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1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2011 Phase Field Modeling of Microstructure Evolution in Thermal Barrier Coating Systems Karim Ahmed Follow this and additional works at the FSU Digital Library. For more information, please contact

2 THE FLORIDA STATE UNIVERSITY THE GRADUATE SCHOOL PHASE FIELD MODELING OF MICROSTRUCTURE EVOLUTION IN THERMAL BARRIER COATING SYSTEMS By KARIM AHMED A Thesis submitted to the Graduate School in partial fulfillment of the requirements for the degree of Master of Science Degree Awarded: Fall Semester, 2011

3 Karim Ahmed defended this thesis on October 27, The members of the supervisory committee were: Anter El-Azab Professor Directing Thesis Anke Meyer-Baese Committee Member Sachin Shanbhag Committee Member Xiaoqiang Wang Committee Member The Graduate School has verified and approved the above-named committee members, and certifies that the thesis has been approved in accordance with university requirements. ii

4 To my family and friends. iii

5 ACKNOWLEDGEMENTS I would like to acknowledge the incredible support I had throughout my graduate studies at FSU. First of all, I would like to express my sincere gratitude to Anter El-Azab, my advisor, for the chance to work on such a challenging project. His support, patience, and guidance are appreciated. I would like to thank all the members of computational materials science group at FSU for helpful discussions and good times. Finally, I would like to thank my committee members, Prof. Meyer-Baese, Prof. Shanbhag, and Prof. Wang for devoting some of their valuable time for reviewing this work. iv

6 TABLE OF CONTENTS List of Tables... vi List of Figures... vii Abstract...x 1. INTRODUCTION MICROSTRUCTUE ASPECTS OF THERMAL BARRIER COATING SYSTEMS TBC Systems Manufacturing of TBC Systems TBC Performance Microstructure Evolution of TBC Systems PHASE FIELD MODELING OF SINTERING IN TBC SYSTEMS An Introduction to Phase Field Modeling of Microstructure Evolution Phase Field Modeling of Sintering in Polycrystalline Materials Determination of Model Parameters Numerical Implementation RESULTS AND DISCUSSION Concurrent Densification and Grain Growth Sintering of Idealized Columnar TBC Structures Simulating Actual TBC Microstructures Simulations of 3D Test Cases CONCLUSION AND FUTURE WORK...60 REFERENCES...63 BIOGRAPHICAL SKETCH...71 v

7 LIST OF TABLES 1 Physical properties of YSZ Model parameters set for YSZ Modified model parameters set for YSZ ( 1000 ) Reference scales (YSZ) Non-dimensionalized model parameters...38 vi

8 LIST OF FIGURES 1 TBC structure indicating the underlying microstructure of an electron-beam physical vapor deposited TBC (taken from Ref [1]) Plot of the thermal expansion coefficient vs. the thermal conductivity showing YSZ as the best candidate coating material for nickel-based substrate (taken from Ref [7]) Lifetime of TBC vs. its composition (taken from Ref [5]) Yttria-Zirconia phase diagram (taken from Ref [7]) Schematic illustration of an air plasma spray (APS) system (taken from Ref [71]) Cross-section of a TBC produced by air plasma spray (APS) showing the plate-like pore morphology (taken from Ref [6]) Schematic illustration of an electron-beam physical vapor deposition (EBPVD) system (taken from Ref [72]) Cross-section of a TBC produced by electron-beam physical vapor deposition (EBPVD) showing the columnar structure of the ceramic top-coat (taken from Ref [6]) Schematic illustration of a zigzag structure produced by rotating the substrate during deposition (taken from Ref [7]) Spallation of TBC ceramic top-coat (taken from Ref [6]) Microstructure evolution of a columnar TBC structure produced by EBPVD (taken from Ref [6]) Evolution of Young s modulus of EBPVD coating with time at temperature 1473K (taken from Ref [89]) Evolution of thermal conductivity of APS coating with time at different temperatures (taken from Ref [90]) Sharp interface (a) vs diffuse-interface (b) (taken from Ref [71]) Schematic illustration of (a) phase field variables and (b) their change across the interfaces (e.g., free surfaces and grain boundaries) (taken from Ref [136]) vii

9 16 Schematic illustration of a flat grain boundary between two semi-infinite grains Schematic illustration of the diffuse interface width Schematic illustration of a flat free surface between solid and vapor Strain generated in an epitaxial thin film system (taken from Ref [175]) Snapshots of the evolution of a sinusoidal surface with short wavelength; perturbation damps Snapshots of the evolution of a sinusoidal surface with long wavelength; perturbation grows Evolution of strain energy, surface energy, and total free energy with time for a sinusoidal surface with (a) short wavelength (b) long wavelength Snapshots of the evolution of field for two circular grains with different crystallographic orientations; m: minute h: hour Evolution of (a) the area of grains and (b) the total surface, grain boundary, and excess free energies Effect of (a) grain boundary migration mobility and (b) curvature on the kinetics of grain growth Snapshots of the evolution of density field of an idealized columnar TBC structure Snapshots of the evolution of orientation field of an idealized columnar TBC structure Evolution of the porosity in an idealized columnar TBC shown in figures (26 and 27) Snapshots of the evolution of field for large diameter columnar TBC grains Effect of columns width on the sintering of columnar TBC grains Effect of columns heighet on the sintering process of TBC columnar grains Effect of initial separation distance between columns on the sintering of TBC columnar grains Evolution of an idealized zigzag structure with short wavelength Evolution of an idealized zigzag structure with long wavelength Effect of strain on the sintering of TBC...54 viii

10 36 Steps for constructing the computational domain from actual TBC structure Evolution of an actual TBC structure Snapshots of the evolution of orientation field for two spherical grains Snapshots of the evolution of density field for a 3D columnar TBC structure Snapshots of the evolution of orientation field for a 3D columnar TBC structure...59 ix

11 ABSTRACT The development of robust thermal barrier coating (TBC) systems is crucial in many high-temperature applications. The performance of a TBC system is significantly limited by microstructural evolution mechanisms, such as sintering at elevated temperatures. Sintering reduces the porosity of TBC and makes it denser which eventually increases the thermal conductivity and reduces the strain compliance of TBC. Understanding how sintering proceeds in TBC systems is thus important in improving the design of such systems. An elaborate phase field model was developed in order to understand the sintering behavior of columnar TBC structure. The model takes into account different sintering mechanisms, such as volume diffusion, grain boundary diffusion, surface diffusion, and grain boundary migration, coupled with elastic strain arising from the thermal expansion mismatch in thermal barrier coating system. Direct relations between model parameters and material properties were established. Such relations enable us to quantitatively study the sintering process in any material of interest. For example, here the simulations were conducted for Yttria Stabilized zirconia (YSZ) since it is the most common TBC material. The model successfully demonstrates a strong dependence of the sintering in TBC on the initial morphology and dimensions of coatings. Furthermore, by studying actual representative TBC microstructures by means of image processing techniques, the model was able to capture the concurrent densification (e.g., coating shrinkage) and coarsening (e.g., concurrent pore and grain growth). These results are important since they may be used to tailor TBC microstructures with high sintering resistance. Simple 3D examples are given for the sake of demonstrating the fact that expanding the model into the general 3D case is straightforward. x

12 CHAPTER ONE INTRODUCTION Thermal barrier coatings (TBCs) are commonly used to provide thermal, oxidation, erosion, and hot corrosion protection to metallic components from the hot gas stream in gasturbine engines used for aircraft and marine propulsion and in power generation [1-11]. The use of TBC systems has enabled modern gas-turbine engines to operate at higher temperatures, and thereby improving the engine efficiency and performance. TBC systems have thus become an indispensable part of the design of structural metallic components for high temperature environments. The efficiency, performance, and durability of these coatings are strongly affected by microstructural evolution processes that can be triggered at high temperatures [1-11, 84-87], a particular process of which is the sintering [96-110]. The sintering process is usually driven by reducing the free energy of materials, and it is mainly carried out by surface and grain boundary diffusion. Sintering reduces the porosity of TBC and makes it denser which inevitably increases the elastic modulus and the thermal conductivity of the TBC layer [88-95]. The modulus increase reduces the strain tolerance, which is detrimental to the entire TBC system, while the increase in conductivity lowers the ability of the TBC system to protect the alloy substrate. So studying the sintering process will pave the way to produce a robust TBC, which is the main motivation behind the present study. Most of the existing models that were developed to study the sintering in TBC systems [ ] are based on the varitional principles [ ]. In such models, simple neck geometries are assumed and kinetic evolution equations of the parameters describing these geometries are derived based on energy minimization concept. Nevertheless, these models have two evident limitations that need to be addressed. First, these models assume simple geometries that do not represent the actual microstructure of TBC. Second, these models are not able to study the sintering of initially separated grains like the case of columnar TBC structure produced by electron beam physical vapor deposition (EBPVD). These limitations can be alleviated by using phase field models. 1

