barrier (eφ B ) for electronic conduction is formed in this case. χ Depletion Region eφ VL eφ B E C E g E f E V Core Grain Interior Figure 2-6: Typical space charge distribution. Solute and point defect segregation driving by the electrostatic energy was explored by Chiang and his coworker in doped TiO 2 systems [14, 42]. In these studies, emphasis was placed upon the space charge over the elastic aspect of segregation by determining the relative values of these two driving forces. Samples were doped or codoped with aluminum and niobium at concentrations no greater than 2 mol%. Aluminum and niobium were chosen as dopants because they have similar ionic size as titanium and have decent solid state solubility in TiO 2. Interesting experiments were designed using various combinations of the dopant levels. Based on reasonable assumptions of the individual formation energies of the intrinsic defects, space charge models were established to predict the barrier height and solute excess at the grain boundaries. Since

49 37 electrostatic energy was the only driving force in their models, the barrier height (space charge potential) was found only related to the net dopant concentration in the bulk. Chiang proposed a very good space charge model that agreed well with their experimental discovery. Having both driving forces exist in the same system could complicate the model since the segregation behavior depends on the relative values of the two energies. The way how the two driving forces control the segregation behavior is also significantly different from each. Elastic driving force is a high-magnitude interaction energy that is only effective in a much shorter range when compared to the electrostatic driving force. In this case, the grain boundary charge is not solely determined by the net dopant concentrations. Two limiting cases have been discussed in the literature: the elastic energy is much smaller, as is the Al+Nb doped TiO 2 [42], or much greater than the electrostatic energy, as in the Ca or Ce doped TiO 2. If these two driving forces are comparable, the elastic driving force may be large enough to change the details of electrostatic potential φ and therefore the excess solute density, Γ. For most ionic systems, there are very few data available for the individual defect formation energies, and thus it is difficult to determine the sign or the magnitude of the space charge potential [42]. Complex space charge models have to be built based various assumptions that will be discussed in more details in the following chapters.

50 Numerical solutions of grain boundary segregation Intrinsic defect concentration and Boltzmann distributions The total free energy of a system can be expressed in terms of enthalpy and entropy. In our system, the enthalpy is related to the energies of individual intrinsic and extrinsic defects. Assuming pure TiO 2 without existence of Ti 3+ ion, the total free energy is expressed by Eq [8, 15]: 1 F = [ n Ti ( x) gti nv ( x) gv nv ( x) gv ρ ( x) φ( x)] dx TS C 2.36 i i Ti Ti O O 2 where n i (x) is the concentration of intrinsic defect i as a function of distance x from the grain boundary, g i is the individual defect formation energy that is assumed to be independent of distance x, ρ(x) is the local charge density, T is the absolute temperature and S c is the configurational entropy and also a function of x (Eq. 2.42). The charge density is given by Eq and the configurational entropy can be evaluated by Eq ,,,, { 4[ Ti ]( x) + 2[ V ]( x) + p( x) 4[ V ]( x) n( x) } ρ ( x) = e Zi ni ( x) = en i O Ti 2.37 i S C = K B N i! ln Ω = K B ln( Πω i ) = K B ln 2.38 i ( N n )! n! i i i i Considering that the number of defect (n i ) is much less than that of the lattice site (N i ) of the same type of cation, the configurational entropy, Eq. 2.39, can be further simplified using the Stirling s approximation. S C = K B i N >> n N i N i n i i i N i N i ln + ni ln K B ni ln 2.39 N n n n i i i i i

51 Eq. 2.40: Then the partial differential of the system free energy can be expressed by 39 δf L = dx[ gti δnti + gv δnv + gv δn i i O O Ti V + 1 2δ ( ρφ)] 0 Ti TδS c 2.40 where the electrostatic energy term and configurational entropy term can be specifically given based on the partial differentials of intrinsic defect numbers, Eq and Eq L δ ( 1 2 ρφ) = e dxφ( x)( ZTi δnti + ZV δnv + Z i i O O V δn 0 Ti V ) 2.41 Ti L N 2N N δ S C = k dx δnti ln + δnv ln + δnv ln i O Ti nti nv n i O VTi The most stable system status corresponds to the minimum point of the free energy where δf = 0. From Eqs. 2.40, 2.41 and 2.42, the equilibrium intrinsic defect concentrations at temperature T can be expressed using Kroger-Vink notation by Eqs. 2.43, 2.44 and 2.45 [42]: [ Ti gti + 4e i ]( x) = exp kt φ... i ( x) 2.43 [ V g ]( x) = exp 4e V φ,,,, Ti Ti kt ( x) 2.44 [ V gv + e x = O 2 ]( ) 2exp kt φ.. O ( x) 2.45 Eqs known as the Boltzmann distributions, are the intrinsic defect concentrations in the space charge region starting from the first atomic layer beyond the

52 grain boundary core. Also note that the electrostatic potential φ (x), for convenience, is referenced to zero at the grain boundary ( φ = 0 ) and is a constant in the bulk x=0 40 φ = φ ). x = ( From defect reactions Eq and Eq concentrations of electrons and holes can be expressed in terms of that of titanium vacancy and oxygen partial pressure, from the reaction constant given by Eq and Eq K i = np [ Tii ] n PO 2 K = 2.47 The values of reaction constants (K i = cm -6, K 1 = atm/cm 15 at 1300ºC) were obtained by interpolation from the experiment results of Baumard and Tani at 1200ºC and 1350ºC [42, 58], Extrinsic defect concentrations When the system is doped with solutes, extrinsic defects reside in the lattice substitutionally and contribute to the total free energy by adding a n s (x)g s term in Eq. 2.36, where n s is the concentration of the extrinsic defects and g s is defined as the formation energy (or driving forces) of the extrinsic defects, which is also referenced to the grain boundary core. The free energy is minimized using similar procedures. The extrinsic defect concentration n i can be derived mathematically using Lagrange multiplier and expressed

53 41 by Eq. 2.48, based on the assumption that the grain size is much larger than the width of space charge region [7], Z eziφ + eziφ [ ] ( x ns = C ) i exp 2.48 kt where C the nominal dopant concentration. Eq is suitable for the case of Al or Nb doped TiO 2, where only the electrostatic energy -ez i φ is the driving force. When the dopant cation is much larger than Ti 4+, such as yttrium, the defect formation energy is now composed of two interdependent terms, electrostatic energy -ez i φ and elastic interaction energy U s (x) defined by Eq Thus the yttrium concentration is given by Eq [15]: [ Y eφ e exp, φ Ti ]( x) = CY ( x) U kt s ( x) 2.49 where yttrium concentration is not only related to the electrostatic energy eφ(x) but also subject to the elastic interaction energy U s (x) that originates from ionic misfit. C Y is a constant which equals the total dopant concentration for micro-sized grains Poisson s equation and numerical solutions Eq. 2.50, The total charge density ρ(x) as given by Eq satisfies the Poisson s equation, 2 φ( x) ρ( x) = 2 x εε

54 where ε is the relative dielectric constant of bulk TiO 2 and ε 0 is the permittivity of vacuum. Because of the anisotropy of tetragonal rutile structure, average polycrystalline TiO 2 dielectric constant ε = 120 was used [42]. Since the grain size (~µm) is much larger than the width of space charge layer (~nm) for our system, the electrostatic energy will converge to a constant bulk value φ at a distance far away enough from the boundary. This validates three boundary conditions Eq to which the Poisson s equation Eq is subject. 42 φ φ x= 0 x= φ x = 0 = φ x= = The bulk value of electrostatic potential can be calculated from the bulk electroneutrality condition, Eq ,,,, { 4[ Ti ] + 2[ V ] + p 4[ V ] n + Z C } = 0 ρ 2.52 = en i O Ti Substituting Eq Eq into Eq. 2.52, the value of φ is solved. Using yttrium-doped TiO 2 as an example, procedures of obtaining numerical s s solution of Poisson s equation are shown. Defining the Debye length εε kt δ = 0 and 2 e N making substitutions : x s = and δ eφ z =, Poisson s Eq can be simplified to Eq kt 2 z 2 = s f ( z) 2.53

55 and Eq gti i f ( z) = 4exp kt PO K i K 1 4 gv + Ti 4 exp kt If function z is digitized to an array of discrete numbers z(i) with step size h, Eq becomes Eq gti i exp exp 4kT exp exp V ( O 4z) 4exp exp( 2z) K PO s ( z) + C exp z + exp( z) 1 4 g 4kT 1 Tii ( 4z) + exp exp( z) 2 Y g kt U 2.54 kt z ( i + 1) 2z( i) + z( i 1) h ( z( )) = f i or Eq ( i + 1) + z( i 1) h f ( z( i) ) z z( i) = For any starting values of array z(i), a new set of values is calculated from Eq and Eq This process is iterated until the difference between two adjacent sets of arrays is very small (usually less than 10-6 ). Therefore, the discrete values of function z are solved and substituted back to Eq Eq so that the defect concentrations are calculated. The program is written with MATLAB. The efficiency of the program is determined by the choices of step size h (usually h=1) and convergence criteria (10-6 ). If smaller values are used, it takes significantly longer time to run the program.

