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Extreme Liquid Superheating and Homogeneous Bubble Nucleation in a Solid State Nanopore Extreme Liquid Superheating and Homogeneous Bubble Nucleation in a Solid State Nanopore The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Levine, Edlyn Victoria. 2016. Extreme Liquid Superheating and Homogeneous Bubble Nucleation in a Solid State Nanopore. Doctoral dissertation, Harvard University, Graduate School of Arts & Sciences. Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:33493497 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Extreme Liquid Superheating and Homogeneous Bubble Nucleation in a Solid State Nanopore a dissertation presented by Edlyn Victoria Levine to The School of Engineering and Applied Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Applied Physics Harvard University Cambridge, Massachusetts April 2016 ©2016 – Edlyn Victoria Levine all rights reserved. Thesis advisor: Professor Jene A. Golovchenko Author: Edlyn Victoria Levine Extreme Liquid Superheating and Homogeneous Bubble Nucleation in a Solid State Nanopore Abstract This thesis explains how extreme superheating and single bubble nucleation can be achieved in an electrolytic solution within a solid state nanopore. A highly focused ionic current, induced to flow through the pore by modest voltage biases, leads to rapid Joule heating of the electrolyte in the nanopore. At sufficiently high current densities, temperatures near the thermodynamic limit of superheat are achieved, ultimately leading to nucleation of a vapor bubble within the nanopore. A mathematical model for Joule heating of an electrolytic solution within a nanopore is presented. This model couples the electrical and thermal dynamics responsible for rapid and extreme superheating of the electrolyte within the nanopore. The model is implemented nu- merically with a finite element calculation, yielding a time and spatially resolved temperature distribution in the nanopore region. Temperatures near the thermodynamic limit of superheat are predicted to be attained just before the explosive nucleation of a vapor bubble is observed experimentally. Knowledge of this temperature distribution is used to evaluate related phenomena including bubble nucleation kinetics, relaxation oscillation, and bubble dynamics. In particular, bubble nucleation is shown to be homogeneous and highly reproducible. These results are consistent with experimental data available from electronic and optical measurements of Joule heating and bubble nucleation in a nanopore. iii Contents Abstract iii Table of Contents v List of Figures vii Dedication viii Acknowledgments ix 1 Introduction 1 1.1 Solid State Nanopores ............................... 3 1.2 Experimental Investigation of Joule Heating in a Nanopore ........... 3 1.3 The Superheated Liquid: A State of Metastable Equilibrium ........... 7 1.4 Organization of Thesis ............................... 14 2 The Dynamics of Joule Heating in a Solid State Nanopore 17 2.1 Aqueous 3M Sodium Chloride Solution ...................... 18 2.2 Electrodiffusion of Ions: Applying a Voltage Bias to the Electrolyte . 23 2.3 Electroosmotic Flow and the Convection of Ions . 26 2.4 Entropy Balance of Thermoelectric Phenomena . 27 2.5 Summary of Governing Equations ......................... 39 3 Model of Joule Heating in a Nanopore: Finite Element Method (FEM) 41 3.1 Simplifying the Governing Equations ....................... 42 3.2 Geometry and Boundary Conditions of the Nanopore System . 45 3.3 Material Properties ................................. 47 3.4 Numerical Implementation: Meshing and Solvers . 49 4 FEM Model Results and Discussion 51 4.1 Electrical Conductivity of the Superheated Electrolyte . 52 iv 4.2 Extreme Superheating within the Nanopore .................... 55 4.3 Comparison with the Limit of Superheat of Pure Water . 57 4.4 Induced Charge Densities Near the Nanopore ................... 59 5 Analytical Approaches to the Joule Heating Problem 61 5.1 Dimensionless Form of the Governing Equations for a Simplified Model . 62 5.2 Green’s Function Solution ............................. 66 5.3 Comparison of Analytical Solution with Numerical Results . 74 5.4 Remarks in Summary ............................... 76 6 Kinetics of Bubble Nucleation in the Nanopore 78 6.1 Surface Tension and the Size of the Critical Nucleus . 79 6.2 Fluctuations Driving Bubble Nucleation ..................... 84 6.3 The Kinetics of Phase Transformation ....................... 88 6.4 Homogeneous Nucleation of the Vapor Bubble in the Nanopore . 92 6.5 Heterogeneous Nucleation ............................. 95 6.6 Experimental Confirmation of Homogeneous Nucleation . 