13 Phase field models are powerful and popular techniques that are widely used for simulating microstructure evolution in materials [131,132]. They were successfully applied to model various microstructural evolution processes, such as solidification, solid-state phase transformations, martensitic transformations, solid-state sintering, precipitate growth and coarsening, crack propagation, grain growth, and electromigration [ ]. The appeal of phase field models stem from the diffuse-interface description of boundaries between different phases. This description allows us to capture the evolution of complex microstructures without explicitly tracking moving boundaries. Few phase field models were developed to study the sintering in solid materials [ ]. Nevertheless, all these models are qualitative in nature, i.e., model parameters were arbitrarily selected and kinetic equations were solved in non-dimensional forms. Here we present an elaborate phase field model of sintering that is able to quantitatively capture the microstructure evolution in polycrystalline materials. The model parameters were uniquely determined by establishing direct relations between the model parameters and material properties. The model takes into consideration multiple sintering mechanisms, including surface diffusion, grain boundary diffusion, volume diffusion, and grain boundary migration. Furthermore, the model accounts for the elastic strain effects, which arise because of the thermal expansion mismatch between the coating and substrate, on the sintering process in TBC. By studying simple cases, such as the sintering of spherical grains, the model was able to capture the basic features of sintering in polycrystalline materials. One of such features is the concurrent densification and grain growth that takes place in such materials. The model was then applied to study the sintering in columnar TBC structures. At first we used idealized configurations such as cylindrical grains to represent the columnar structure of TBC produced by EBPVD. The idealized configurations successfully captured important characteristics of the sintering in such systems, such as the strong dependence of the sintering rate on the initial morphology and dimensions of coatings. Such results are important since they may be used to tailor TBC microstructures with high sintering resistance. We then used image processing techniques to simulate actual TBC microstructures in order to add more insight to the understanding of sintering process in real systems. Simulations of actual structures showed that in addition to densification, the coatings experience coarsening (i.e., concurrent pore and grain growth) as well. This result tells us that a reliable model for studying the sintering in TBC must 2

14 be able to take into account these concurrent processes because otherwise the densification will be exaggerated. This thesis is organized as follows. In chapter two, we provide the reader with the general features of thermal barrier coating systems, such as the structure of coatings, the requirements for material selection, the common processing methods, the degradation mechanisms, etc. Chapter three gives first a brief background about the phase field method and then it elaborates the developed phase field model of sintering in TBC. In chapter four, we present and discuss the results obtained from the proposed model. Finally, chapter five summarizes the work, highlights the key results, and gives suggestions for future work. 3

15 CHAPTER TWO MICROSTRUCTURE ASPECTS OF THERMAL BARRIER COATING SYSTEMS This chapter provides the reader with the general features of thermal barrier coating systems (TBCs). Section 2.1 gives a detailed description of the structure of the TBC system, its benefits, and the criteria and requirements of materials selection for the TBC. Section 2.2 presents the different processing methods for manufacturing the TBC and the different microstructure features of TBC produced by the most common methods. Section 2.3 briefly discuses the performance of TBC highlighting the key factors that affect the lifetime of TBC systems and the common degradation mechanisms observed in typical TBC systems. Finally, section 2.4 is devoted to present the microstructure evolution of TBC systems during operation with much attention given to the sintering behavior of TBC since it is considered as the process by which most of the TBC degradation mechanisms are initialized especially at high enough temperatures. 2.1 TBC Systems Thermal barrier coatings are advanced material systems that are widely used in aero, marine, and industrial gas turbine, gasoline, and diesel engines [1-11]. Foremost of the benefits of the TBCs is the extension of lifetime of expensive substrate super alloys by protecting them from high temperature, rapid temperature transients, oxidation, erosion and corrosion [1-11]. The use of TBCs allows huge reductions in the surface temperature (100 to 300 C) of super alloys [1]. This temperature reduction has enabled us to operate modern gas-turbine engines at gas temperatures higher than the melting temperature of the super alloys, and hence improving the efficiency and performance of the engine. A specific study of a diesel engine [11] showed a significant effect of the TBC on the fuel economy and engine performance. The specific fuel consumption was decreased by 20% and the engine power increased by 8%. TBC could also be 4

16 used to decrease the internal cooling requirements for turbine blades and vanes. It was reported that a TBC of 125 micrometer thickness was able to reduce blade cooling requirements by 36% [2] Structure of a Typical TBC System A typical TBC system consists of four layers (see Fig1) i) the nickel or cobalt-based super alloy substrate; ii) the metallic bond-coat layer of µm in thickness made of platinum or nickel aluminide or of MCrAlY alloy (M = Ni, Co, Fe). The primary goal of this layer is to protect the substrate from oxidation that could happen because of oxygen diffusion from the ceramic top coat during service operation. So it is engineered to ensure that the thermally grown oxide (TGO) layer forms as α-al 2 o 3 which has a very low oxygen diffusivity so its growth is slow, uniform, and defect-free; iii) the thermally grown oxide (TGO), mainly α- Al 2 o 3, with a thickness ranging from 1 to 10 µm, which is created by the oxidation of the bondcoat during ceramic top-coat deposition and grows during service operation; and iv) the thermal insulator ceramic top-coat with µm in thickness, where 6 8 wt% of yttria-stabilized zirconia (YSZ) layer is the most common used [1-11]. In addition to the major reduction of heat transferred into the substrate, the YSZ top-coat improves the substrate hot corrosion resistance since it has a limited solubility to molten salts and sulfur oxides [2]. The hot corrosion resistance is crucial in aggressive environments such as the ones encountered in marine applications where efficiency and performance of gas turbine engines are known to depend not only on the super alloys high temperature strength, but also on their hot corrosion resistance [12-20]. Figure 1: TBC structure indicating the underlying microstructure of an electron-beam physical vapor deposited TBC (taken from Ref [1]). 5

17 2.1.2 Materials Selection for TBC A robust TBC must satisfy some principal requirements including low thermal conductivity to decrease the heat transferred into the substrate, high melting point to withstand the hot gases temperature, chemical inertness, no phase transformation between room temperature and operating temperature, and excellent adherence to the underlying substrate. High strain compliance is also required in order to accommodate the stresses arising from the thermal expansion mismatch between the coating and the underlying super alloy. Additionally, low sintering rate due to the fact that the pores reduce the thermal conductivity and increase the strain tolerance of the coating, Furthermore, for rotating components such as blades, low density is desirable to reduce centrifugal loads [1, 6]. YSZ is the most widely employed thermal barrier coating [1-10] due to the fact that it simply fulfills almost all the requirements for a robust TBC mentioned above. it has excellent mechanical, chemical and thermal properties, it is considered to be one of the lowest ceramic thermal conductors at high temperature ( 2.3 W/m K at 1000 C) (see Fig 2) because of its high concentration of point defects (e.g. oxygen vacancies and substitutional solute atoms) that scatter heat-conducting phonons (lattice waves). YSZ also has a high thermal expansion coefficient that is close to that of nickel-based super alloys (see Fig 2) which reduces the stresses that could arise from the thermal expansion mismatch between the ceramic top-coat and the underlying substrate. As early as mid-1970s [5], it was reported by the National Aeronautics and Space Administration (NASA) that the addition of 6-8wt% of Y 2 O 3 to the ZrO 2 matrix gives the optimum performance of a TBC for advanced gas turbine applications (see Fig 3). The high performance of TBC at such concentration of Yttria was later attributed to the formation of the metastable tetragonal zirconia phase known as t -tetragonal (see Fig 4). This particular phase is desirable since it yields a sophisticated microstructure (containing twins and antiphase boundaries) that has good crack propagation resistance [7], and it does not undergo martensitic transformation that is usually accompanied by volume change which is detrimental to the integrity of the TBC system [1,6,7]. Despite the success of YSZ, many studies have been conducted in order to develop advanced thermal barrier coatings with even lower conductivity and higher thermal stability. The studies showed that there are three possible ways for achieving that goal. The first is by modifying the current YSZ coating microstructures and porosity [21-25]. The second is by using 6

18 alternative oxide ceramic compounds [26-32]. The third is by using the concept of multicomponent doping, where multi-component paired rare-earth oxide cluster dopants were added to conventional YSZ oxide systems [33-36] in order to increase the percentage of point defects or even create large immobile defect clusters which will eventually lead to lower thermal conductivity and higher sintering resistance. For further information about these methods, we invite the readers to check the references we mentioned in the text since the details of these different methods could be the subject of another thesis. Figure 2: Plot of the thermal expansion coefficient vs. the thermal conductivity showing YSZ as the best candidate coating material for nickel-based substrate (taken from Ref [7]). 7

19 Figure 3: Lifetime of TBC vs its composition (taken from Ref [5]). Figure 4: Yttria-Zirconia phase diagram (taken from Ref [7]). 8