56 Improved space charge model with core segregation The aforementioned space charge model is capable of predicting the segregation behavior of solute and point defects at high temperatures when the solute is similarly sized to the host cation. It has been utilized to pure and stoichiometric, reduced, Nbdoped TiO 2 and achieved reasonable agreement between experiments and theory. However, it does not take into account the part of solute that may segregate to the grain boundary core, which, as a result, will significantly alter the boundary charge and thus the distributions of point defects in the space charge region while the global ENC is retained. Based on the same iteration method, new space charge model will be outlined in the Chapter 4 along with an extensive discussion between the theoretical calculation and experiment results. 2.3 Experimental Background Energy dispersive X-ray spectroscopy General knowledge about EDXS There are three major techniques currently used to measure the amount of segregation at an interface: Energy Dispersive X-ray Spectroscopy (EDXS), Electron Energy Loss Spectroscopy (EELS), and the atom probe field ion microscopy approach

57 45 [69]. The third method approach is unfortunately limited to metallic and metal-ceramic interfaces so we will only discuss the EDXS and EELS that were used in this study. EDXS is a powerful characterization technique that provides compositional information of the specimens at the nanometer scale. As illustrated in Fig. 2-7, X-rays are generated when high-energy incident electrons penetrate through the specimen. If an incident electron ionizes an atom by exciting a shell electron, characteristic X-rays will be emitted during the relaxation whose energies correspond to the variations in the quantum-mechanic energy levels that are unique to the specific atom type. If an electron is decelerated by the nucleus, a continuum of Bremsstrahlung X-rays is produced in the low energy regimes. High Energy (200kV) Electron Probe EDXS HAADF g EELS Figure 2-7: Illustration of EDXS and EELS.

58 46 An EDXS detector is usually a piece of cylinder-shaped reverse-biased p-i-n diode based on silicon-lithium semiconductor. Ohmic contacts by gold are made on both sides of the semiconductor. During the interaction of the incident X-rays with the Si (Li), thousands of electron-hole pairs are generated within the intrinsic region (referred to as the active layer ) that are separated by an applied bias between the two ohmic contacts. The p and n regions are usually called dead layers because most of the electron-hole pairs created in these regions do not contribute to the spectra observed. Thus charged pulses of electrons are formed and can be measured at the bottom ohmic contacts, which are subsequently amplified by the low-noise FET devices before being stored in the appropriate energy channel of the multi-channel analyzer (MCA). While the incoming voltage pulses are measured by a pulse processor, the detector is effectively switched off for the period of time termed as the dead time, which can be calculated by Eq Rout Dead time in % = (1 ) 100% 2.57 R where R out is the output count rate while R in is the input count rate. In general it is preferred to have an appropriate value of the dead time since a high dead time means the detector is swamped with X-rays while the collection becomes inefficient. The resolution R of the detector can be defined as Eq [70] in = P + I X 2.58 R + where P equals to the full width at half maximum (FWHM) of a randomized electronic pulse generator, X is the FWHM equivalent attributable to detector leakage current and

59 incomplete charge collection, and I is the intrinsic line width of the detector and is given by Eq [70] 47 I 1 2 = 2.35( FεE) 2.59 where F is the Fano Factor, ε is the energy needed to create an electron-hole pair, and E is the X-ray line energy. The resolution of a detector can be practically measured by measuring the width of Mn K α peak or Cr K α peak. The typical energy resolutions for the type of detectors discussed are on the order of 140 ev. The detector collection angle (Ω) is an important factor in determining the quality of EDX analysis, which is the solid angle subtended at the analysis point on the specimen by the active area of the detector front face. The desired collection angle is set by the collimator placed in front of the detector crystal that is also used to prevent unwanted or spurious X-rays from entering the detector. The collection can be given by Eq [70] Acosδ Ω = S where A is the active area of the detector, S is the distance from the analysis point to the detector, and δ is the angle between the normal to the detector and a line from the detector to the specimen. Another important parameter associated with EDXS is the take-off angle, which is the angle between the specimen surface (at 0 tilt) and a line to the center of the detector. The take-off angle needs to be optimized so that the count rate is maximized. In addition, spatial resolution is also a very critical parameter in the X-ray microanalysis, which is defined as the smallest distance (R) between two volumes from

60 48 which independent microanalyses can be obtained. Since the spatial resolution is controlled by the interaction between the electron beam and specimen, it is a function of the incident beam size (d) defined as the FWHM of the Gaussian electron intensity and the beam broadening (b) that is caused by elastic scattering of the electrons within the specimen. If the beam emerging from the sample also retains a Gaussian distribution, spatial resolution can be given by Eq [70] 2 2 = b d 2.61 R + Beam broadening, in theory, can be calculated by the single-scattering model that assumes each electron is only scattered once when traveling through the sample. The extent of beam broadening (b) can be given by Eq [70] b = Z E 0 ρ A t 3 2 (cm) 2.62 where Z is the atomic number, E 0 is the incident beam energy, ρ is the density, A is the atomic weight and t is the thickness. For a typical sample thickness (100 nm), the beam broadening for rutile is about 8.5 nm at 200keV incident beam energy Quantitative X-ray microanalysis by Cliff-Lorimer technique EDX spectra collected by the detector can be quantified by the Cliff-Lorimer ratio technique to obtain the relative concentrations between two elements from the integrated EDXS peak intensities. When assuming that the sample satisfies the thin-foil criterion, which implies that it is thin enough so that any absorption or fluorescence effects are

61 49 negligible, the Cliff-Lorimer method for a binary or multi-component system is given by Eq and Eq [71] C J I = J kja C A I 2.63 A + J C C = A where C is the atomic percentage, I is the integrated EDXS peak intensity, and k JA is the Cliff-Lorimer sensitivity factor between elements J and A. The k factor is not a constant but a parameter that varies with microscope conditions such as acceleration voltage. The k factor is also related to the atomic number correction factor (Z) considering that the effects of absorption and fluorescence are negligible. There are two methods to obtain the k factor: experimental determination using standards or theoretical calculations based on the first principle. Since the results from calculation are generally not very reliable [70], the theoretical method is normally used for find quick answers when accuracy is not important. So in this study, experimentally determined k factors from known standards will be adopted (see Chapter III for details). Sometimes a standard containing the two elements of interest cannot be found, it is possible to obtain the k factors relative to a third element by Eq [70]. K AC K AB = 2.65 K BC Characteristic X-rays can be absorbed by thick specimens, which affect the magnitude of k factor used in Cliff-Lorimer equation. Corrections are usually necessary

62 50 in such cases, where the zero-thickness k-factor is related to the actual k-factor at thickness t by Eq J A µ µ ρt csc( α ) ρ ρ sp 0 t sp 1 e k ( ) JA = kja A J 2.66 µ µ ρt csc α ρ sp ρ 1 e sp where (µ/ρ) is the mass absorption coefficient, ρ is the density of the specimen, α is the take-off angle and t is the local thickness where the beam is placed. However, this correction method requires knowledge of exact specimen thickness, which may be very hard to obtain. An alternative method was developed in our experiments, as described in chapter In general, the Cliff-Lorimer method provides a relatively straightforward approach to determine the relative concentration of specific elements at grain boundaries. In order to achieve satisfying results, sufficiently high counts in the characteristics peaks need to be collected. Specimen drift, contamination and beam damage are necessary to be taken into account as well. The spectra should also be collected in the weak diffraction condition to avoid the channeling effects. Details about applying this method to our system will be discussed in the next chapter. The interfacial excess of the dopant (yttrium or niobium), Γ M, can be calculated according to Eq [28, 29] Γ M gb grain C M CM = N Ti W 2.67 CTi CTi

