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Defects in Two Dimensional Colloidal Crystals Defects in Two Dimensional Colloidal Crystals by Lichao Yu B.Sc., University of Science and Technology of China; Hefei, China, 2010 M.Sc., Brown University; Providence, RI, 2013 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Department of Physics at Brown University PROVIDENCE, RHODE ISLAND May 2015 c Copyright 2015 by Lichao Yu This dissertation by Lichao Yu is accepted in its present form by Department of Physics as satisfying the dissertation requirement for the degree of Doctor of Philosophy. Date Xinsheng Sean Ling, Ph.D., Advisor Recommended to the Graduate Council Date John Michael Kosterlitz, Ph.D., Reader Date Robert Pelcovits, Ph.D., Reader Approved by the Graduate Council Date Peter M. Weber, Dean of the Graduate School iii Vitae Lichao Yu was born in the city of Nanchang, Jiangxi, P. R. China, on July 12th 1990. He obtained his Bachelor of Science from University of Science and Technology of China in 2010, and a Master's degree in Physics from Brown University in 2013. iv Acknowledgements Time flies. Five years went really fast. I still have a vivid recollection of the day I stepped into the campus of Brown. It's sunny, warm breeze carasses my face. At that moment, I know I would, with no regret, spend the best five years of my life, here at Brown, dedicating to Physics research, the great journey of exploration. This is meant to be a wonderful experience throughout my life. First of all, I would like to give my thanks to my advisor Professor Xinsheng Sean Ling for his professional advice and guidance on research. His patience, conscien- tiousness and dedication, has made huge impact on my development as a physicist. Besides that, he also serves as a mentor providing career advices and mental sup- ports, and a friend to share with, happiness or sadness, successes or regrets. Second, I thank Professor John Michael Kosterlitz and Professor Robert Pelcovits for taking interests in my research and reading my Phd dissertation. They happily became my defense committees and provided valuable advices. Then I thank my lab mates Sungcheol Kim, who guided and taught me experi- mental and programming techniques, Wang Miao, Helen Hanson, Xi Wang, visiting scholars and interns: Hongwen Wu, Liping Liu, Yin Di, Stephanie Huang, etc. I also wish to mention all my friends, colleagues, staffs and faculty who offered me help v during the past five years. I am grateful to everyone for being part of my graduate school memory. I would also thank my family: my parents, my wife and our newborn boy Max. They support me during the dark ages and share with me all the happy moments. Last but not least, I acknowledge the financial support from Brown University, the Physics Department, the National Science Foundation, etc. Thank you. vi Abstract of \ Defects in Two Dimensional Colloidal Crystals " by Lichao Yu, Ph.D., Brown University, May 2015 We use digital video microscopy to study the defects in two-dimensional colloidal crystals (2DCC). A crystalline solid, different from its liquid state, preserves long- wavelength shear rigidity and broken symmetry. Questions about how shear rigidity and long-range order disappear during melting, are unresolved in terms of the com- plication of defect structures and their roles in crystal melting, especially in two dimension. Colloidal crystals (CC) serve as a promising model system to directly observe the defects under optical microscope. In our first study, we report the effects of vacancies and interstitials on the phonon modes in a 2DCC. By applying the equipartition theorem, we extract the dispersion relation of the lattice vibrations using real-time video microscopy. We find that both longitudinal and transverse modes in the spectrum are softened by the existence of point defects. Second, we investigate the diffusion process of interstitials in a 2DCC. The mo- tion is viewed as gliding of both edge dislocations along one of the crystalline axes. The microscopic process is equivalently a point mass overcoming Peierls barrier with an exponential escaping time. We also establish a new criterion to determine the er- godicity of a defect system and discover the nonergodic behavior of di-interstitials. Contents Vitae iv Acknowledgments v 1 Introduction 1 2 Theory 4 2.1 Two-dimensional Melting . 5 2.1.1 Introduction . 5 2.1.2 Kosterlitz-Thouless conjecture . 9 2.1.3 KTHNY theory . 12 2.2 Lattice Dynamics in 2DCC . 17 2.2.1 1D atom chain . 17 2.2.2 Lattice dynamics . 18 2.2.3 Dynamical matrix and normal modes . 20 3 Experimental Techniques 22 3.1 Sample Cell Preparation . 23 3.2 Suspension Preparation and Cleaning . 