20 2.2 Manufacturing of TBC Systems TBC systems may be produced by various methods such as air plasma spray (APS) [37-47], electron beam physical vapor deposition (EBPVD) [48-55], solution precursor plasma spray (SPPS) [56-60], conventional chemical vapor deposition (CVD) [61-66], and enhanced chemical vapor deposition (ECVD) [67-70]. In addition to the feasibility of economical production, the microstructure features of the produced TBC are taken into consideration in selecting one method over another due to the fact that the microstructural features of the TBC (e.g. porosity content and pore morphology) affect its performance and durability. Since covering even the basic details of these different methods is beyond the scope of the thesis, we limit ourselves to present a brief review of the features of first two processing methods which are the most common methods used to produce TBC systems [1-8] and encourage interested readers to view the references for further details Air Plasma Spray (APS) Method In this process the material to be deposited (typically as a powder) is introduced into the plasma plume emanating from a plasma torch (see Fig 5). In the plume, where the temperature is on the order of 16,000K, the material is melted and propelled towards a substrate to build up the coating [2, 4, 37-39]. There are a large set of technological parameters that may affect the interaction of the particles with the plasma plume and the substrate and therefore the deposit properties. Such parameters include, powder injection location and angel, powder particle size and distribution, plasma gas composition and flow rate, energy input, torch offset distance, and substrate cooling [4, 42-47]. In addition to the feasibility of economical production [4], the microstructure of TBC produced by APS has a plate-like porosity (see Fig 6) that is the pores are parallel to the substrate (normal to the heat flux). This pore morphology significantly decreases the heat transferred into the substrate leading to an effective thermal conductivity of the coating of W/m k, which is much less than the YSZ bulk theoretical thermal conductivity W/m k [6-8]. On the other hand, this pore morphology tends to decrease the strain compliance of the coatings which makes APS coatings more suitable for industrial applications rather than aerospace or marine applications. 9

21 Figure 5: Schematic illustration of an air plasma spray (APS) system (taken from Ref [71]). Figure 6: Cross-section of a TBC produced by air plasma spray (APS) showing the plate-like pore morphology (the dark veins in the center of the micrograph) (taken from Ref [6]). 10

22 2.2.2 Electron Beam Physical Vapor Deposition (EBPVD) Method In this method, the power to heat and evaporate a ceramic material (usually in the form of ingots, see Fig 7) is provided by high-energy electron beam guns [2, 4, 48-50]. As the beam of electrons bombard the surface of the ingot, the surface temperature of the ingot increases leading to the formation of liquid melt. Then the liquid ingot material evaporates under vacuum and travels along the line of sight to the substrate where it condenses and forms the coating. The microstructure of TBC produced by EBPVD has characteristic columnar grains (see Fig 8) with micrometer scale elongated pore channels separating the grains and nanometer scale porosity within the grains [1, 2, 4-8, 48-50]. This pore morphology does not allow much reduction of heat transferred into substrate since the pores are aligned normal to the substrate (parallel to the heat flux) which is the reason why TBCs produced by EBPVD have higher effective thermal conductivity than their APS counterparts [1, 6-8]. On the other hand, thanks to their columnar structure TBCs produced by EBPVD have much higher strain tolerance than their APS counterparts which makes them more durable and qualifies them to operate in harsh environments like the ones encountered in aerospace and marine applications [1, 6-8]. Since the TBC microstructure and pore morphology have significant effect on the coating thermal conductivity [6-8], one may propose to alter the morphology of the TBCs produced by EBPVD in order to decrease their effective thermal conductivity. In principle this is possible since it is well known that The morphology of vapor deposited coatings may be controlled by a large set of variables including adatom kinetic energy [73, 74], adatom angle of incidence [75, 76], substrate temperature [77, 78], deposition rate [77, 79], the presence and nature of the surrounding gas [77, 80, 81], elemental compositions of the adatoms [82], and substrate roughness [83]. Unfortunately, controlling many of these variables in EBPVD system is often limited by other engineering aspects of coating design [7]. Nevertheless, engineers were able to produce EBPVD TBCs with lower thermal conductivity by controlling just one variable, namely, the angle of incidence. By allowing the substrate to rotate during the deposition [7, 8], we can control the angle of incidence and hence change the TBC morphology. A TBC with Zigzag structure (see Fig 9) produced by that technique has 40% less thermal conductivity than the one produced by usual EBPVD [8]. The lower thermal conductivity of the Zigzag structure is attributed to two reasons. First, the fact that the columns and hence the elongated pores 11

23 separating the columns are now inclined to the substrate makes them able to reduce the heat transferred into substrate [7, 8]. The Second reason is the low sintering rate and hence high sintering resistance of such structure. Figure 7: Schematic illustration of an electron-beam physical vapor deposition (EBPVD) system (taken from Ref [72]). 12

24 Figure 8: Cross-section of a TBC produced by electron-beam physical vapor deposition (EBPVD) showing the columnar structure of the ceramic top-coat (taken from Ref [6]). Figure 9: Schematic illustration of a zigzag structure produced by rotating the substrate during deposition (taken from Ref [7]). 13

25 2.3 TBC Performance Multiple factors may impact the performance and durability of thermal barrier coatings. These factors include composition, microstructure, density, thickness, oxidation resistance of the bond-coat, and thermal expansion mismatch between the top-coat and substrate. Many investigations were conducted to determine the principal degradation mechanisms that control the lifetime of TBC systems [1-8]. Such investigations showed that oxidation of the bond-cot, spallation of the ceramic top-coat, and sintering of the ceramic top-coat are the main degradation mechanisms that lead to the failure of TBC systems Degradation Mechanisms of TBC Systems Oxidation mechanism. As we mentioned earlier the oxidation resistance of TBC systems is provided by the metallic bond coat which is tailored to form adherent aluminum oxide layer that has very slow growth rate, thereby inhibiting additional oxidation [1-8]. So if the concentration of aluminum at the bond-coat surface falls below a critical value, aluminum oxide will no longer be the thermodynamic preferred phase and other oxides may form. The other oxides do not form such a protective layer and hence the alloy oxidizes faster. That made early investigations in the field [2, 3] to link between the TBC failure and the oxidations of the bond-coat and suggest oxidation criteria as basis for the prediction of the TBC average lifetime [3, 6]. Nevertheless, the investigations failed to explain the wide distribution in failure lives [6] Spallation mechanism. Another studies suggested the spallation mechanism (a mechanism in which the top part of coating buckles and spalls away from the underlying super alloy, see Fig 10) as the life-limiting factor for TBC systems [6,84-87]. This spallation takes place because of the compressive stresses developed at the interface between the TBC and the underlying metal substrate due to the thermal expansion mismatch. Considering the spallation as the main failure mechanism in TBC systems can indeed interpret the wide distribution of TBC failure lives. For example, the fact that the EBPVD coatings are more durable than their APS counterparts [1, 6-8] can now be attributed to the fact that the columnar structure of the TBC produced by EBPVD has higher strain compliance than the lamellar structure of the TBC produced by APS and hence it has higher spallation resistance and longer lifetime. Sintering. Sintering of the ceramic top-coat is also an important factor that affects the performance and durability of TBC systems. Similar to porous ceramic materials, one can expect 14

26 that the ceramic top-coat of TBC will densify in order to decrease the surface energy associated with the excess surface area of the pores. This densification behavior was shown to alter the TBCs mechanical and physical properties [88-95]. Many investigations showed that elastic modulus [88-93], strength [88-89], and work of fracture [89] of as-sprayed TBCs increased because of sintering of coatings at high temperatures. Thermal conductivity has been reported to increase as well [88, 90, 93-95]. The increase in the coatings mechanical properties mentioned above inevitably decreases their strain tolerance which will ultimately decrease their spallation resistance and shorten their lifetime. The increase in the coating thermal conductivity will defeat the purpose of using the coating as a thermal barrier. Figure 10: Spallation of TBC ceramic top-coat (taken from Ref [6]). 15

27 2.4 Microstructure Evolution of TBC As we mentioned above, oxidation of the bond-coat, spallation of the ceramic top-coat and sintering of the ceramic top-coat are considered as the main degradation mechanisms that control the TBC lifetime. Although Spallation of the ceramic top coat dominates the life of the current TBC systems [1-6], there are strong indications [96-103] that spallation is related to the sintering and stiffening of the ceramic top-coat. Furthermore, it is expected that the sintering and densification of the ceramic top-coat to control the life of future coatings for higher temperature usage [6]. Since our primary goal in this thesis is to model the sintering process in TBC systems, we limit ourselves here to briefly review the experimental facts about the sintering of TBC systems. At the end of the section we also briefly discuss the modeling techniques presented in literature to simulate the sintering process in TBC systems Experimental Facts of TBC Sintering Several experimental studies were conducted to study the sintering of TBC at high temperatures [ ]. Sintering in general is driven by the reduction of the surface energy and it is mainly carried out by diffusion. In a particular study of coatings produced by EBPVD [102], it was shown that the as-deposited feathery morphology (see Fig 11) of individual columns smoothes out quickly by surface diffusion. Then necks form between the closely-spaced columns which will generate force that tends to pull the columns together and hence causes coating densification. The densification will increase the elastic modulus of coatings (see Fig 12) and hence decrease their strain tolerance. Furthermore, decreasing the volume fraction of pores will inevitably increase the coatings thermal conductivity (see Fig 13) and hence reduce their ability to protect the underlying super alloys. Figure 11: Microstructure evolution of a columnar TBC structure produced by EBPVD (taken from Ref [6]). 16

28 Figure 12: Evolution of Young s modulus of EBPVD coating with time at temperature 1473K (taken from Ref [89]). Figure 13: Evolution of thermal conductivity of APS coating with time at different temperatures (taken from Ref [90]). 17

