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Text of Relevant Applications --- Alloys, Batteries, Fuel Cells, Etc 315: Phase Equilibria and Diffusion in Materials Professor Thomas O. Mason February 24, 2021 Contents 1 Catalog Description3 2 Course Outcomes3 3 315: Phase Equilibria and Diffusion in Materials3 4 Introduction4 5 Gibbs-Duhem Eq.7 5.1 Degrees of Freedom.......................... 10 5.2 Phase Rule............................... 10 5.3 Classifying Phase Diagrams..................... 14 6 Type I Phase Diagrams 16 6.1 Ellingham Diagrams ......................... 16 6.2 Two More Type I Phase Diagrams.................. 25 7 Type II Phase Diagrams 27 7.1 Liquidus Lines............................. 28 7.2 Liquidus and Solidus Lines ..................... 32 7.3 Estimating Solvus Lines ....................... 35 7.4 Activity vs. Composition Plots ................... 38 7.5 Free Energy vs. Comp......................... 46 7.6 Regular Solution Predictions..................... 53 7.7 PO2 vs. Comp.............................. 54 8 Type III Phase Diagrams 59 8.1 Ternary Lever Rule.......................... 60 1 CONTENTS CONTENTS 8.2 Degrees of Freedom.......................... 61 8.3 Liquidus Projections ......................... 62 8.4 Hummel’s Rules............................ 65 8.5 Isothermal Sections.......................... 76 8.6 Crystallization Paths and Microstructure Evolution . 83 8.7 Isothermal Sections of “Real” A-B-C Systems........... 86 8.8 Examples................................ 89 9 Fick’s Laws of Diffusion 96 9.1 Fick’s First Law............................ 97 9.2 Random Walk Diffusion ....................... 100 9.3 Steady State Diffusion ........................ 102 9.4 Fick’s Second Law........................... 103 10 Applications of Fick’s Laws 106 10.1 Homogenization ........................... 106 10.2 Finite Systems............................. 109 10.3 Semi-Infinite Systems......................... 111 11 Atomistics of Diffusion 122 11.1 Interstitial Diffusion ......................... 123 11.2 Vacancy Diffusion........................... 128 11.3 Interstitialcy Diffusion........................ 134 11.4 Arrhenius Behavior.......................... 136 12 Point Defects 138 12.1 Simmons-Balluffi Expt......................... 138 12.2 Vacancy Formation in fcc Metals .................. 141 12.3 The Bauerle and Kohler Experiment . 142 12.4 Vacancy Motion............................ 147 13 Transport in Ceramics 148 13.1 Kröger-Vink Notation ........................ 148 13.2 Point Defect Reactions ........................ 150 13.3 Point Defect Thermo ......................... 160 13.4 Brouwer Diagrams .......................... 164 14 Electrical Conductivity 180 14.1 Electronic Conductivity ....................... 180 14.2 Ionic Conductivity .......................... 182 15 315 Problems 186 16 315 Labs 197 16.1 Lab Schedule.............................. 198 16.2 Safety Guidelines........................... 199 2 3 315: PHASE EQUILIBRIA AND DIFFUSION IN MATERIALS 16.3 Worksheet 1 .............................. 201 16.4 Worksheet 2 .............................. 202 16.5 315 Lab 1: Binary Phase Diagrams . 206 16.6 Worksheet 3 .............................. 209 16.7 315 Lab 2:Bi-Sn Alloys ........................ 211 16.8 Worksheet 4 .............................. 214 16.9 Worksheet 5 .............................. 215 16.10315 Lab 3: Pack-Carburization.................... 218 Nomenclature 221 1 Catalog Description MAT_SCI 315 covers, broadly, two topics: phase equilibra and diffusion in materials. In the first half of this course, we concentrate on foundational thermodynam- ics. Namely, the application of thermodynamics to the prediction and inter- pretation ofphase diagrams. The level of presentation assumes that students have a background in the laws of thermodynamics - especially in the area of solution thermodynamics (MAT_SCI 314). We’ll build from these foundations so that students can apply thermodymics to Type I, II, and III phase diagrams. In the second half of the course we’ll concentrate on the foundations of diffu- sion in solids. We’ll introduce the atomistic descriptions of diffusion and in- troduce the physical laws (Fick’s laws) that govern how atoms are transported in solids. We’ll apply these behaviors in engineering scenarios. Prerequisite: MAT_SCI 314-0 or equivalent. 2 Course Outcomes 3 315: Phase Equilibria and Diffusion in Materials At the conclusion of the course students will be able to: 1. Classify, interpret, and analyze Type I, II, and III phase diagrams. 2. Construct schematic phase diagrams from elementary thermodynamics. 3 4 INTRODUCTION 3. Navigate binary and ternary phase diagrams to assess phase equilibrium of mixtures. 4. Utilize ternary phase diagrams to follow crystallization paths and pre- dict microstructure evolution. Utilize an understanding of the role of point defects in diffusion and atomistic behavior of solids. 