THE MEASUREMENT OF SOLID-LIQUID INTERFACIAL ENERGY IN COLLOIDAL SUSPENSIONS USING GRAIN BOUNDARY GROOVES by RICHARD B. ROGERS Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Dissertation Advisors: Prof. K.P.D. Lagerlöf Prof. B.J. Ackerson Department of Materials Science and Engineering CASE WESTERN RESERVE UNIVERSITY May, 2006 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the dissertation of Richard B. Rogers candidate for the Doctor of Philosophy degree*. Approved by Prof. K. Peter Lagerlöf (chair of the committee) Prof. Bruce J. Ackerson Prof. Frank Ernst Prof. J. Iwan Alexander Prof. Pirouz Pirouz (date) January 19, 2006 * We also certify that written approval has been obtained for any proprietary material contained therein. To the praise of Jesus Christ, the Master Materials Engineer In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning. Through him all things were made; without him nothing was made that has been made. John 1: 1-3, NIV (New International Version) TABLE OF CONTENTS Chapter 1 Introduction ....................................................................................................... 1 1.1 Research Goal ..................................................................................................... 1 1.2 Motivation........................................................................................................... 1 1.3 Approach............................................................................................................. 5 1.4 Dissertation Overview........................................................................................ 6 Part I: In-situ Crystallography of Ordered Colloids using Optical Microscopy Chapter 2 Parallel-Beam Technique .................................................................................. 8 2.1 Introduction......................................................................................................... 8 2.2 Background....................................................................................................... 10 2.2.1 Technique Overview................................................................................. 10 2.2.2 Analysis Procedure................................................................................... 12 2.3 Experimental Procedure.................................................................................... 18 2.3.1 Samples..................................................................................................... 18 2.3.2 Experimental Equipment .......................................................................... 18 2.3.3 Calibration................................................................................................. 20 2.3.4 Image Processing and Data Analysis........................................................ 21 2.4 Results............................................................................................................... 21 2.5 Discussion......................................................................................................... 26 2.5.1 Real Space................................................................................................. 26 2.5.2 Reciprocal Space....................................................................................... 30 2.6 Summary and Conclusions ............................................................................... 43 Chapter 3 Divergent-Beam Technique ............................................................................ 45 3.1 Introduction....................................................................................................... 45 3.2 Background....................................................................................................... 46 3.2.1 Technique Overview................................................................................. 46 3.2.2 Analysis Procedure................................................................................... 50 3.3 Experimental Procedure.................................................................................... 55 3.3.1 Sample....................................................................................................... 55 3.3.2 Experimental Equipment .......................................................................... 55 3.3.3 Image Processing and Data Analysis........................................................ 55 3.4 Results............................................................................................................... 57 3.5 Discussion......................................................................................................... 62 3.6 Conclusions....................................................................................................... 75 Chapter 4 Comparison of Techniques.............................................................................. 76 Part II: Grain Boundary Groove Techniques Applied to Colloids in a Gravitational Field Chapter 5 Theory ............................................................................................................. 80 5.1 Introduction....................................................................................................... 80 5.1.1 Literature Review......................................................................................80 5.1.1.1 Grain Boundary Groove Techniques .................................................... 80 5.1.1.2 Hard Sphere Colloids............................................................................ 95 i 5.2 Isotropic Theory for Colloids in a Gravitational Field ................................... 107 5.2.1 Development and Solution of General Equation for Interface Shape..... 107 5.2.2 Application to Hard Sphere Colloids in a Gravitational Field................ 113 5.2.2.1 General Expressions for Π(φ) and y(φ)............................................... 114 5.2.2.2 yr(φ) and φ(yr) for the Fluid Phase ...................................................... 116 5.2.2.3 yr(φ) and φ(yr) for the Solid Phase ...................................................... 118 5.2.2.4 ΔΠr(yr) and fr(yr) ................................................................................. 119 5.2.2.5 Equation for the Interface Shape......................................................... 122 5.3 Summary and Conclusions ............................................................................. 130 Chapter 6 Experiments................................................................................................... 132 6.1 Introduction..................................................................................................... 132 6.2 Experimental Procedure.................................................................................. 132 6.2.1 Experimental Equipment ........................................................................ 132 6.2.2 Samples................................................................................................... 137 6.2.2.1 Particle Size Characterization............................................................. 138 6.2.2.2 Particle and Fluid Density Characterization ....................................... 139 6.2.2.3 Sample Preparation............................................................................. 140 6.2.3 Image Processing.................................................................................... 142 6.3 Results............................................................................................................. 145 6.3.1 Interface Cross Sections.......................................................................... 147 6.3.2 Grain Boundary Grooves ........................................................................ 148 6.3.3 Grain Tips............................................................................................... 156 6.4 Discussion....................................................................................................... 165 6.4.1 Interface Cross Sections.......................................................................... 165 6.4.2 Grain Boundary Grooves ........................................................................ 166 6.4.2.1 Groove Stability.................................................................................. 167 6.4.2.2 Crystal Orientation.............................................................................. 169 6.4.2.3 Analysis using Isotropic Theory ......................................................... 173 6.4.2.4 Analysis using the Capillarity Vector Approach ................................ 182 6.4.2.5 Interfacial Energy Anisotropy............................................................. 185 6.4.3 Grain Tips............................................................................................... 186 6.4.3.1 Tip Stability........................................................................................ 188 6.4.3.2 Tip Orientation.................................................................................... 191 6.4.3.3 Analysis using Isotropic Theory ......................................................... 193 6.4.3.4 Interfacial Energy Anisotropy............................................................. 200 6.4.4 Comparison with Literature Results ....................................................... 200 6.5 Summary and Conclusions