29 2.4.2 Modeling of Sintering Process in TBC Systems Modeling the sintering process is a classical problem that has been widely studied, starting from the classical work of Herring [111] who formulated the scaling laws in Scaling laws consider the effect of change in scale (e.g., the particle size) on the sintering rate regardless of the specific geometry of the problem (e.g. the particle shape, the neck shape). Other early work [ ] developed analytical models that relate the neck radius and shrinkage to transport parameters and geometrical variables. These models were restricted to the case where a single transport mechanism (e.g., surface/grain-boundary/lattice diffusion) dominates the sintering process. Furthermore, the analytical models were based upon greatly oversimplified geometry. Such limitations of analytical models made them suitable only for describing the early stage of sintering. Later, more sophisticated models for the sintering of spherical particles via two competing transport mechanisms were developed and their solutions were obtained by numerically solving appropriate partial differential equations formulated based upon mass conservation [ ]. More recently, attractive numerical methods based on varitional principles were developed [ ]. In these methods, relatively simple neck geometries are assumed and the time rate of evolution of the parameters describing these geometries is obtained from energy rate minimization. Most of the existing models that aimed to describe the sintering behavior of TBC [ ] are based on the varitional principles mentioned above. Nevertheless, such models may be divided into two approaches, namely, free-sintering approach and constrained-sintering approach. The former separates the ceramic coating from the substrate and investigates its sintering behavior as a free-standing piece of ceramic [126, 127]. The later takes into account the possible effects of the substrate on the sintering process such as the stresses that stem from the thermal expansion mismatch between the ceramic coating and the metallic substrate [ ]. Despite the relative success of these models, there are two principal limitations that need to be addressed. First, the models assume idealized geometries that do not represent the actual microstructure of the TBC. Second, these models are usually not able to simulate the sintering of well separated columns like the case of columnar TBC structure produced by EBPVD since some amount of overlapping (e.g. pre-sintered columns) must be assumed. The limitations of such models could be defeated by using phase field models which are able to deal with any complex geometry. 18

30 CHAPTER THREE PHASE FIELD MODELING OF SINTERING IN TBC SYSTEMS This chapter is devoted to present an elaborate phase field model of sintering in polycrystalline materials. Section 3.1 gives a general introduction to phase field modeling of microstructure evolution. Section 3.2 presents a detailed description of a proposed phase field model that is able to simulate the sintering process in TBC. Section 3.3 establishes a direct relation between the model parameters and thermodynamic and kinetic properties of the material under consideration. Section 3.4 first depicts the numerical scheme that was used to solve the equations, and then presents simple test cases for demonstration. 3.1 An Introduction to Phase Field Modeling of Microstructure Evolution Phase field methods have become important and powerful techniques for modeling microstructure evolution in materials at the mesoscale level [131, 132]. In contrast to the conventional sharp interface methods (see Fig 14), the interfaces between domains in the phase field methods are defined by a continuous variation of the properties (the phase fields) within a narrow region (the diffuse interface width). This diffuse-interface description is what gives the phase field method its appeal since the position of the interfaces is implicitly given by contour values of the phase fields and hence the explicit tracking of the interface position is no longer required. This characteristic of phase field methods has enabled us to simulate the microstructure evolution of complex structures that are typically observed in real systems. In addition to the success in simulating solidification [131] and solid-state phase transformations [132] processes, phase field models were successfully applied for modeling grain growth [133, 134], solid-state sintering [ ], dislocation dynamics [ ], crack propagation [143,144], electromigration [145], and vesicle membranes in biological applications [146, 147]. It was reported that the concept of diffuse-interface was first introduced by van der Waals more than a century ago [148]. He modeled a liquid-gas system using a density function that varies continuously across the interface. Approximately 50 years later, Ginzburg and Landau 19

31 used a complex order parameter and its gradient to study the transition from conductor to superconductor state in metals [149]. In late 1958, Cahn and Hilliard presented a general thermodynamic formulation for treating the thermodynamic properties of heterogeneous systems [150]. Their formulation automatically gives rise to diffuse interfaces between different domains and it is considered as the basic reference for most of existing phase field models. (a) Figure 14: Sharp interface (a) vs diffuse-interface (b) (taken from Ref [71]). (b) In general, one can divide phase field variables into two categories conserved and nonconserved variables. Conserved variables describe conserved quantities such as concentration, density, etc. Non-conserved quantities may represent different crystallographic orientations in polycrystalline materials, different spin directions in magnetic materials, etc. A typical phase field model starts with an expression for the total free energy functional of the system which may take the general form, where F is the total free energy of the system, F F Fint (1) bulk F LR F bulk is the bulk or chemical free energy which is the homogeneous free energy of the system if we neglect the interfaces, F int is the interfacial free energy of the system, and F LR is the long range interaction energy such as elastic or electrostatic energy. Such terms are usually written in terms of the phase field variables. Then the kinetic equations that describe the evolution of the phase field variables are derived according to 20

32 nonequilibrium (irreversible) thermodynamic principles which usually lead to Onsager-type kinetic equations [151]. 3.2 phase Field Modeling of Sintering in Polycrystalline Materials In order to fully resolve the microstructure of a polycrystalline material, we need to have both conserved and non-conserved phase field variables. The conserved variable represents the fractional density of the solid ρ(r,t) which gives the distribution of solid materials over the entire material domain. Hence the density field variable is taken to be 1 in the solid phase and 0 in the amorphous (e.g., pore/vapor) phase. Non-conserved phase field variables η α (r,t;α = 1,2,..., p) are used to distinguish different solid regions (e.g., grains with different crystallographic orientations), such that η α equals to 1 in α -th grain and 0 otherwise (see Fig 15). Since these variables vary slowly in bulk or pore regions and rapidly across the interface between these regions, their evolution gives the position of interfaces as a function of time, and hence, reveals the time evolution of microstructure. (a) Figure 15: Schematic illustration of (a) phase field variables and (b) their change across the interfaces (e.g., free surfaces and grain boundaries) (taken from Ref [136]). (b) 21

33 As we previously mentioned in order to derive the kinetic equations of phase field variables, the free energy of the system must be given in terms of these variables. Here the free energy functional of the system takes the form F f p , e d r, where the first term represents the bulk (homogeneous) free energy density. The gradient terms give the energy contribution from free surfaces and grain boundaries. The gradient coefficients are material constants. P is the total number of grains of different crystallographic orientations in the solid. The last term is included to describe the strain energy in TBC systems that stems from the thermal expansion mismatch between the ceramic top-coat and the underlying metallic substrate. The bulk (homogeneous) free energy density is the nonequilibrium free energy density that defines the homogeneous coexisting phases, e.g., the amorphous phase (pore or vapor) and the polycrystalline solid phase. A particular form for the bulk free energy can be given by a Landau-type polynomial expansion in the phase field variables [152]. Here we adopt a form that was suggested by Y.U. Wang [138] to simulate the sintering of powder compact, f, B 1 C (3) where B and C are constants. The Landau-type potential of equation (3) is so constructed that it gives rise to P+1 minima, i.e., this free energy form requires that the free energy is minimum in the amorphous (vapor/pore) phase and in each grain of the solid phase. The strain energy arising from the thermal expansion mismatch in the TBC system is given by the usual form, where and e 1 2 e 2 e e E E E ii ij ij are the two Lame constants which are taken here to be function of the density field in order to take into consideration the inhomogenity of the domain. strain tensor which has the form E e ij (2) (4) e E ij is the elastic E, (5) ij ij 22

34 where E ij is the total strain tensor and ij is the inelastic (stress-free) strain tensor. Please note that the famous Einstein summation notation is used to express the tensor equations in a convenient compact form. In order to calculate the strain energy, the strain tensor must be obtained first. This can be done by solving the mechanical equilibrium equation which states that in the absence of body forces the stress tensor must be divergence free, r 0 (6) Where the stress tensor is related to the elastic strain tensor by the generalized Hooke s law as ij e r c re r Here the fourth order elastic constant tensor expressed in terms of Lame constants as By substituting (5), (8) into (7), we get c ijkl (7) ijkl r ij kl 23 kl c ijkl r for elastically isotropic materials is (8) ik r E r r E r r jl 2 (9) ij For small strains, the total strain is related to the total displacement by, where u u E kk kk r u r u r ij i, j j, i ij il jk 1 (10) 2 i i, j and i,j= 1,2,3 (in 3D). By substituting (9), (10) into (6), we end up with x j u u u u u 2, (11) i, ij j, ii, i i, j j, i, j i, i ii, j ij, i which may be written for any curvilinear coordinate system as Where ii 2 u u u u T u tr ij ij 2 tr is the trace of the second order inelastic (Eigen) strain tensor, i.e., the summation of its diagonal elements, u T is the transpose of the second order tensor u. This equation is considered as a generalized Navier equation [153,154] where in addition to the first two terms that are typical in Navier equation of pure elastic medium, the third and fourth terms appeared to account for the heterogeneity of our domain. The right hand-side of equation (11or12) gives the source term for the stresses in the elastic medium which is here the inelastic (stress-free) strain arising from thermal expansion mismatch in TBC system. (12)