25 3.3 Imaging with Optical Microscope . 30 3.4 Image Processing and Particle Tracking . 34 4 Effect of Point Defect on Phonon Modes 38 4.1 Introduction . 39 4.2 Method . 41 4.3 Results . 46 4.4 3D Visualization of Phonon Bands . 51 4.5 Conclusion and Future Directions . 52 5 Microscopic Process and Nonergodicity in Defect Diffusion 56 5.1 Introduction . 57 vii 5.2 Method . 57 5.3 Results . 59 5.4 Peierls Barrier . 61 5.5 Nonergodicity . 64 5.6 Monte Carlo Simulation . 66 5.7 Conclusion and Future Directions . 67 A Kosterlitz-Thouless Transition 69 A.1 Model System . 70 A.2 Scaling Equations . 73 A.3 Real System . 77 A.3.1 X-Y Model . 77 A.3.2 2D solids . 78 A.4 Asymptotic Solutions . 79 A.5 Correlation Length . 86 B Diffusion 88 B.1 Random Walk to Diffusion . 89 B.2 Kramer's Escape Rate . 91 C IDL Scripts and C++ Code 93 C.1 IDL Scripts . 94 C.2 IDL Source Code for Particle Tracking and Phonon Mode Analysis . 96 C.3 C++ Code for Monte Carlo Simulation . 121 viii List of Tables ix List of Figures 2.1 Summary of KTHNY theory. Note the vanishing behavior of stiffness KR(T ) and Frank constant KA(T ) is not the exact solution. 15 2.2 1D atom chain model. Atoms are interacting via ideal springs. 18 3.1 Sample cell for 2DCC experiment (not to scale). Side-view (top): shaded area is the optical adhesive which bonds both substrates. In silica disk (top substrate), a channel is designed for the flow. A less than µm uniform gap between the plateau and coverslip provides space for 2DCC. Top-view (below): gray area shows the flow pattern of the colloid suspension. A very small amount of the solution travels through the plateau (inside center circle). 24 3.2 Cross-flow diafiltration setup (not to scale). The colloidal suspension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. The cartridge used is MidGee Microfiltration Hollow Fiber Cartridges together with Poly- sulfone membrane with 0.2 micron pore size (CFP-2-E-MM01A). A conductivity meter (Amber Scientific Model 1056 Digital Conductiv- ity Meter) is used to monitor the ionic strength. DI-water buffer so- lution is stored in a squeeze bottle and transferred into the reservoir continuously. 27 3.3 Cross-flow diafiltration performance. Conductivity is measured with Amber Scientific Model 1056 Digital Conductivity Meter. Red repre- sents the readings before dilution (refill) and blue for readings after dilution. 28 3.4 Ion-exchange deionization setup to prepare 2DCC. The colloidal sus- pension is stored in a sealed Telfon container. The circuit is connected with Galtek Integral Ferrule fittings. The suspension is pumped out by a peristaltic pump (Fisher Scientific Variable-Flow Peristaltic Pump), providing smooth feed into the cartridge filter. Mixed-bed ion-exchange resin beads (Fisher Chemical Rexyn I-300 (H-OH) Beads) are stored in the ion exchange column. A flow-through conductivity meter (Am- ber Scientific Conductivity Micro Flow Cells) is used to monitor the ionic strength. Sample cell is mounted above the oil-immerse objective lens. 29 x 3.5 Colloidal suspension under optical microscope. Top: near the circular channel. Bottom: far from the channel. 31 3.6 Side view of sample cell showing the maximum spacing between sub- strates to achieve one -ayer. Solid circle represents colloid surface. Dashp circle is the Debye layer (Electron Double Layer). D ∼ 2λD + d + 3a=2 ' 2 µm............................. 32 3.7 Two-layer colloidal crystal . 33 3.8 Mono-layer colloidal crystal with different volume fractions and thick- ness. 35 3.9 Histogram of lattice spacings and angles with equilibrium positions. Top: histogram over 200 frames. Bottom: fitting with standard Gaus- sian distribution. 36 3.10 Fast Fourier Transform (FFT) of a cropped snapshot of 2DCC. 37 4.1 Optical micrograph and delaunay triangulation diagrams of a two di- mensional colloidal crystal. (a) (b) One microscope snapshot and its delaunay triangulation diagram: defect-free lattice. The inset shows ~ ~ primitive vectors (~a1, ~a2), its reciprocal lattice vectors (b1, b2), and the first Brillouin zone (not to scale) of perfect triangular lattice with standard labels of the symmetry points. (c)-(f) Lattice with point defects: mono-vacancy, mono-interstitial, both mono-vacancy and in- terstitial and lattice with tri-vacancy. Red and blue dots stand for 5 and 7-coordinate disclinations. 42 4.2 Hexagonal lattice of 2DCC with equilibrium positions. The density ~ function ~r(t) is a superposition of delta functions on n sites Rn. We can measure the real-time displacement ~u(Rn) as the difference be- tween real time position (green dots) and the equilibrium position (blue dots). 44 4.3 Reciprocal space of 2DCC.
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