63 51 where N Ti is the Ti site density ( cat/cm 3 for rutile), W is the width of grain boundary analyzed (13 nm for our experiments), (C M /C Ti ) gb and (C M /C Ti ) grain are cation ratios from GB and bulk, which can be calculated via Eq The yttrium peak intensity was not detected in the bulk although the bulk dopant concentration determined by PIXE (Table 3-1 ) may be used as (C M /C Ti ) grain in Eq Electron energy loss spectroscopy General knowledge about EELS Electron energy loss spectroscopy (EELS) is the general title and acronym for techniques in which an electron beam interacts with the specimen and the scattered electrons are spectroscopically analyzed to form the energy spectrum of electrons after the interaction. Compared to EDXS, EELS has its unique advantages in microanalysis, with which one can perform fully quantitative microanalysis across the periodic table with good accuracy (±5 atom%), avoid peak overlaps in EDXS, such as Ba(L)-Ti(K) and Ti(L)-O(K), measure direct specimen thickness from the zero loss peak (ZLP), and obtain local coordination and bonding information from energy-loss near-edge structure (ELNES) and extended energy-loss fine structure (EXELFS). The basis of EELS originates from the interactions between incident electrons and specimen atoms. There exist two types of scattering mechanisms: elastic and inelastic scattering. Elastic scattering is the major scattering mechanism where no changes in the energy of electrons are involved. It usually takes place as Bragg diffraction in crystalline

64 52 materials. Inelastic scattering generally involves changes in both energy and momentum of the electrons during the electron-electron interactions. The amount of changes are collected and analyzed in the form of energy-loss spectrum. Ranging in the order of electron energy-loss, the major inelastic interactions include phonon excitations, plasmon excitations, inter and intra-band transitions, and inner-shell ionizations, which correspond to different regions of a typical energy-loss spectrum. There are three regions in a typical EEL spectrum: zero-loss, low-loss and coreloss region, ranging from low to high energy-loss. The zero-loss region, originating from the unscattered and elastically scattered electrons and phonon excitations (<<1eV energy loss), contains a sharp narrow peak usually call zero-loss peak (ZLP). Since the ZLP is very intense, the detector can be easily saturated thus the ZLP is usually collected with a very small collection time (5 ms). The energy resolution of EELS can be measured by the FWHM of the zero-loss peak (~0.1 ev). The low-loss region corresponds to electron energy-loss up to about 50 ev. It is generated by plasmon excitations and inter- and intra-band transitions. In a plasmon excitation, electron excites oscillations of the electron gas in a solid material, with a typical energy loss of 5~25 ev. The plasmon peak is the second most intense peak in the spectrum after the ZLP. An incident electron may also transfer enough energy to a core electron so that it moves within the same or a higher orbital state. A piece of useful information provided by the low-loss region is the direct measurement of the sample thickness, which can be calculated by Eq [70] I ZL t = λ ln 2.68 IT

65 53 where I ZL is the integrated intensity of ZLP, I T is the total intensity the ZLP and plasmon peaks, λ is the mean free path of the plasmon that is usually on the order of 10 nm depending on the type of material at normal TEM operation voltage. When the sample is very thin, the low-loss spectrum contains only one plasmon peak, while multiple peaks may present due to plural scattering in a thick sample. Eq is a convenient approach to estimate the thickness. However, the accuracy measurement of sample thickness depends on determination of the mean free path using a standard with unknown thickness. The most important part of an EEL spectrum for this study is the core-loss region which results from the ionization of inner shell electrons to unoccupied states. It is usually a high energy process (>50 ev). As discussed in chapter , characteristic X- rays are emitted when the ionized atom decays back to its ground state. So the core-loss spectrum and EDX are different aspects of the same phenomenon. The ionization of K- shell electron is in the 1s orbital and gives rise to a single K edge. For the L-shell, if a 2s electron is excited, an L 1 edge is obtained, and a 2p electron produces either an L 2 or L 3 edge depending on the spin quantum number of the electron. The L 2 or L 3 edges are sometimes called L 2,3 edge because they may be hard to resolve at lower ionization energies. Similarly, M 1, M 2,3 and M 4,5 originate from the M-shell electrons and so on so forth. For each orbital, there is a minimum energy, termed as the critical ionization energy E c, which is required to exceed the ionization threshold. When the energy is higher than E c, the possibility of ionization becomes less due to the decreasing value of the cross section. Consequently, the ionization-loss spectrum has a triangular-shaped

66 54 edge profile where a sharp rise to a maximum intensity at E c followed by a slow decay in intensity beyond the threshold. The ideal triangular shape is called a hydrogenic ionization edge that is only found in spectra from isolated hydrogen atoms. A real edge shape near E c may be more complicated due to the bonding effects from crystal lattice, and is called as the electron energy-loss near edge fine structure (ELNES), which tells about local valence state and coordination. Small variations in intensity may be also detected at more than 50 ev after the edge, termed as extended energy-loss fine structure (EXELFS), because of the diffraction effects from the atoms surrounding the ionized atom. The EXELFS gives information such as long/short range order, bond distance, etc Quantitative microanalysis by EELS The integrated intensities of ionization edges can be used for quantitative compositional analysis, after removing the plural scattering background. The absolute number of the atoms (N) contributing to the edge can be calculated by Eq [70], I K ( β, ) N = 2.69 I ( β, ) σ ( β, ) 0 where I k (β, ) is the integrated intensity of ionization edge above background, I 0 (β, ) is the intensity of ZLP and low-loss peak included in the integration window with the same width, and σ (β, ) is termed as partial ionization cross section that is function of collection semi-angle β and width of the integration window, as shown in Fig. 2-8.

67 55 I 0 (β, ) Intensity (a.u.) Gain I k (β, ) Energy Loss (ev) Figure 2-8: Illustration of integration window and peak intensity. [70] Therefore, the atomic ratio of two elements A and B can be calculated by Eq N N A B = I I A K B L B ( β, ) σ L ( β, ) = k A ( β, ) σ ( β, ) K A( K ) B( L) I I A K B L ( β, ) ( β, ) 2.70 which is analogical to the Cliff-Lorimer Eq It is critical to choose the same width of integration window for each element since the theoretical cross section is a function of. When the width has to be different, a correction factor based on the ZLP intensities needs to be incorporated into Eq The partial ionization cross-section is an important parameter in the Eq that can be determined either via theoretical calculation or experimental approach with known standard spectra. There are a few types of theoretical models for calculating the

68 56 ionization cross section. Hydorgenic model is the simplest where the edge is fit by an ideal spectrum from single scattering from an isolated hydrogen atom. More complex models, such as Hartree-Slater model or atomic-physics approaches, calculate the cross section in a more realistic way than the hydrogenic model and therefore are better for the more complex L and M edges. Although such models provide good result for K and L edges, the theory does not agree with each other or experiment very well for the M shell [70]. In such cases, experiment approach may be utilized to determine the k factor in Eq with a standard sample instead of calculating it theoretically. However, the standard and unknown must have the same thickness and the same bonding characteristics, and the EEL spectra must be collected under identical microscope conditions as well as β and Impedance spectroscopy General knowledge about IS Impedance spectroscopy (IS) is a powerful technique where microstructural information such as defect chemistry, barrier height, and grain boundary segregation can be obtained from electrochemical measurements in macroscopic scales. Compared to the TEM approach, impedance experiments usually involve much simpler equipment, faster spectra collection and easier specimen preparations. It has been broadly applied to a variety of ceramic materials such as ceria [26], ziconia [24, 25, 72] and titanates [34-36].