35 As we mentioned earlier, the elastic moduli (Lame constants) and the inelastic (Eigen/stress-free) strain are usually expressed in terms of the phase field variables [ ]. The elastic moduli here take the form, a s a (13) a s a (14) a where and a s are the elastic moduli of the amorphous (e.g., pore/vapor) phase, and the elastic moduli of the solid phase, and s are is an interpolation function that interpolates the elastic moduli and hence the stresses across the interfaces. Such function takes the form , (15) which gives the value of 1 in the solid phase ( 1) and the value of 0 in the amorphous (pore/vapor) phase ( 0). This specific form was shown [153] to reduce the elasticity part in the phase field model to its counterpart in the sharp interface model. This formulation which assigns small but non-zero values for the elastic moduli of the amorphous phase is adopted [154] in order to avoid numerical instabilities. The stress-free strain due to the thermal expansion mismatch is an equal biaxial strain (since there are no normal stresses through the thickness of the coating) in the solid phase which is expressed as, 0 0 r 0 0 (16) where the same interpolation function was used to assure that the stress-free strain vanishes in the amorphous phase. is a constant that characterizes the amount of Eigen (stress-free) strain in the TBC system which is given by here T, TBC SUB TBC and SUB are the thermal expansion coefficients of the TBC and substrate, respectively, and T is the temperature. Finally, kinetic equations for the phase field variables must be developed in order to reveal the microstructure evolution. Since the normalized density field is a conserved variable, it evolves according to the continuity equation which is written as, (17) 24

36 J t where J is the mass flux which is, according to irreversible thermodynamics, expressed as (18) J M, (19) here M is the chemical mobility and is a generalized chemical potential that represents the driving force for the evolution. The generalized chemical potential is the varitional derivative of the free energy functional with respect to the conserved phase field variable which gives, f, de F 2 The chemical mobility is related to the diffusion coefficient (D) by, d (20) D M m (21) RT where m is the molar volume, R is the universal gas constant, andt is the temperature. In order to describe different diffusion mechanisms such as surface, grain boundary, and lattice diffusion the diffusion coefficients must be formulated as a function of the phase field variables. Here we adopt the method by, Gugenberger et al, that was proposed to describe the surface diffusion in phase field models [156] and shown to recover the sharp interface limit. Following that method, we can describe the different diffusion mechanisms by using three diffusion tensors as follow here s D is the surface diffusion tensor, under consideration, and s s s D d T, (22) s d is the surface diffusion coefficient of the material s T is the surface projection tensor, which guarantees that the surface diffusion is confined along the surface, and it has the form, T s I n n (23) where I is the identity matrix, the symbol represents the dyadic product, and n s is the unit normal to the surface which is obtained by This form makes the surface diffusion tensor s s n s (24) s D anisotropic with zero eigenvalue in one of the three spatial eigen directions (the one perpendicular to the surface). Similarly, we can confine the 25

37 grain boundary diffusion along the grain boundaries by constructing the grain boundary diffusion tensor as where again gb D is the grain boundary diffusion tensor, coefficient of the material, and T gb gb gb gb D d T, (25) gb d is the grain boundary diffusion is the grain boundary projection tensor, which guarantees that the grain boundary diffusion is confined along the grain boundary of two different grains, and it takes the form, T gb I n n (26) gb gb where again I is the identity matrix, the symbol represents the dyadic product, and n gb is the unit normal to the grain boundary between two different grains which is obtained by [134] n gb The lattice diffusion tensor is constructed as, i j (27) i j D l l d I (28) where l D is the lattice (volume) diffusion tensor, material, I is the identity matrix, and l d is the lattice diffusion coefficient of the is a function which guarantees that the lattice diffusion exists only inside the solid phase and vanishes in the amorphous phase. Here it takes the form [138] (29) In the solid phase 1, this formulation assumes isotropic lattice diffusion since the lattice diffusion tensor is a diagonal matrix with a constant value l d on the diagonal. Finally, the total diffusion tensor (the one appears in equation (21)) is the sum of these different diffusion tensors, s gb l D D D D (30) The tensorial formulation of the diffusion coefficients and hence the chemical mobilities describes correctly the different diffusion mechanisms that are active during the sintering process, and therefore captures the right evolution of the microstructure. By substituting (19) and (20) into (21), we get F M, t (31) 26

38 which is also known as the Cahn-Hilliard non-linear diffusion equation [157]. By solving equation (31), the evolution of the conserved density field can be tracked. The evolution of the non-conserved phase field variables is governed by Allen-Cahn [158] (time-dependent Ginzburg-Landau [149]) structural relaxation equation which has the form, t F L f L & 1,2,... p (32), 2 where L is a constant (for isotropic materials) that characterizes the mobility of grain boundary migration. Allen and Cahn showed that the grain boundary migration mobility in diffuse interface can be related to its counterpart in sharp interface by expressing the velocity of interface in each model in terms of its parameters [158]. For example, in the sharp interface the interface velocity is given by where gb gb is the grain boundary mobility, gb gb 1 1 gb, (33) R1 R2 gb is the grain boundary energy, and R1 and R2 are the radii of curvature. In phase field models, the interface velocity is given by and hence, 1 1 gb L, (34) R1 R2 L gb gb (35) This direct relation between the phase field and sharp interface parameters is important since the sharp interface parameters are physical properties of materials that could be directly determined from experiments. By solving equation (32) simultaneously with equation (31), we can track the evolution of both the conserved and non-conserved fields and hence capture the microstructure evolution. Nevertheless, in order to get quantitative results of the microstructure evolution we still need to determine the parameters of the free energy functional in our model. In principle, the phase field parameters can be determined by investigating their equilibrium solution profiles for simple cases such as their equilibrium solution profiles across a flat interface. These investigations, as 27

39 we will show in details in the next section of this chapter, will lead to direct relations between the model parameters and material properties such as grain boundary energy and surface energy. 3.3 Determination of Model Parameters As we mentioned above, studying the equilibrium solutions of phase field variables could be used to fix the model parameters. Since we have two types of interfaces, namely, free surfaces and grain boundaries, we need to conduct this study on two steps. First we study the phase field profiles in equilibrium across a flat grain boundary, and then we repeat the study for a flat free surface Phase Field Profiles in Equilibrium across a Flat Grain Boundary Here we study the phase field profiles across a flat grain boundary at x=0 between two semi-infinite different grains with orientations i and j (see Fig 16). Across a grain boundary, the change in the density field is very small 1and can be neglected (review Figs 15 and 16). gb Hence by Following Cahn and Hilliard approach [150], the specific grain boundary energy is given by the integral, gb f di j 1,, dx i j 2 dx 2 2 d dx where for simplicity an isotropic grain boundary energy was assumed (i.e., this assumption lead to only one gradient coefficient). x is the coordinate perpendicular to the grain boundary, and f i j 1,, is the bulk free energy density (see equation (3)) at the grain boundary between grain i and grain j which reads, ,, C i j i j i j 2 (36) f (37) For a grain boundary in local equilibrium, the profiles x x and must adopt a shape which minimizes the functional (36) and satisfies the following boundary conditions (see Fig (16)) i 1 and 0 for x, (38a) i j 0 and 1 for x, (38b) i j j d dx i d dx j 0 for x (38c) 28

40 1 1 1 i 1 j 1 j 0 i 0 0 x Figure 16: Schematic illustration of a flat grain boundary between two semi-infinite grains. According to the principles of calculus of variations, the functions x x extremize the functional (36) must satisfy the Euler equations, namely, f f 1,, 1,, j i i i j j and that 2 d i 0, 2 (39a) dx 2 d j 0, (39b) 2 dx or, equivalently, the integrated equation (where the boundary conditions were taken into account) f 2 d dx di j 1,, 0 i j 2 dx Across a flat grain boundary, the following relation exists (see Fig 16), d j di (41) dx dx by integrating the above equation taking into account the boundary conditions, we simply get j x 1 x, i 2 i j (40) (42) which is readily seen from the above figure. This immediately modifies equation (41) to 29

41 which also gives, d j di, (43) dx dx d j d i 1 Rearranging Eq (40) taking into account the above equations (42-44) and the boundary conditions, gives us d dx i 1, i, j, (44) f (45) d j f 1, i, j dx Now if we substitute by Eq (40) into Eq (36), we get but from Eq (42), we have x 1 x skipping some algebraic steps, we end up with, f 1,,, (46) gb 2 f dx (47) j i i j, so by plugging this relation into Eq (37) and 2,, 1 f 12C 2 1 (48) 1 i j i i i i Substituting by Eq (45) and Eq (48) into Eq (47) gives, gb C i i i d C Equation (49) establishes a direct relation between some of the model parameters and a material property which is the specific grain boundary energy. Nevertheless, we still need to develop other relations to fix all the parameters. Another relation that we can develop is the relation between the physical grain boundary thickness and the diffuse interface width across a flat grain boundary since in principle they are the same. The diffuse interface width across a flat grain boundary can be estimated from the model parameters by (see Fig 17), d j dx x0 tan 1 (49) (50) 30

42 1 j x Figure 17: Schematic illustration of the diffuse interface width. But from Eqs (46), (48) and the figure above, we have and then, d j dx f i 0.5 3C 4 x0, (51) where 4 gb, 3C l (52) gb l is the grain boundary thickness of the material. Equation (52) provides another relation between the model parameters and another material property which is the grain boundary thickness. There are still two parameters in the free energy functional to be determined, namely, B and. These parameters can be obtained from the phase field profiles in equilibrium across a flat free surface Phase Field Profiles in Equilibrium across a Flat Free Surface Here we study, without loss of generality, the phase field profiles across a flat free surface at x=0 between a semi-infinite solid grain and semi-infinite vapor phase (see Fig 18). In this case both fields change across the interface and hence the specific surface energy is calculated from the integral, 31