95 7 Temperature and Bubble Dynamics in the Nanopore After Nucleation 100 7.1 Dynamics of Heating and Relaxation Oscillation . 102 7.2 The Equations Governing Bubble Dynamics . 104 7.3 Approximating the Initial Growth of the Bubble . 109 7.4 Exploring the Complete Bubble Dynamics . 111 8 Conclusion 113 8.1 Summary of Results Presented in this Thesis . 114 8.2 The Future for Joule Heating in Nanopores . 115 Appendix A The Debye-Huckel¨ Theory 117 Appendix B Equations of Continuity and Conservation 125 Appendix C Solution of Joule Heating with Separation of Variables 141 Appendix D From Navier-Stokes to Rayleigh-Plesset 152 References 167 v Listing of figures 1.1 Experimental Setup: Conductance Measurement ................. 4 1.2 Experimental Results: Nanopore Conductance .................. 5 1.3 Experimental Results: Nucleation Events ..................... 6 1.4 PV Diagram, van der Waals Fluid ......................... 9 1.5 Gibbs free energy, van der Waals Fluid ...................... 11 1.6 PT Diagram, van der Waals Fluid ......................... 13 2.1 Debye Length for NaCl Solutions of Different Conductivities . 22 3.1 Nanopore Geometry for FEM Model ....................... 46 3.2 Temperature Dependence of the Material Properties of H2O . 48 4.1 Temperature Dependence of the Electrical Conductivity, 3M NaCl . 53 4.2 Nanopore Conductance, Data and Fit ....................... 54 4.3 Temperature Distribution in the Nanopore .................... 56 4.4 Maximum Temperature in the Nanopore ..................... 57 4.5 Charge Densities In and Near the Nanopore .................... 60 5.1 Radial Green’s Function vs Time ......................... 71 5.2 Radial Green’s Function vs Radius ........................ 72 5.3 Temperature Distribution from Green’s Function Method . 73 5.4 Analytical and Finite Element: T vs r ....................... 74 5.5 Analytical and Finite Element: T vs t ....................... 75 5.6 Joule Heating Source Term ............................ 76 6.1 Temperature Dependence of Surface Tension ................... 82 6.2 Temperature Dependence of Critical Radius ................... 83 6.3 Energetic Barrier to Nucleation .......................... 86 6.4 Nucleation Rate as a Function of T ........................ 92 6.5 Nanopore Conductance and Bubble Nucleation . 93 vi 6.6 Nucleation Rate Inside the Nanopore ....................... 94 6.7 Optical Experimental Setup ............................ 96 6.8 Optical Transmission Data ............................. 97 6.9 Homogeneous Nucleation: Delay between Ionic and Photodiode Current . 98 7.1 Experimental Results: Nucleation Events . 101 7.2 Temperature Dynamics of Relaxation Oscillation: Maximum Temperature . 102 7.3 Temperature Dynamics of Relaxation Oscillation: Spatial Distribution . 103 7.4 Rayleigh-Plesset Dynamics of Bubble Growth . 110 C.1 Analytical Solution for Temperature Distribution . 149 vii To my parents viii Acknowledgments Achievements are not attained in isolation, and I can thankfully say this is true for my the- sis. I have had the great fortune of having supportive family, friends, and mentors who have contributed to my growth and indeed to the development of this work. I am particularly grateful to my parents, Drs. Barry and Arlene Levine, and to my brother, Lionel, for their loving and caring support throughout my life. Thank you for cheering me on in all my diverse and various endeavors, both in physics, and in other aspects of life. I would like to thank my advisor, Prof. Jene Golovchenko for his guidance and mentorship throughout my time in graduate school, and for fostering my development as a physicist. Many thanks as well to Dr. Mike Burns and Prof. Lene Hau for consistent advice and feedback. I would like to give credit to Gaku Nagashima for conducting all of the experiments I discuss in this work, and am thankful to him for his friendship over the years. I am also grateful to Dr. David Hoogerheide, Aaron Kuan, Tamas Szalay, Dr. Ryan Rollings, Dr. Lu Bo, Don Dressen and Stephen Flemings for many interesting discussions about physics and of course, conversations about life in general. Finally, I want to thank Eric Brandin, Peter Frisella, Ray Aubut, and Jane Salant for everything they have done in support of this research. ix The single, all-encompassing problem of thermody- namics is the determination of the equilibrium state that eventually results after the removal of internal constraints in a closed, composite system. Herbert B. Callen 1 Introduction We recently discovered that an aqueous electrolyte can be strongly superheated within a solid- state nanopore by electrical
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