5. Describe the equilibrium thermodynamics of point defects in both crys- talline solids. 6. Use thermodynamics and computational tools to predict and interpret phase equilibria in simple unary and binary systems. Examine the role of phase equilibria and diffusion in the context of relevant applications --- alloys, batteries, fuel cells, etc. 7. Prepare alloy specimens for microstructural observation and measure- ment of hardness profile. Assess experimental results within the context of phase equilibria/diagrams and diffusion. 4 Introduction “Thermodynamics” and “kinetics” are fundamental skill sets/tool boxes re- quired of all materials scientists/engineers. Thermodynamics tells us which phase - or assemblage of phases - has the absolute lowest free energy, and therefore represents the equilibrium state. (Note: there may be other metastable states at higher energies.) Kinetics tells us much more: how fast those phases will form, and the paths they will take along the way. Together, thermodynamics and kinetics determine the phase assemblages/microstruc- tures that can be obtained, and how to obtain them. In the materials science and engineering paradigm of Fig. 4.1, thermodynamics and kinetics come pri- marily into play in the first “chain link” between “Processing” and “Structure.” Figure 4.1: The Materials Science and Engineering Paradigm The interplay between thermodynamics and kinetics can be illustrated with two case studies. The first involves the Fe-C phase diagram, which is intro- duced in virtually all introductory Materials Science and Engineering courses. 4 4 INTRODUCTION A schematic of the Fe-rich end (small concentrations of carbon: XC <2%) of this diagram is given in Fig. 4.2. Figure 4.2: Schematic of the Fe-rich end of the Fe-C phase diagram If we solutionize the eutectoid composition austenite (g-phase) at the point indicated in Fig. 4.2, and then cool it (follow the arrow) below the eutectoid temperature Te(thermodynamics), various microstructures can result, depend- ing upon the rate of cooling (kinetics). For example, slow cooling can result in discrete coarse-grained phases (a-phase ferrite and cementite, or Fe3C). This assemblage of phases is not very strong or hard. On the other hand, by cooling more rapidly, we can produce a layered structure of ferrite and cementite, re- ferred to as pearlite for its “mother of pearl” appearance under the microscope. This microstructure is found to be quite strong and hard. This is a prime ex- ample of how the processing-structure “chain link” can influence the resulting structure , properties “chain-link” in the Materials Science and Engineering paradigm (Fig. 4.1). The second example involves the oxidation of silicon to silicon dioxide through the reaction of equation 4.1. Si(s) + O2(g) SiO2(s) (4.1) Later in this course we will learn about Richardson-Ellingham diagrams (for simplicity, these will be referred to as Ellingham diagrams). An Ellingham di- agram is just a superposition of lines representing the free energy of oxidation 5 4 INTRODUCTION for a large number of metals. A schematic of the Ellingham diagram for silicon alone is given in Fig. 4.3 (later graphs, like Fig. 6.2 show much more data). Figure 4.3: Schematic Ellingham diagram for the oxidation of silicon using O2 The top left of the diagram is at zero T = 0oC and at DGo = 0. (Note that the letter “M” and “ M ” refer to the melting points of the metal (Si) and oxide (SiO2), respectively.) As we will see, the free energies of formation of most ox- ides, including silica, are strongly negative. This means that reactions like that in Eq. 4.1 have a strong tendency to go to the right, that is, to produce their oxide at the expense of the corresponding metal. For example, if we throw an iron bar out in the “elements”, we know that it rusts (forms the oxide1) quite readily. However, an aluminum object, in spite of having an even larger neg- ative free energy of oxidation than either iron or silicon, will hardly corrode under the same conditions. That is due to the formation of a coherent “pas- sive” oxide film on the surface, through which diffusion is extremely slow. The same shenomenon takes place on silicon, as illustrated in Fig. 4.4. The passive SiO2 film that forms is extremely important to the microelectronics at work in many computers. (Note: silica has been largely replaced [ca. 2007] by oxides with larger dielectric constants [such as hafnia], to ensure that the miniaturization necessary to keep extending Moore’s Law, i.e., the observation 1To be more accurate, rust is a complex mixture of hydrated iron oxides and oxide-hydroxides, but you get the point. 6 5 GIBBS-DUHEM EQ. Figure 4.4: Passive oxide film formation on silicon, which can be
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