43 sf f d 2 dx 2 d 2 dx, dx 2 (53) Figure 18: Schematic illustration of a flat free surface between solid and vapor. Where f, is the bulk free energy density (see Eq (3)) when only one solid grain is present, f, B 1 C , (54) from the figure above, the following relation is valid which, after applying the boundary conditions (see Fig 18), gives and hence, d d, (55) dx dx x x, (56) d d (57) dx dx which can be readily seen from the figure above (in this particular case both fields are actually redundant). Substituting by Eq (56) into Eq (54) gives 2 2, f B 7C 1. f (58) 32

44 Then by following the same procedure conducted for the grain boundary energy calculations taking into account the above equations and boundary conditions, the surface energy can be derived as sf 2 B 7C (59) 6 Also by following the same steps that were taken to estimate the diffuse interface width across a flat grain boundary in terms of the model parameters, the diffuse interface width across a flat free surface can be calculated from where sf l 8 sf l, (60) B 7C is the physical free surface thickness of the material which is usually approximated as half of its grain boundary thickness, i.e., the free surface is considered as a one-sided grain boundary. The four equations, namely, (49), (52), (59), and (60) that were derived above relate the four unknown free energy functional parameters in the model to four known material properties. Hence the model parameters are uniquely determined Model Parameters Set for Yttria Stabilized Zirconia (YSZ) As we mentioned earlier in this thesis, YSZ is the most common TBC material. Hence the relations developed in this chapter are used to relate the model parameters to the YSZ physical properties. These properties are listed in table (1) where surface energy, grain boundary energy, and diffusion coefficients are given for a temperature of 1400 Celsius. Then the model parameters that were calculated based on such data set are listed in table (2). The sharp interface grain boundary migration mobility gb, which appears in Eq (35) that is used to calculate its diffuse interface counterpart, is related to the familiar rate constant of grain growth (k) by then equation (35) becomes, gb k, (61) 2 gb gb k L gb (62) 2 s Finally, the different chemical mobilities (e.g., surface M /grain boundary M gb l /lattice M ) are simply obtained from the corresponding diffusion coefficients by means of equation (21). 33

45 Table (1): Physical properties of YSZ Name of material property Value of material property [Reference taken from] SI unit sf Specific surface energy ( ) [159] 2 J / m Specific grain boundary gb energy ( ) Rate constant for parabolic grain growth (k) [159] 2 J / m [160] m 2 / s Molar volume ( m ) [160] m 3 / mol 9 First Lame constant ( ) [161] Pa (Pascal) Second Lame constant (Shear modulus) ( ) Thermal expansion coefficient ( TBC ) [161] Pa [7] K -1 Lattice diffusion coefficient l ( d ) Grain boundary diffusion gb coefficient ( d ) Surface diffusion coefficient s ( d ) Grain boundary thickness ( l gb ) [162] m 2 / s [162] m 2 / s 13 [126] m 2 / s [160] m 34

46 Table (2): Model parameters set for YSZ Model parameter Value SI unit B J / m C J / m J / m J / m l M M M gb s m / J s m / J s m / J s 5 L m / J s 3.4 Numerical Implementation This section is devoted to describe in details the implementation of the numerical scheme that was used to solve the kinetic equations of the phase field model. First, we discuss the problem of computational limitations on the system size which is a common problem in all phase field models that arises due to the fact that on one hand the physical interface width is usually in nanometers and on the other hand the system size of interest for mesoscale simulations is in micrometers. Then we present the methodology of the numerical scheme and how different parts (e.g., diffusion and strain energy) of the problem were treated. Finally, we present a simple test case to demonstrate the validity of the numerical scheme Increasing Length Scale of Phase Field Model Numerical stability requires, when solving the phase field kinetic equations, few mesh points to be located within the interfacial regions. Since the physical interface width is in nanometers, our grid spacing is limited to be around 1 nanometer. In this case mesoscale simulations with a system size of hundreds of microns are not affordable even with the use of state-of-the-art parallel computers. A remedy for this problem is to introduce an artificially diffuse interface at the length scale of interest by modifying the model parameters without 35

47 altering the thermodynamic driving forces or the kinetics [ ]. This can be done by manipulating some of the model parameters and modifying the rest accordingly. For example, it can be readily seen from equations (49, 52, 59, 60) that if we increase the gradient coefficients by times, i.e., and, and at the same time decrease the free energy density parameters by the same factor, i.e., B B / and C C /, the diffuse interface will increase by the same factor while the surface and grain boundary energies will be left unaltered. In order to preserve the kinetics, the kinetic parameters in the phase field kinetic equations must be modified accordingly. According to equation (20), if we decrease the free energy density by times, the generalized chemical potential which is the driving force for diffusion decreases by the same factor. Hence according to equation (31) the Cahn-Hilliard mobility must be increased by the same factor to reserve the same kinetics. On the contrary, according to equation (34) the velocity of an interface between two different grains depends only on the gradient coefficient. Hence if we increase by times, the Allen-Cahn mobility L needs to be decreased by the same factor in order to keep the interface velocity and therefore the kinetics unchanged. Here we use 1000 which scales up our mesh spacing from 1 nanometer to 1 micrometer. Table (3) lists the model parameters after modification. Table (3): Modified model parameters set for YSZ ( 1000 ) Model parameter Value SI unit B J / m C J / m J / m J / m l M M M L gb s m / J s m / J s m / J s m / J s 36

48 3.4.2 Numerical Scheme In solving the kinetic equations of phase field model, second order centered finite difference approximation was used for all spatial derivatives and an explicit forward Euler scheme was used for time derivatives. In order to determine the stable mesh size and time step, the kinetic equations were non-dimensionalized using reference energy, length, and time scales. The reference scales and their corresponding values calculated for YSZ are listed in table (4). In calculating the reference time scale, the surface mobility was used since the surface diffusion is the fastest diffusion mechanism. Table (4): Reference scales (YSZ) Reference parameter Form Value for YSZ (SI unit) Reference energy density ( ) Reference length scale ( ) Reference time scale ( 50 C J / m 3 25C t ) t M 2 s s 6 m By using the above reference scales, the non-dimensional forms of the kinetic equations may be written as F ~ ~ ~ ~, ~ ~ ~ 2, M f d e M M d (63) s F L 2 ~ f, M s L ~ ~. ~ 2 (64) 37

49 t ~ Where is the non-dimensionalized time, is the non-dimensionalized Del operator, * t ~ f, ~ e is the f, is the non-dimensionalized bulk free energy density, e non-dimensionalized strain energy. Table (5) lists the non-dimensional model parameters and their corresponding values calculated for YSZ. Table (5): Non-dimensionalized model parameters Non dimensionalized model parameter B ~ C ~ ~ ~ M ~ M ~ M ~ l gb s ~ B B ~ C 0.18 C ~ ~ ~ M l 2 M M 2 L ~ ~ L l s Form Value for YSZ (nondimensional) gb ~ gb M 0.02 s M M s ~ s M 1 s M L M s M 7 Since the Cahn-Hilliard partial differential equation is a fourth order while the Allen- Cahn equation is a second order, Cahn-Hilliard equation controls the upper limit for the time step 38

50 given the mesh size. Here we take the mesh size x 1micrometer. Hence the upper limit time step may be estimated by (see Eq (63)), ~ s d M ~ 1 ~ (65) 4 x 2 where ~ x x / 1is the non dimensional grid size and d is the dimension of the problem. Nevertheless, the time steps that were used here are usually taken as half of this upper limit since this formula was derived for linear partial differential equations. For 2D simulations, the stable time step was For 3D simulation a stable time step of 0.02 was used. Note that since the reference time scale was calculated to be 5 seconds for YSZ (see Table (4)), the physical time of the time step is simplyt t. Consequently, each time step corresponds to an actual physical time of 0.2 second for 2D simulations and 0.1 second for 3D simulations. The boundary conditions for the evolution equations are simply periodic boundary conditions in the horizontal direction and zero flux boundary conditions in the vertical direction. When the strain energy is present, the kinetic equations are solved in a staggered manner, i.e., we first obtain the displacement field from solving the modified Navier equation ( Eq (11)) given the phase fields and then add the elastic energy to the kinetic equation and update the phase fields. The underlying philosophy of this strategy is the fact that the mechanical equilibrium is reached much faster than the chemical equilibrium. The modified Navier equation may be solved using conjugate gradient [153], fixed iterative point schemes [154], or iterativeperturbation scheme [155]. Here we used the simple Jacobi iterative scheme since it is straightforward to include into the finite difference scheme. Following [154], small but non-zero values were assigned to the elastic moduli of amorphous phase in order to avoid numerical instabilities Test Case The familiar Asaro-Tiller-Grinfeld instability [166,167] may be studied in order to examine the strategy of the proposed staggered technique. Theoretical studies [ ] based on simple linear stability analysis of this problem have demonstrated that an initially flat surface of a stressed solid is unstable with respect to morphological variations in surface shape. Under specific conditions such morphological variations are able to drive the surface toward a shape with cusp-like surface valleys [ ]. An engineering application that is subjected to such phenomenon is epitaxial thin film. 39