69 57 Resistance (R) of a material is defined by the Ohm s law as the quotient of DC voltage over DC current as measured at this voltage. The impedance is a similar concept which is defined when AC signals are applied and measured. The impedance of a material can be expressed in terms of a complex number Z(jω)=Z +jz, which is a function of the frequency of the applied AC voltage. For common analog components (resistor, capacitor and inductor), the impedances can be expressed by Eqs. 2.71, 2.72 and 2.73, Z R = R Z C = 2.72 j ω C Z L = jωl 2.73 where R is the resistance, C is the capacitance, L is the inductance, ω is the angular frequency and j is the imaginary number. If an electrical circuit is composed of a combination of such basic units, the total impedance can be calculated from the impedance of each component when they are in series (Eq. 2.74) or in parallel (Eq. 2.75). Z( jω) = Z ( jω) 2.74 i Z( jω) 1 1 = Z ( jω) 2.75 i

70 58 The real part Z and imaginary part Z are actually interdependent since both are functions of frequency ω. The plot of Z versus Z, known as cole-cole plot [73], constructs continuous curves in the complex plane, whose exact expression can be explicitly derived via Eq and Eq Electrical equivalent circuits (EEC) Once the spectra are collected, electrical equivalent circuit models are used to analyze the data. Each microscopic feature (interface, boundary, precipitation, etc.) is electrically equivalent to one or combination of the three basic analog units (R, C, L). Values of such units are then obtained through data fitting to provide reasonable representation between the EEC models and actual spectra. The brick layer model (Fig. 2-9a) is a most widely accepted concept for polycrystalline dielectric materials. In this model, the variations in the grain geometry are nullified and all grains are represented by cubes with the nominal grain size. The grains are regularly separated by flat grain boundaries. In general, grains with more electrically resistive grain boundaries can be represented electrically by a parallel connection of a resistor and a capacitor termed as a RC branch. A combination of several of such RC branches represents a polycrystalline ceramic. The number of branches in the equivalent circuit is equal to the number of different microstructural components or phases in the materials, such as grains, grain boundaries, electrodes and precipitates. In the absence of any precipitates or electrode polarization effects, two RC branches representing only grain bulk and grain boundaries exist. In the case of blocking grain boundaries, the two

71 59 RC branches form a series combination, whereas for grain boundaries having higher conductivity compared to the bulk, they form a parallel combination. Further, in practical cases the capacitance in the equivalent circuit is replaced by a constant phase element (CPE) due to the nonideal Debye dielectric relaxation processes in the material (Fig. 2-9b). The resistor-capacitor (RC) parallel combination appears as a semicircle in the Cole- Cole impedance plots, however, with CPE replacing the capacitors the center of the semicircle is suppressed below the x-axis. The depression of the semicircle is a measure of the deviation from ideal capacitance behavior due to the chemical or geometrical inhomogenities in the polycrystalline material of interest [47, 48]. A typical spectrum corresponding to the EEC is shown in Fig. 2-9c, where the solid curves stand for the ideal RC circuits while the dotted ones represent the effects of constant phase elements. Impedance spectra were fit using the EEC used on the brick layer model and the resistances and capacitance were extracted for each microstructure or phase, from which the resistivity, carrier concentration, electrostatic barrier height and dielectric constant can be calculated. Detailed equations are described in the next chapter.

72 60 Grain Boundary Electrode R B R GB d (a) δ (b) CPE B CPE GB Z (c) R b R b +R gb Z Figure 2-9: (a) Brick-layer model for polycrystalline materials where d is the average grain size and δ is the GB width. (b) Electrical equivalent circuit (EEC) with RC branches in series. (c) The spectra corresponding to the EEC shown in (b).

73 61 Chapter 3 Experimental Procedures 3.1 Specimen Preparations Starting materials All TEM and IS experiments were carried out using high-purity polycrystalline TiO 2 specimens fabricated via ceramic powder processing and high-temperature thermal treatments. Since the control of impurity level is one of the most important concerns during sample preparation, all the operations were performed in a class-10 clean room. Anti-contamination control was emphasized throughout every step of the process to minimize impurities from containers and environment. The starting material was ultrahigh quality (99.999%, Sigma-Aldrich Co., St. Louis, MO) titania powder composed of mixed phases of anatase and rutile, as confirmed by the x-ray diffraction (XRD) results. All peaks for both phases were identified without any extra peaks as shown in Fig After sintering, the solidified pellets consisted only of the rutile phase as shown in the XRD results in Fig. 3-2.

74 62 Figure 3-1: X-ray diffraction spectrum of TiO 2 starting powder where A presents anatase phase peak and R presents rutile phase peak. Figure 3-2: XRD scan of a sintered pellet displaying a single phase of rutile.

75 3.1.2 Solute doping and pellet fabrication 63 The doping process was handled by labware (bottles, beakers, mixtures and droppers) made of polypropylene rather than glass to avoid silicon and silicate contaminations. Before each usage, all labware underwent a standard cleaning procedure that usually took about two hours and a half. They were first cleaned by high-purity alcohol, acetone and methanol in sequence for about ten minutes at each solution to degrease and remove organic compounds. Then the interior of each container was rinsed thoroughly using 18-MΩ deionized (DI) water for at least five times. An acid-washing procedure using high-purity hydrochloric, nitric and hydrofluoric acid followed. Containers were filled with each type of those acids in the order and left undisturbed for about half an hour. Thorough DI water rinsing was scheduled in between the change of each acid and upon the completion of the acid-cleaning process. All labware was then blow-dried using a nitrogen gun before used for doping procedure. Solutes were mixed into the titania powders in the form of solid or solution. In some of our early experiments, solid powder mixture was used, where high-purity oxide powders, Y 2 O 3 (99.999% purity, Alfa Aesar Co., Ward Hill, MA) and Nb 2 O 5 (99.999% purity, Alfa Aesar Co.) were added to TiO 2 powders at the specified molar ratios in a suspension of methanol. The liquid was completely expelled at around 100 C on a hot stage, agitated using a low speed magnetic stirrer. The solution doping, as the major method, produced much more favorable results in terms of homogeneity and particle size. The dopant sources were chosen to be % Y(NO 3 ) 3 4H 2 O (Sigma-Aldrich Co.) and % Nb(OC 2 H 5 ) 5 (Alfa Aesar

76 64 Co.). The yttrium nitride was dissolved into DI water and the niobium ethoxide into anhydrous glycol (decompose in water) to form precipitate-free solutions at specific molarities that were pre-calculated with objective doping levels and atomic mass of titania. About 100g TiO 2 powders were weighed for each batch of the samples and suspended in 500 grams DI water or glycol in a 1 liter container placed on a precise electronic balance. The required amount of dopant solution was then added into the system by polypropylene droppers. One or two drops of BYK -156 (BYK-Chemie Co., Wallingford, CT) were introduced as a dispersant to increase the stability of suspension. This dispersant was chosen because it was a solution of an ammonium salt of an acrylate copolymer free of metallic ions, which would be completely removed by subsequent thermal treatments. The doped suspension was then sealed and shaken gently and then ultrasonically agitated for 30 minutes to break any large titania agglomerates. A favorable result showed suspensions with no indication of sedimentation after a period of time (approximately 15 minutes) long enough to sustain the following freeze-drying process. The suspension was transferred immediately to another pre-cooled wide-mouth container placed in a dewar full of liquid nitrogen. It took approximately five to ten minutes for the suspension to completely solidify, depending on the amount of suspension. After the completion of solidification, the container was quickly moved into a pre-cooled Labconco Freezone freeze dryer, where the mixture was freeze-dried for about 48 hours while maintaining -50 C within the chamber. The freeze-dried powder, which was fluffy and had a white color, was then calcined in a platinum crucible at 300 C for 2 hours in constant air flow to expel all nitrate and organic substances.

77 65 About 0.5 grams of such calcined powders were weighed for each pellet and uniaxially compacted in a 13mm-diameter titanium-coated stainless steel die, whose surfaces were thoroughly pre-cleaned using organic solutions. Moderate hand pressure was applied onto the die to compress the pellets and make sure extra uniaxial internal stress was not introduced. Each green body was transferred into an individual polypropylene bag that were evacuated and sealed tightly. All such packages were isostatically pressed at 30,000 psi. Isostatic pressing did not introduce inhomogeneous stress distributions thus significantly reduced the occurrence of internal cracks. The compressed pellets, measured at around 10 mm in diameter and 5 mm in thickness, were removed from the package with a Teflon tweezer. Then the skins of the pellets were carefully scraped off by a PTFE-coated scraper to remove the residues and contaminations from the die and bags. Embedded under titania packing powders of the same dopant concentration, the pellets were placed at the center of a platinum crucible and sintered in a horizontal tube furnace outfitted with a high-purity alumina tube. Only pellets of the same dopant type and concentration were placed together for each sintering process to avoid cross contaminations. The furnace was programmed to provide a 5 C/min ramping rate and a 1.5~14 hour sintering time at 1300 C depending on the experiments. Some samples were air-quenched by rapidly removed from the furnace into the room-temperature atmosphere. Since the TiO 2 stoichiometry is sensitive to oxygen partial pressure, the heat treatments were performed under constant air flow and controlled oxygen partial pressure (0.21 atmosphere at 1300 C), monitored by an oxygen sensor that continuously measured the O 2 partial pressure around the specimens. The finished pellets were in ivory (yttrium-