51 A stress-free strain is usually developed in an epitaxial thin film system due to the lattice mismatch between the thin film and substrate (see Fig 19). In such system, the elastic strain energy and the surface energy are the dominant thermodynamic forces that may derive morphological evolution. The theoretical studies of this problem have shown that for a slightly perturbed sinusoidal solid surface with short wavelength, the chemical potential is higher at a surface peak than at a surface valley, and hence the process of surface diffusion tends to transport matter from peaks to valleys which smoothes out the perturbed surface. On the contrary, the chemical potential of a slightly perturbed surface with long wavelength is lower at a surface peak than at a surface valley, and hence the surface diffusion transports matter from valleys to peaks which magnifies the initial perturbation and roughens the surface. Figure 19: Strain generated in an epitaxial thin film system (taken from Ref [175]). This problem can be readily studied using the model developed in this chapter. The stress free strain is modeled by means of equation (16) then we solve Navier equation (Eq (11)) and couple the solution to the kinetic equations as explained above. Note that in this particular case we solve the kinetic equations in a non-dimensional form without specifying a particular material since the qualitative behavior of the process mentioned above is the same regardless of the material under consideration. As we expect, if we start our simulation with a perturbed sinusoidal surface with short wavelength (see Fig 20), the surface diffusion will smooth out the perturbation by transporting material from peaks to valleys. In this case the system increases its strain energy, but at the same 40

52 time it decreases its surface energy (i.e., by decreasing the surface area) with higher rate and hence the total free energy decreases with time (see Fig 22a). On the contrary, if we start with a perturbed sinusoidal surface with long wavelength (see Fig 21) the perturbation will grow by transporting material from valleys to peaks via surface diffusion. In this case the system increases its surface energy by increasing the surface area associated with the formation of cusped surfaces, but at the same time it decreases its strain energy (i.e., the cusped surface structure relives the stress along the surface) with higher rate and hence the total free energy decreases with time (see Fig 22b).. (a) t=0 (b) t=15000 (c) t=25000 (d) t=50000 (e) t=75000 (f) t= Figure 20: Snapshots of the evolution of a sinusoidal surface with short wavelength; perturbation damps. 41

53 (a) t=0 (b) t=50000 (c) t= (d) t= (e) t= (f) t= Figure 21: Snapshots of the evolution of a sinusoidal surface with long wavelength; perturbation grows. (a) Figure 22: Evolution of strain energy, surface energy, and total free energy with time for a sinusoidal surface with (a) short wavelength (b) long wavelength. (b) 42

54 CHAPTER FOUR RESULTS AND DISCUSSION In this chapter, we discuss the results obtained by applying the phase field model proposed in the previous chapter to the sintering process in thermal barrier coatings. Section 4.1 presents the basic features of sintering model by studying the sintering of two circular grains. Despite the simplicity of that problem, it shows an important characteristic of the sintering process in polycrystalline materials which is the competition between densification and grain growth. In section 4.2, we investigate the sintering of idealized columnar grains that were chosen to present the columnar grain structure of thermal barrier coatings (TBCs) produced by electron beam physical vapor deposition (EBPVD). Section 4.3 demonstrates the power of phase field model by simulating the sintering of an actual TBC microstructure obtained by incorporating a micrograph captured by scanning electron microscope (SEM) into the model by means of image processing techniques. Finally, in section 4.4 we give simple 3D examples for the sintering of spherical and cylindrical grains. All the simulations presented in this chapter were conducted based on the model parameters for Yttria Stabilized Zirconia (YSZ) calculated in chapter three. 4.1 Concurrent Densification and Grain Growth Studying the sintering of two circular grains with different radii and crystallographic orientations (see Fig 23) is a simple case that gives us a deep understanding of the sintering process in polycrystalline materials. If the two grains are close enough to each other, a neck will be formed via surface diffusion that is able to help the material to span the small distance between the grains. The two grains would like to from a neck and hence a grain boundary due to the fact that a grain boundary is less energetic than two free surfaces as long as the specific grain boundary energy is less than double the specific surface energy of the material. This is readily seen in figure (24b) where the system increases its grain boundary (i.e. by creating a grain boundary) but at the same time decreases with higher rate its surface energy and hence decreases its total excess free energy (i.e., the sum of grain boundary and surface energies). 43

55 (a) t=0 (b) t=5m (c) t=10m (d) t=45m (e) t=1.5h (f) t=10h (g) t=20h (h) t=25h Figure 23: Snapshots of the evolution of field for two circular grains with different crystallographic orientations; m: minute h: hour. (a) Figure 24: Evolution of (a) the area of grains and (b) the total surface, grain boundary, and excess free energies. (b) 44

56 Since the two grains have different radii, the formed grain boundary will be curved toward the small grain. Due to the chemical potential difference, a curved surface will move toward its center of curvature and hence the large grain grows while the small grain shrinks (see Fig 23 and 24). Figures (23 and 24a) capture an important characteristic of the sintering process in polycrystalline materials which is the fact that densification happens simultaneously with grain growth. This is evident in the figures by the concurrent increase of neck size (densification) and decrease of small grain area (grain growth). The area of grains and the different free energies could be easily calculated numerically from the contour values of the phase field variables. The area of a grain with orientation i is calculated from the integral, 2 r d r The total excess free energy is given by the integral, F f A i i (66) 2 (67) , d r 1 The total grain boundary energy is estimated from the integral F gb f 2, d r whenever 0, then the total surface energy is simply the difference between the total excess free energy and the total grain boundary energy. The factors that affect the interface velocity (grain boundary migration) and hence the grain growth kinetics are the gradient energy coefficient, the grain boundary migration mobility, and the mean curvature of the interface (see Eq (34)). The last two effects are illustrated in figure (25) where the shrinkage of the small grain is taken as an indicator for the kinetics of grain growth. In investigating the curvature effect, higher interface curvature was obtained by increasing the difference in the grains radii. The radius of large grain (30 microns) was kept the same while the radius of small grain was decreased from 20 microns to 15 microns. In investigating the effect of grain boundary migration mobility, the mobility was increased five times while all other model parameters were unaltered. All the results presented in this section are in agreement with the results obtained in a similar phase field model that was proposed to study the sintering of two unequal size grains [139]. (68) 45

57 (a) Figure 25: Effect of (a) grain boundary migration mobility and (b) curvature on the kinetics of grain growth. (b) 4.2 Sintering of Idealized Columnar TBC Structures The columnar grain structure of TBC produced by EBPVD techniques can be approximated by cylindrical grains as in figures 26 and 27. Figure 26 represents the evolution of the conserved density field where the cylindrical grains are attached to the substrate. Here we neglect the inter-diffusion that may happen between the substrate and columns (i.e., we assume both are from the same solid material) since the only goal of adding the substrate is to assure that these columns are constrained at the bottom. Since the non-conserved structural order parameters represent different grains with different crystallographic orientations, the grain boundaries between different grains and the equilibrium dihedral angel at the triple junctions can be visualized by plotting that field. A standard way of visualizing that field in a phase field model is by plotting the function, r p Since the substrate itself does not evolve with time, there is no need to assign an order parameter for it and hence it does not appear in figure 27. The relative porosity is taken here as a quantitative indicator of the densification of coating. The relative porosity between two representative columns i and j is estimated by 46 r 2 (69) 1

58 where gb A t A 1, (70) P t P t is the relative porosity, A gb t representative columns which is calculated from, and A is the grain boundary area between the two t r, t r, t d r (71) gb 2 i j A is the initial pore (vacant) area between the two columns which is simply the product of the horizontal separation distance between the columns and the height of columns. Equation (70) says that the relative porosity is inversely proportional to the grain boundary area between columns. Hence as a grain boundary forms and grows, the relative porosity decreases. As we can see from figures 26-28, the sintering process in columnar TBC grains may be divided to four stages. At the first stage, the sintering starts at the top of columns where the curvature is highest. The edges at the top of columns become rounded to decrease the surface area and hence the surface energy. No densification is pronounced during the first stage since the relative porosity is constant (see Fig 28). The relative porosity is constant due to the fact that no grain boundaries were formed during this stage. In the second stage, after the top columns become rounded the chemical potential gradient will derive material from top to bottom by diffusion. The material transported to the bottom starts to from grain boundaries between different grains and hence the relative porosity starts to decrease. In this stage the relative porosity decreases linearly with time (see Fig 28) as more material is transported from top of columns to bottom. The third stage begins when necks start to form at the top of columns. The necks form at the top of columns via surface diffusion that helps material to span the distance between columns. At this stage the relative porosity decreases exponentially with time (see Fig 28) because of the accelerated rate of the increase in grain boundary area between different columns after formation of necks at the top of columns (see Fig 27 d-f). Finally, the fourth stage starts after the accelerated increase of grain boundary area reach a steady state. This steady state is experienced after the formation of an equilibrium dihedral angle between different grains (see Fig 27f). The equilibrium dihedral angle depends on the ratio between the surface and grain boundary energies. The relative porosity is almost constant during this steady state stage (see Fig 28). 47