78 66 doped) or dark gray (niobium-doped) color with no appearance of inhomogeneity. The air-quenched samples look slightly darker in color due to the loss of oxygen during the process. The dopant information is summarized in Table 3-1, as stated in units that were used throughout the paper: mole percent (mol%) or part per million (ppm), noting that 1 mol% = 10 4 ppm. It is noticeable that the lowest Y-doped specimen was made via powder doping, by mixing the Y 2 O 3 and TiO 2 powders. The experimental results showed, however, no significant difference in homogeneity between the different doping sources. All finished pellets were analyzed by x-ray diffraction, which indicated the presence of only rutile TiO 2. The actual dopant concentrations in all pellets were examined by the particle-induced X-ray emissions (PIXE) method carried out at Element Analysis Corporation, Lexington, KY, which allowed trace impurity detection of yttrium at a level of 18 ppm. In this non-destructive technique, a model FN tandem Van De Graaff particle accelerator was used to produce protons that bombard a sample of compacted powder and the X-rays given off are collected with a detector. This method was much more sensitive to trace element detection since the Bremsstrahlung background was greatly reduced compared with the electron probe in the TEM. The PIXE analysis results, as shown in Fig. 3-3, indicated that the specimens were free of major impurities such as Fe and Si (below the detection limits of 13.3 and ppm, respectively). The measured doping levels, converted from mass% to mol% using the result from PIXE analysis (Fig. 3-3), are also presented in Table 3-1 with estimated measurement errors.

79 Table 3-1: Starting material details and dopant concentrations. Dopant Y 3+ Ionic Radius 1.02 Å [74] Nb Å [42] Y 3+ Nb 5+ - Alfa Aesar Ti Å [75] Source Initial Form Purity Doping Level (mol%) nominal real (PIXE) Alfa Aesar Y 2 O % ± Sigma-Aldrich Y(NO 3 ) % ± Sigma-Aldrich Y(NO 3 ) % ± Alfa Aesar Nb 2 O % ± Y(NO 3 ) 3 Nb(OC 2 H 5 ) % % Sigma-Aldrich TiO % solvent - 67

80 Figure 3-3: A sample PIXIE analysis result of 243ppm Y-doped TiO 2 powder. 68

81 3.1.3 TEM sample preparations 69 TEM sample preparation followed conventional wedge-polishing procedures. Portions of the densified pellets were sliced into small mm 3 pieces using a lowspeed diamond saw. After mounted onto a wedge tripod polisher by crystal bond, such a small piece was mechanically thinned with diamond polishing films down to less than 10 µm in thickness and 0.1 µm surface roughness. The thinned section was mounted by M- bond TM 610 adhesive onto a 3 mm copper support grid that contains a hole in the center and left in air for at least 12 hours for the glue to solidify completely. The sample attached to the copper grid was then carefully placed into a Fishion precision ion miller and polished at liquid nitrogen temperature using argon ion beams at 4 kev and 10 incident angle for about 1~2 hours until a small hole with thin regions appeared near the center of the specimen The samples were then ion milled at 5 kev beam energy and 15 incident angle for 20 minutes and plasma cleaned for 10 minutes to remove contaminants IS sample preparations A sintered pellet bisected along its diameter was carefully sliced to about 1 mm thick in the direction parallel to the surface by a low-speed diamond saw. It was then mechanically polished down to less than 300 µm. Electrodes, composed of an inner layer of 20nm-thick chromium and an outer layer of 200nm-thick platinum, were deposited on both sides of the pellets using thermal evaporation. The films were deposited on a 2.0 mm diameter circular area at the center of one side of the titania pellet while the other

82 70 side was coated completely with the metal films to avoid edge effects. The thickness of the pellets were kept small (200~300 µm) compared to the electrode area to keep the electric field lines parallel between the electrodes. All plated pellets were annealed on a Pt foil in constant air-flow at 600 C for about 6 hours to improve the bonding between electrodes and the pellet. These thermally treated samples with electrodes plated did not show any hint of reduction or oxidation and were highly stable during the elevated temperature impedance measurements. The effective electrode areas measured by an optical microscope are listed in Table 3-2. Note that the experiment error associated with the thickness and area measurements was estimated at 10~20%. Table 3-2: Geometries of specimens used for impedance measurements. Dopant (Y) Conc. (mol%) Thickness (µm) Effective Elec. Area (mm 2 ) Grain Size (µm) ± ± ± Density and Grain Size Measurements The densities of pellets were characterized using the Archimedes method with distilled water as the immersion fluid. The pellets were then sliced, polished on one side and thermally etched at 1300 C for 1 minute in the same tube furnace system. Utilizing an optical microscope, multiple images were taken of the microstructure of all pellets. These images were analyzed using the linear intercept technique to measure

83 the average grain size. The microstructures of the pellets taken at this stage showed equiaxed grains with low porosity and no signs of abnormal grain growth. The average grain size can be calculated by Eq. 3.1 [76] C D = MN where D is the average grain size, C is the total length of line used, M is the magnification, and N is the number of intercepts. This equation contains a proportionality constant which is a correction factor taking into account that random slices were made through a system containing tetrakaidecahedrally shaped grains [77] Scanning Transmission Electron Microscopy Based Chemical Analysis Energy dispersive X-ray spectroscopy (EDXS) TEM and EDX data acquisition TEM experiments were performed on a JEOL 2010F field-emission TEM operated at 200 kv, equipped with a high-angle annular dark field (HA-ADF) detector, scanning unit, Gatan Enfina post-column EELS and Oxford EDXS detector. A lowbackground beryllium Gatan double-tilt holder was used to reduce the production of spurious x-rays. The detector (Oxford Instruments, Inc.) itself contained a Si (Li) crystal and had a resolution of 136 ev determined upon installation. The X-rays had a take off angle of 40 at 0 sample tilt.

84 72 The TEM was aligned and then put into the dark-field STEM mode. The amount of solute segregation was measured at multiple grain boundaries, using established experimental protocols based on STEM and EDXS [14]. The real-time STEM image was taken and displayed on a CRT screen at high magnification (400kX for our experiments). The EDX signals were collected from three identical well-defined scanning windows, one on the grain boundary with the interface sitting in the middle of the scanning box, and the other two inside each adjacent grain interiors around 60 nm apart from the boundary in the direction perpendicular to the boundary, as illustrated in Fig The electron beam was rastered within these specific regions in order to prevent beam damage to the sample that could occur if the beam was left at one position for an extended period of time. The grain boundary STEM image was rotated parallel to the longer side of the rectangular scanning area. Considering an approximate probe size of 0.7 nm, the dimensions of the scanning windows were nm 2 and the EDX data was only collected from those regions. Samples were tilted, if necessary, such that the grain boundary plane of interest was approximately parallel to the incident electron beam direction.

85 73 Grain Electron Beam Direction GB Grain Figure 3-4: Schematics of EDX data collection regions (not to scale) [23]. EDX data were acquired using the ES Vision TM software that was calibrated initially. A 10 ev/channel with 0 ev offset acquisition parameter was chosen. The dead time recorded for every grain boundary ranged within 10~15%. Signal collection times were maintained constant (400 second live time) for each measurement while obtaining good counting statistics (e.g., 10 4 to 10 5 counts for the Ti Kα peak and ~10 2 for the Y and Nb Kα peak) to ensure comparability of multiple measurements. The grain boundary position was monitored continuously to correct sample drift. Since TiO 2 has a tendency to reduce under the electron beam [78], if any indications of beam damage were observed during the analysis period, those data were excluded from final results. This procedure was continued until the completion of data collection for subsequent grain boundaries.

86 74 A hole count was recorded after all the boundaries have been analyzed, where the beam probe was positioned directly through the central hole of the specimen. An EDX spectrum was collected for the same time interval as the previous spectra EDX data analysis Experimental results concluded that x-ray data should obey Gaussian statistics and the accuracy could then be determined using simple statistics [70]. EDXS data were then exported and analyzed using Desktop Spectrum Analyzer (DTSA) software available from NIST [79]. This program was specifically designed for accurate analysis of x-ray spectra generated by focused electron beam bombardment of specimens. In this program, all the detector and experimental parameters were entered such as the thickness and composition of the detector window as shown in Fig The values of the parameters used were all given by the manufacturer, Oxford Instruments, of the detector.

87 75 Figure 3-5: Detector input parameters for the DTSA program. DTSA software allows to import up to eight spectra taken from the same sample and to do the peak fitting in one group so that each spectrum is handled in exactly the same manner. The ROIs (region of interest) of peak and background were then manually selected. The background was subtracted using a digital bandpass filter that convoluted a top-hand function (peak=1 and background=0) with the spectra. The actual peak intensities were then determined by a multiple linear least squares fit (MLLSQ). This procedure needed reference peaks from standards for each peak family, which were used by the software to determine the shape of each peak since in some cases there may be distortion from a pure Gaussian curve due to charge effects, microscopic conditions, or detector hardware. A reference was made up of segments of well-characterized spectra that had good counting statistics (higher than 10,000), no overlaps for the peaks of

88 interest, and from which the background could be easily subtracted. Reference peaks for 76 the Ti K α were obtained from pure polycrystalline TiO 2 samples, the Y K α and the O K α peaks from Y 2 O 3 powders, and Nb K α and K β peaks from Nb 2 O 5 powders used in the doping procedures also under the same microscopic conditions. The background was automatically subtracted when the MLLSQ fit was executed. The integrated peak intensity of each peak of interest (e.g, Ti K α, Y K α, Nb K α ) was thus obtained along with a counting statistic error. Quality of the fit could be determined from the residual output file where artifacts or residual intensity can be seen. Note that oxygen was not included in the analysis due to the overlapping O K and Ti L peaks, which may be quantitatively determined using EELS (chapter 3.3.2) K-factor determination As discussed in chapter , the k-factor used in Cliff-Lorimer method could be obtained from standards. A single-phased Y 2 Ti 2 O 7 specimen was made as a standard to serve this purpose while a Ti 2 Nb 10 O 29 specimen was commercially available to obtain the k-factor between Y-Ti and Nb-Ti. Following the suggested processing methods [43, 80], high purity Y 2 O 3 and TiO 2 powders were homogenously mixed at 1:2 molar ratio and sintered in air at 1600ºC for 8 hours. It was confirmed by x-ray diffraction that the specimen only contained a single Y 2 Ti 2 O 7 phase. For a foil thickness beyond the thin-film limit an absorption correction [43, 81] is used to modify the k factor, which involves precisely known detector information, mass absorption coefficients and sample thickness. In this study we

89 77 observed a linear relationship between the k YTi factor and EDXS count rate, which was proportional to sample thickness at given identical microscope conditions, as shown in Fig In the Cliff-Lorimer equation, we therefore used the appropriate k YTi factor for each grain boundary to relate the measured intensities to relative concentrations KYTi Average EDXS Count Rate (counts/sec) Figure 3-6: The EDXS sensitivity factor, k YTi, measured from an Y 2 Ti 2 O 7 standard, was found to vary linearly with average EDXS count rate which is proportional to the specimen thickness Electron energy loss spectroscopy (EELS) EELS data collection Although EDXS is very powerful in quantitative compositional analysis, it cannot provide any information about in valance state or coordination. Nor is it able to handle

90 78 the peak overlap for some elements such as Ti(L)-O(K) and Ba(L)-Ti(K). EELS is another type of useful technique that utilizes the energy loss of transmitted electrons to provide local chemical information whose theoretical background has been discussed in details in chapter The TEM was aligned in STEM mode and the EEL spectra were collected from the thin areas of specimens. The beam size was carefully chosen to be 0.5 nm for good spectra intensity and negligible beam damage. 10 cm camera length and 3 mm entrance aperture were used, which corresponded to a mrad collection semiangle for the optimum collection condition. Using DigitalMicrograph software provided by Gatan company, a zero-loss peak was first collected at an energy dispersion of 0.1 ev/div starting at -20 ev, which gave information about the energy resolution and the local specimen thickness. Then the starting energy was offset to a value slightly lower than that of the edge of interest, where the core loss spectra were acquired with a collection time (10~60 Sec) and frame number (1~3) that resulted in good counting statistics. Appropriate energy dispersion was important because a lower value of dispersion was usually required to achieve better energy resolution while a higher value yielded larger viewable range of the spectra EELS data analysis The calibration of EEL spectra is usually necessary to regulate a common energy loss scale. For TiO 2, it is appropriate to align all the spectra with respect to the onset of oxygen K edge in order to detect any chemical shift associated with the Ti edge. After the

91 79 completion of calibration, a background window was defined, whose width should be greater than 10 channels and less than 30% edge energy. Then the background was fitted by a power function, in which the background intensity was assumed proportional to a negative power of the energy. Sometimes plural scattering deconvolution was necessary when the sample thickness was high (greater than about 100 nm), where the experimental core-loss spectrum of each ionization edge was deconvolved using the zero-loss peak by Fourier transform to produce a single-scattering spectrum. However, deconvolution always increased the statistical noise of the spectrum, even after smoothing, so the precision of the integrals was worsened. An integration window was chosen for each ionization edge of interest, which was wide enough to cover the fine structure of each edge. Although the choice of onset of an ionization edge can be arbitrary, a common way is to start from the energy where the intensity is half of the maximum edge intensity. The width of the integration window was kept constant for each peak because the theoretical cross-section is a function of the window width; otherwise corrections may be necessary [70]. A 29 ev integration window width was used in this study for both Ti L 2,3 and O K edges. A partial ionization cross section must be chosen from three available models from theoretical calculations: Hartree-Slater, Hydrogentic and Hydrogentic with white lines. The integrated intensity of each peak was thus calculated using Eq and given along with the quantification error associated with counting statistics that was generally no less than 5% [70].

92 3.4 Impedance Spectroscopy Equipment and data collection The impedance measurements were performed using a computer controlled Hewlett Packard 4192A-LF impedance analyzer. Pellets were mounted on the test fixture where both sides of the pellets were securely contacted with platinum electrodes and leads. The test fixture was placed and heated in air ambient in the center of a horizontal quartz tube furnace. The test temperatures were chosen between 450~750 C so that the impedance of the specimens fell in the measurable range of the instrument (up to 1.2MΩ). In addition, this temperature range is much lower than the sintering temperature (1300 C), which represented the equilibrium state. Ionic defects redistribution at this temperature range was negligible during the measurement cycles. Impedance spectra were acquired at temperature steps of 25 C for the pure specimens or 50 C for the mol% and 0.1 mol% Y-doped specimens during heating and cooling cycles. Data were collected at approximately 20 minutes after each preset temperature was achieved to allow sufficient thermal equilibration. An AC signal of 90 mv amplitude was applied on the samples over an exponentially decreasing frequency range of 13 MHz to 5 Hz. The total impedance was resolved into real, Z, and imaginary, Z, parts and Cole-Cole plots [82] constructed for analyzing the data. In specific experiments, a mix signal of DC voltage and the AC signal were applied to the specimens. The specimens, thermally stabilized at 550 C before the introduction of the mix signal, went through a DC voltage ramp from 0 to 3 V at 0.1 V

93 81 intervals. Three spectra were acquired for each bias level at 5 minutes time intervals, which were compared later to ensure the stability of data with time. Our impedance measurements were performed over a wide frequency range. It was difficult to obtain high accuracy of measurements due to errors caused by cables and test fixture that became even more pronounced at high frequency (>1 Mhz). Thus specialized reference and normalization techniques were required to reduce or eliminate the parasitic effects and obtain accurate and repeatable results. A low loss reference capacitor of a known capacitance value was used to provide a completely flat response (constant capacitance) across the entire frequency range. Although there was always be some parallel resistance in the reference capacitor even though this could be an extremely high value, the difference between an ideal and nonideal capacitor was sufficiently small for our calibration purposes. The impedance response of the reference capacitor was obtained over our measurement frequency range at room temperature and compared to its ideal response. The deviation, due to the parasitic inductance and capacitance of cables and test fixture, was measured and recorded by a C-based calibration program. For each spectrum obtained from our samples, the calibration program was applied to eliminate the parasitic capacitive and inductive elements introduced by the fixture and wires Data fitting and analysis The calibrated data was imported into ZView software version 2.2 obtained from the Scribner Associates Inc. [83]. The software is fully functional of performing complete modeling or fitting on the impedance data in the Z domain or frequency domain. The R-