59 (a) t=0 (b) t=3h (c) t=10h (d) t=12h (e) t=15h Figure 26: Snapshots of the evolution of density field of an idealized columnar TBC structure. (a) t=0 (b) t=5h (c) t=8h (d) t=10h (e) t=12h (f) t=15h Figure 27: Snapshots of the evolution of orientation field ( ) of an idealized columnar TBC structure. 48

60 Figure 28: Evolution of the porosity in an idealized columnar TBC structure shown in figure 26. A comprehensive study of the effect of geometrical variables of the columnar TBC microstructure was conducted in order to develop a deep understanding of the sintering process in columnar TBC grains. The variables include the width (diameter) of columns, the height of columns, and the separation distance between columns. This study confirmed a strong dependence of the sintering process on the initial microstructure of coating. The effect of columns width (diameter) on sintering was investigated by studying two sets of columns with different diameters. The width of columns has a strong impact on the sintering as illustrated in figures 29 and 30. The increase of columns width retards the sintering process since it decreases its rate. As we can see from figure 30, the time at which neck formation at the top of columns starts is very sensitive to the columns width. All the four sintering stages are pronounced in the case of the columns with small width, while in the case of columns with large width the third stage which is characterized by the neck formation at the top of columns was not reached during the same simulation time. This could be attributed to the fact that the surface diffusion which is the mechanism that helps solid material to span the distance between columns and forms neck is curvature dependent. Hence columns with larger diameter will have lower curvature and therefore lower deriving force for neck formation at the top of columns. 49

61 (a) t=0 (b) t=8h (c) t=13h (d) t=26h Figure 29: Snapshots of the evolution of field for large diameter columnar TBC grains. Figure 30: Effect of columns width on the sintering of columnar TBC grains. The effect of the height of columns on the sintering process is illustrated in figure 31. The height of columns was shown to have a strong effect on all the sintering stages. Similar to the width of columns, the sintering rate is inversely proportional to the height of columns. This is because in the case of long columns more amount of time is required for the solid material to be transported from the top of columns, where the curvature is highest, to the bottom. The effect of separation distance between columns is illustrated in figure 32. The initial separation distance between columns inversely affects the sintering rate. Such behavior is 50

62 attributed to two reasons. First, when we increase the gaps between columns, more amount of material is needed to fill these gaps. Second, by increasing the initial separation distance between columns, the chance of neck formation at the top of columns will be decreased. Figure 31: Effect of columns heighet on the sintering process of TBC columnar grains. Figure 32: Effect of initial separation distance between columns on the sintering of TBC columnar grains. 51

63 Another columnar TBC microstructure of interest is the zigzag structure (review Fig 9). This microstructure is produced by rotating the substrate during the deposition process. This zigzag structure was reported to have 40% less thermal conductivity than the usual columnar one [7, 8]. The decrease in thermal conductivity was attributed to the low sintering rate of this structure [8]. The evolution of idealized zigzag structures is illustrated in figures 33 and 34. At the beginning of the sintering process, the sharp edges on the sides of columns smooth out by diffusion in order to decrease the surface energy. Then as the sintering process proceeds, the initial zigzag structure becomes similar to the usual vertical columnar structure. After that the zigzag structure experiences the same four sintering stages as the columnar structure. Hence we can expect that zigzag structures have lower sintering rate than the usual columnar ones since these structures smooth out first to the vertical columnar and then sinter in the same manner. Zigzag structures with short wavelength smooth out faster (see Fig 33) because of their higher curvature, while zigzag structures with long wavelength form necks earlier (see Fig 34) because they retain some roughness (undulations) at the top that helps neck formation. (a) t=0 (b) t=15m (c) t=30m (d) t=1.5h (e) t= 6h (f) t=17h Figure 33: Evolution of an idealized zigzag structure with short wavelength. 52

64 (a) t=0 (b) t=5m (c) t=20m (d) t=3h (e) t=9h (f) t=15h Figure 34: Evolution of an idealized zigzag structure with long wavelength. Before we conclude this section we present the investigation of effect of strain on the sintering process in columnar TBC structures. As we mentioned in chapter two, the TBC system is subjected to stress free strain because of the thermal expansion mismatch between the coating and the substrate. This stress free strain could be easily incorporated in the phase field model as explained in the previous chapter. Due to the fact that YSZ coatings have thermal expansion coefficient that is close to most of nickel based super alloys, the amount of strain produced in the system is in the range of 1%. As in the case in most of constrained sintering problems, the strain tends to retard the sintering process. The effect is readily seen in figure 35 which says that the sintering rate decreases as the amount of strain present in the system increases. Although the effect of 1% strain on the sintering of TBC is almost negligible which is in agreement with some experimental studies [102], an increase in the strain beyond that level drastically affects the sintering rate. The strain increases the time required for neck formation at the top of columns and hence slows down the sintering process. Our simulations even suggested a threshold value of strain that could completely prevent the neck formation at the top of columns, but that value (in the range of 5% or higher) is very high and it cannot present in any realistic systems. 53

65 Figure 35: Effect of strain on the sintering of TBC 4.3 Simulating Actual TBC Microstructures This section is devoted to study the sintering process in actual TBC structures. Instead of using idealized configuration to represent the initial morphology of TBC structure, one may use actual micrographs of TBC microstructure as the initial configuration for simulation. The procedure is as follows, first we pick from experimental literature a clear scanning electron microscope (SEM) micrograph that represents the actual morphology of TBC (see Fig 36).Then we use edit software to distinguish different phases and different grains by assigning them different colors (see Fig 36). Note that in assigning different colors to different columns, no adjacent columns can have the same color since it will lead to quick unphysical coalescence. The edited figure can be converted into date file with different numerical values for different colored domains using image processing software. Here we used the simple image processing tool in MATLAB. Finally, we construct our simulation domain based on the numerical values included in the data file. Studying actual TBC structures reveals important facts about the actual sintering behavior in TBC systems. In contrast to idealized configurations, concurrent densification and 54

66 grain growth was observed in the sintering process of actual TBC structures (see Fig 37). Such behavior is due to the fact that the morphology and dimensions of different columns vary in actual structures. The irregularity of columns morphology and dimensions develops nonuniform sintering behavior that is similar to the experimentally observed behavior. The densification of coating microstructure is evident by the shrinkage of columns, while grain growth is clearly observed from the decrease of number of grains with time. The occurring of grain growth decreases the densification rate since it provides the microstructure with other way by which it can decrease its interfacial energy. In addition to grain growth, the grain boundary migration leads to pore growth (see Fig 37(g)) as well. Similar to the solid material, the pores will prefer to sinter together in order to decrease the interfacial area and hence interfacial energy. The concurrent grain and pore growth is usually called coarsening and it is a characteristic of microstructure evolution in ceramic materials. Hence by simulating actual TBC structures, we can capture more realistic features of the sintering process in such systems. Image Processing Software Figure 36: Steps for constructing the computational domain from actual TBC structure (the micrograph is taken from Ref [50]). 55

67 (a) t=0 (b) t=2h (c) t=10h (d) t=14h (e) t=20h (f) t=23h (g) t= 28h (h) t=30h (i) t=40h Figure 37: Evolution of an actual TBC structure. 56

68 4.4 Simulations of 3D Test Cases In this section, we present simple 3D examples in order to demonstrate that expanding the model into 3D is straightforward. A simple case to start with is the sintering of two different spherical grains. The grains differ in radius and crystallographic orientation. Similar to the sintering of two circular grains in 2D, the grains will form necks and hence a grain boundary since a grain boundary is less energetic than a free surface. After a grain boundary is formed, a chemical potential gradient is developed across the boundary. From atomistic point of view, the chemical potential of atoms inside the grain with smaller radius, and therefore higher curvature, is higher. Hence atoms tend to diffuse from the small grain to the large grain across the grain boundary causing the boundary to move in the opposite direction. This grain boundary migration is the mechanism by which grain growth occurs. In figure 38, we visualize the grains by plotting the isosurface (level set) of (a) t=0 (b) t=10m (c) t=1h (d) t=2h (e) t=4h (f) t=5h Figure 38: Snapshots of the evolution of orientation field for two spherical grains. 57

69 Another test case that we can conduct is the sintering of idealized columnar TBC structure. The columnar structure of TBC may be idealized in 3D as cylindrical grains. By analogy with the sintering of idealized columns in 2D, the sharp edges at the top of columns become rounded to reduce the surface energy. Then material is transported from the top of columns, where the curvature is highest, to the bottom. The sintering rate is then enhanced by the neck formation at the top of columns since it provides a bridge for diffusion. Diffusion allows solid material to fill the gaps between columns causing densification of the coating. In figure 39, we visualize the density field by plotting the level set In figure 40, we visualize the orientation field where grain boundaries can be clearly identified. (a) t=0 (b) t=10m (c) t=1h (d) t=2h (e) t=3h (f) t=5h Figure 39: Snapshots of the evolution of density field for a 3D columnar TBC structure. 58

70 (a) t=0 (b) t=10m (c) t=1h (d) t=2h (e) t=3h (f) t=5h Figure 40: Snapshots of the evolution of orientation field for a 3D columnar TBC structure. 59