94 82 CPE equivalent circuit model (Fig. 2-9) was drawn in the software and initial fitting values of R and C were filled in using empirical estimation of the resistance and capacitance of the material. The fitting of the semicircles were done until all fitting errors, defined by the maximum error between the experimental data and fitted curve, were less than 5%. The resistances for bulk and grain boundary were directly obtained from fitting results while the capacitances were calculated using Eq. 3.2 and Eq. 3.3 formulated for the CPE [82, 84], Ψ [ K( j ) ] 1 Z CPE = ω 3.2 C = 1 Ψ ( RK ) R 3.3 where j is the imaginary unit, ω is the frequency, R is the resistance, K is a fitting parameter that equals to the capacitance of the CPE when it behaves as an ideal capacitor, and ψ is a value between 0 and 1 depending on the suppression angle of the semi-circle. The bulk resistivity (ρ b ), net grain boundary resistivity (ρ gb ), and bulk dielectric permittivity (ε b ) were then calculated based on the following equations. The bulk resistivity (ρ b ), net grain boundary resistivity (ρ gb ), and bulk dielectric permittivity (ε b ) were then calculated based on the Eqs. 3.4, 3.5 and 3.6 A ρ b = R b 3.4 l A ρ gb = R gb 3.5 l

95 C A ε b b = 3.6 l ε 0 where A is the effective electrode area and l is the specimen thickness (Table 3-2). The specific grain boundary resistivity is the average resistivity of one grain boundary, which is theoretically related to the overall grain boundary resistivity ρ gb, grain size d g and grain boundary thickness δ gb as Eq. 3.7 [23]. 83 d sp g ρ gb = ρ gb 3.7 δ gb The actual contribution of a single grain boundary to the entire boundary resistivity is thus closely related to the boundary characteristics such as potential barrier height φ B0 and solute excess at grain boundaries. The grain size was experimentally determined from the average of measurements from more than ten areas for each dopant concentration using an optical microscope or a scanning electron microscope after mechanical polishing and thermal etching. The boundary thickness, δ gb, was determined by the full width half maximum of the electron dispersive x-ray spectroscopy (EDXS) profiles across the grain boundaries [43]. The thickness was experimentally obtained as ~4.5 nm for both 250 and 1000 ppm specimens. Due to the beam broadening effect [70], which widens the electron beam as it penetrates through the thin foils, the actual GB thickness should be less than 4.5 nm. The thickness of the grain boundary core was determined (~2.0 nm) from the deconvolution of EDXS profile by the Gaussian beam profile [43]. In reality, solute distribution across the boundary region is not a step function because the solute may also segregate to the space charge region and still be blocking to the electrical conduction. Therefore, a gradually attenuating profile to the

96 84 bulk concentration presents a more reasonable approximation of the solute distribution. However, it is experimentally difficult to distinguish the grain boundary core from the space charge region because both regions contribute to the grain boundary thickness. Although the exact grain boundary resistivity depends on precise values of the grain boundary thickness from Eq. 3.7, both activation energy and barrier height are independent of the GB thickness.

97 85 Chapter 4 Results and Discussions 4.1 Densification and Grain Growth Density profile of Y-doped TiO 2 Densities of pellets were measured on all the yttrium doped specimens to inspect the role of dopant concentration and densification time in the sintering of titania. The results are shown in Fig. 4-1, which indicates that the density profile follows a second order polynomial fit for each dopant concentration. Each data point represents an average of at least three specimens along with the standard deviation within those samples. In short-time sintering (<10 hours), the densities of doped specimens were lower than that of the pure ones due to the solute dragging effects that deterred the grain boundary diffusion during the process of the densification. When the sintering time was long enough, the densities of all the pellets arrived ~98% theoretical density of titania, which did not rise with the further increment of the sintering time. Therefore, at around 10 hours, the densification was essentially complete and grain growth took over after that.

98 86 % Dense Time (hrs) 0% 0.025% 0.1% 0.3% Figure 4-1: Density profile for sintered pellets of pure and 0.025, 0.1 and 0.3 mol% yttrium doped specimens. Similar sintering experiments have also been carried out for 0.1 mol% Nb and Ce doped titanium [85]. Results showed a 10-hour sintering time reached the maximum relative density.

99 Grain growth of Y-doped TiO 2 Grain size were recorded for pure and all the yttrium doped samples. Microstructure of 10-hour-sintered pellets, Fig. 4-2, shows normal grain geometry and low porosity. The linear intercept method, which requires this grain geometry, was used to determine an average grain size as described in chapter 3 and the error bars stood for the standard deviations coming from multiple measurements of samples sintered for the same times. 20 µm Figure 4-2: Microstructure of 10-hour-sintered pellets shows normal grain geometry and low porosity. The results of grain growth during sintering process are presented in Fig Grain growth is defined as the increase in average grain size of a polycrystalline material during elevated-temperature heat treatment [86], which can be formulated by the theory of grain growth kinetics [87].

100 88 Grain Size (um) % 0.025% 0.1% 0.3% Time (hrs) Figure 4-3: Grain size for sintered pellets of pure and 0.025, 0.1 and 0.3 mol% yttrium doped specimens. It was also observed that the pure samples reached higher grain sizes than yttrium-doped samples for the same sintering time. When yttrium segregated to the grain boundaries, the large impurities ions tended to slow down the movement of the grain boundaries thus lowered the GB mobility. The details may be analyzed by the solute drag theory, where the grain boundary mobility is related to the elastic energy and solute atom diffusivity. However, the study is beyond the scope of this dissertation study and will not be discussed in further details.

101 4.2 TEM Results Interface characterization The grain boundaries in the Y-doped TiO 2 specimens were found to be well crystallized with no presence of amorphous or glassy phases, as indicated by highresolution TEM observation (Fig. 4-4). Z-contrast STEM images which have chemical sensitivity were taken at very high magnifications ( kX) as shown in Fig. 4-5, and clearly showed enrichment of the high-z yttrium solute excess at the grain boundaries by the brighter vertical area near the center of the image. The segregation region was roughly measured to be on the order of a few nanometers from such images. Fig. 4-6 shows another dark-field STEM image taken at relatively lower magnifications to illustrate the three equally sized scanning windows where the EDX signals were collected from the grain boundaries and the grain interiors on both sides. In samples with yttrium concentration higher than 0.1 mol%, yttrium titanate precipitates were detected at some triple or multiple junctions. The precipitates were identified to be Y 2 Ti 2 O 7 from corresponding selected-area electron diffraction patterns and EDX spectra. The onset of precipitation indicated that the solid state solubility of yttrium in TiO 2 was around 0.1 mol%.

102 90 2 nm Figure 4-4: High-resolution TEM image shows clean grain boundary without glassy phase in yttrium-doped TiO 2.

103 91 5nm Figure 4-5: Z-Contrast image of yttrium-doped TiO 2 shows solute segregation at the grain boundary.

104 92 w 500 nm Figure 4-6: Typical dark-field STEM image of grain and boundary area with rectangular scanning regions (not to scale) identified for EDXS data collection EDXS results EDS spectra and profiles Fig. 4-7 is a typical EDX spectrum for yttrium-doped TiO 2, in which the copper peaks originated from the TEM support grid. A close view of the yttrium K peak from both the grain boundary and interior indicated the presence of solute at the grain boundary while no yttrium was detected in the bulk.

105 93 Ti Grain Boundary Grain Interior Intensity (a.u.) O Y Cu Energy (ev) Y Figure 4-7: Typical EDXS spectra from both grain boundary (grey like) and interior (black like) show yttrium segregation at the boundary while no yttrium was detected in the bulk. Similar experiments were performed on the 1 mol% niobium-doped specimens. Since the solid state solubility of niobium is about 2 mol% in titania [14], niobium K peaks were clearly identified in both the bulk and the grain boundaries. EDXS profiles across the grain boundaries were acquired with a 1 nm (for Y) or 3 nm (for Nb) step size to measure the spatial distribution of the solute segregation region. Fig. 4-8 and Fig. 4-9 present typical profiles taken from the grain boundaries in yttrium and niobium specimens, in which the observed EDXS profile widths, defined as FWHM of the curve, are determined to be about 5 nm for Y-doped or 9 nm for Nb-doped TiO 2.

106 94 The actual segregation layer is narrower because of beam broadening effects (see Chapter 2), which depends sensitively on the sample thickness [70]. The EDXS profiles do not indicate any change in the titanium and oxygen concentrations across the grain boundary, which is consistent with the EELS results in the following section. Y 250 Counts ( 10 3 ) O Ti Distance (nm) Figure 4-8: Typical EDX profile across grain boundary in 0.1 mol% yttrium-doped TiO 2.

107 95 Nb(K) 250 Counts ( 10 3 ) Ti O Distance (nm) Figure 4-9: Typical EDX profile across grain boundary in 1.0 mol% niobium-doped TiO EDS quantifications Based on about 10~40 boundaries investigated for each doping level of the Y and/or Nb-doped samples, the average solute excess densities (Γ) at the grain boundary were calculated via Eq in units of cations per grain boundary area (cat/nm 2 ) and summarized in Table 4-1, along with the standard deviations ( Γ) resulting from multiple measurements and average counting statistic errors (Error Γ). Example data from each grain boundary in the yttrium doped samples are given in Table 4-2.

108 96 Table 4-1: Average quantified yttrium interfacial excess density. Doping (mol%) Γ (cat/nm 2 ) Γ (cat/nm 2 ) Error Γ (cat/nm 2 ) 0.022%Y 0.1%Y 0.1%Y (Quenched) 0.26%Y 0.9%Nb Precipitates No No No Yes (Y 2 Ti 2 O 7 ) No 0.1%Y + 1.0%Nb 5.625(Y) 1.285(Nb) 2.385(Y) 0.849(Nb) 0.519(Y) 0.252(Nb) No

109 Table 4-2: Quantified solute excess density from all the grain boundaries analyzed. Normal 0.022% Y Normal 0.1% Y Quenched 0.1% Y Normal 0.26% Y GB Γ error Γ error Γ error Γ error

110 98 For the yttrium doped cases, there was an evident increase in the average value of the solute excess density from to 0.1 mol% doping levels, while no further increment in the 0.26 mol% doped sample was observed. The grain boundaries were saturated at ~5 cat/nm 2 solute excess density under these process conditions, and more dopant only increases the amount of yttrium titanate precipitates. It was also observed that air-quenching did not change, within experimental errors, the average solute excess at the grain boundaries for 0.1 mol% yttrium doped specimens. Ikeda found that the slowcooled Al-doped samples had much higher amount of segregation than the water quenched samples because the mobile aluminum ions could still segregate to the grain boundaries during the prolonged cooling process. However, such a difference was not detected in our experiments. Our samples were quenched in the constant air flow at room temperature to avoid internal fractures. Although the cooling rate is not as great as the water quenching, it is still significantly higher than that of the slow cooled samples. Yttrium ion is much larger than aluminum ion thus has a significantly lower diffusivity in the matrix. Before the solid state solubility was reached, yttrium ions tended to reside at the grain boundaries at the high-temperature equilibriums after the samples were held in the high temperature (1300 C) for a long time. Because of their extremely low mobility, yttrium ions would not easily redistribute during the cooling process. The observations of the consistency of the solute excess density at the GBs between slow-cooled and air quenched samples further validated our assumption that the high-temperature equilibrium states were preserved during the slow cooling process. We did not spend a lot of time on the niobium-doped specimens since electrostatic energy is the only driving force for niobium and it has been thoroughly

111 99 studied by Ikeda and Chiang [14]. But one doping level (0.9 mol%) was used as a reference. The solute excess density results of niobium-doped samples are also listed in Table 4-1 and they agree with those from Iketa [14] within experimental errors. As discussed in the second chapter, there are two segregation driving forces (electrostatic and elastic energy) for yttrium while only one (electrostatic energy) for niobium, 0.1%Y+1.0%Nb doped sample was used to investigate the effects of both driving forces. Based on 15 grain boundaries studied, cosegregation of yttrium and niobium was observed in each boundary. The average excess density of yttrium at the grain boundaries was consistent with the single-doped samples within the experimental errors and that of niobium was found lower than the single-doped case. We have to explain this observation associated with the relative magnitude of the segregation driving forces. On the titanium substitutional site, yttrium bears a net negative charge while niobium bears a net positive charge. When interacting with the electrostatic fields in the shape charge region, one should segregate to while the other deplete from the grain boundaries, as observed in the Al+Nb co-doped TiO 2 cases. However, when a large elastic driving force besides the electrostatic driving force presents in the system, the segregation behavior is not solely controlled by the net bulk dopant concentration any more. The charge / barrier height at the grain boundaries is determined by the strength of both driving forces. For yttrium, the elastic driving force was far more pronounced than the electrostatic driving force in a very short range (~1 nm). Under the influence of the elastic driving force, the majority of the yttrium cations segregated to the grain boundary core and nearby regions (a few nanometers outside of the core) and pinch the core charge to a negative value. This negative core charge in turn attracted the positively charged

112 100 niobium substitutional cations. Nb Ti to the nearby GB regions. Thus cosegregation of yttrium and niobium was observed. Furthermore, the strong segregation of the larger yttrium ions to the GBs may have occupied most of the grain boundary sites but the smaller niobium ions could still segregate to the space charge regions. Thus we observed a lower excess density of niobium at the GBs compared to the single-doped samples. Although the counting statistic errors were only about 10% of the solute excess densities for each doping level, there were large variances in the measured solute excess densities for each sample as evidenced by the large standard deviations in Table 4-1. Such large variations in grain boundary segregation have been reported in previous studies as well [14, 16]. Besides the solute type and concentration, segregation is also a function of grain boundary geometry or misorientation [62]. It was very likely that the relatively large standard deviations originated from the variations in grain boundary geometry, which agreed with the impedance results (chapter 4.3), although quantitative correlation has not been established in TiO 2 [85] EELS results EEL spectra and line profile of Y-doped TiO 2 The stoichiometry and valence state are two important factors associated with the grain boundaries. Both of them were investigated in our experiments using EELS. A series of EEL spectra containing the Ti L 2,3 and O K core-loss edges were acquired across the grain boundaries with a step size of 2 nm considering the typical beam size was 0.7

113 101 nm at our experiment conditions. Typical spectra obtained from the grain boundaries and interiors on both sides of the GBs were plotted together in Fig The onset ( white line ) of the Ti L 2,3 edge and electron energy-loss near edge fine structure (ELNES), which arises from transitions of 2p 3/2 and 2p 1/2 electrons into the empty d and s states, are very sensitive to the Ti valance and oxidation state [88]. In TiO 2, splitting of the L 3 and L 2 peaks originates from ligand-field splitting associated with the octahedrally coordinated Ti ions [89, 90]. Any loss of symmetry will lead to a reduction in this ligandfield splitting [90]. Additionally, as the oxidation state of Ti is reduced, there is an associated chemical shift of the edge onset to lower energies [78, 91]. Quantitative comparison of the grain-interiors and grain-boundary spectra in our Y-doped specimens shows no evidence for cation valence changes at the grain boundaries.

114 102 Ti:O=0.48±0.07 Ti:O=0.48±0.07 Ti:O=0.46±0.06 Figure 4-10: EEL spectra indicate that the majority of Ti ions remain in the 4+ oxidation state at the grain boundary. The spectra were quantified to calculate the titanium to oxygen ratios. As shown in Fig. 4-11, within the experimental error no changes in the Ti:O atomic ratio were detected at the 2 nm scale across the boundary, which agrees with the EDX line profile as shown in Fig. 4-8.

115 103 Ti:O atomic ratio Distance (nm) Figure 4-11: Typical EELS profile across Y-doped TiO 2 grain boundary shows no spatial change in the Ti:O atomic ratio EEL spectra from TiO 2, Ti 2 O 3 and TiO Because TiO 2 has the tendency to reduce to TiO 2-x under reducing atmosphere, it is worthy to obtain reference EEL spectra from titanium oxides at lower valence state, i.e., Ti 2 O 3 and TiO, to make sure our EELS experiments didn t significant change the valence state of our samples since high-energy electron beam can also reduce titanium dioxide. When the beam is placed steadily on a spot of the TiO 2 sample for an extended period of time, TiO 2 gradually loses oxygen and transforms to amorphous Ti 2 O 3 and TiO.

116 104 A series of spectra were recorded at different time spots during the reducing process, which were quantified to obtain the Ti:O ratios. The spectra that gave the closest ratios to 2:3 and 1:2 presented the EEL spectra of Ti 2 O 3 and TiO. Resulting spectra are shown in Fig along with a TEM image showing a burning mark after the sample was reduced by electron beam. The fine splits in the ELNES of both Ti L 2 and L 3 edges started to disappear when Ti valence state fell from 4+ to 3+ and vanished completely at 2+. Similar observations were recognized for the O K edge. Significant chemical shit associated with the Ti L 2,3 edges were also identified during the reduction. Intensity (a.u.) TiO 2 Ti 2 O 3 TiO Energy Loss Figure 4-12: Typical EEL spectra obtained from crystalline TiO 2 and amorphous Ti 2 O 3 and TiO, with inset of TEM image showing beam damage effect.