A Theory of a Self-Assembling Electrovariable Smart Mirror Ryan-Rhys Griffiths (00730937) MSci Chemistry with Molecular Physics, Year 4 Supervisor: Prof. Alexei Kornyshev May 2016 arXiv:1709.05494v1 [cond-mat.mes-hall] 16 Sep 2017 Word Count: 13,066 Abstract A theory describing the forces governing the self-assembly of nanoparticles at the solid-liquid interface is developed. In the process, new theoretical results are derived to describe the effect that the field penetration of a point-like particle, into an electrode, has on the image potential energy, and pair interaction energy profiles at the electrode-electrolyte interface. The appli- cation of the theory is demonstrated for gold and ITO electrode systems, promising materials for novel colour-tuneable electrovariable smart mirrors and mirror-window devices respectively. Model estimates suggest that electrovariability is attainable in both systems and will act as a guide for future experiments. Lastly, the generalisability of the theory towards electrovari- able, nanoplasmonic systems suggests that it may contribute towards the design of intelligent metamaterials with programmable properties. 2 Acknowledgements I would like to express my thanks to Professor Rudi Podgornik and Dr. Debabrata Sikdar for their advice on the theory of Van der Waals forces and nanoplasmonics respectively. Addition- ally, I would like to thank Dr. Yunuen Montelongo, Ye Ma and James Million for discussions on the experimental aspects of the electrovariable mirror project. Most of all, I would like to convey my sincerest gratitude to Professor Kornyshev for his altruistic guidance, patience and encouragement and for impressing on me the point, that while an invaluable tool, technology should never be used as a substitute for one's own judgement in the natural sciences. 3 Contents 1 Introduction 9 1.1 A Brief Overview of Nanoplasmonics . .9 1.2 The Self-Assembling Electrovariable Smart Mirror . 10 1.3 The Choice of Interface . 12 1.4 The Scope of the Present Work . 13 2 Theoretical Methods 14 2.1 The Nature of Van der Waals Forces . 14 2.2 The Choice of Formalism . 15 2.2.1 Hamaker's Method - Pairwise Summation . 15 2.2.2 The Lifshitz Model - Continuum Theory . 16 2.2.3 The Hamaker Hybrid Formalism . 17 2.3 Force Computation . 18 2.3.1 Connecting Forces with Spectra . 18 2.3.2 The Frequency-Dependent Dielectric Function . 18 2.3.3 Imaginary Frequencies . 20 2.3.4 The Hamaker Coefficient . 20 2.4 Additional Comments . 21 3 Nanoparticles in the Bulk 22 3.1 Motivation . 22 3.2 The Van der Waals Force . 24 3.2.1 Hamaker Hybrid Form for Two Nanoparticles . 25 3.2.2 Nonlocality . 26 3.2.3 Size Dependence of the Hamaker Coefficient . 27 3.2.4 The Van der Waals Potential Profile . 28 3.3 The Coulomb Force . 32 3.4 The Total Interaction Energy . 34 4 A Single Nanoparticle at an Interface 36 4.1 The Van der Waals Potential Profile . 37 4.1.1 Gold Electrodes . 37 4.1.2 ITO Electrodes . 40 4.2 The Image Potential Profile . 42 4.2.1 The Image Potential Energy of a Point Particle . 42 4.2.2 The Image Potential Energy of a Nanoparticle . 46 4 5 A Pair of Nanoparticles at an Interface 49 5.1 Van der Waals Interaction . 50 5.2 Image Potential Energy . 52 5.2.1 Point Particle . 52 5.2.2 Nanoparticle . 54 6 Conclusion 56 6.1 Summary . 56 6.2 Outlook . 57 A Supporting Information 63 A.1 Derivation of the Coulomb Potential between Two Nanoparticles in Electrolyte . 63 A.2 Derivation of the Image Potential Energy and Pair Interaction Energy - Point Charge . 65 A.3 Image Potential Energy for Gold and ITO with Hydrocarbon Films . 67 A.4 Point Particle Pair Interaction Energy . 69 A.5 Annotated Program for the Computation of the Hamaker Coefficient . 74 A.6 Drude-Lorentz Model Parameters for ITO . 78 B Epilogue 79 B.1 A Geometrically Accurate Result for a Sphere . 79 5 List of Figures 1.1 Schematic of Nanoparticle Assemblies.9 .......................9 1.2 An Illustration of the Mirror-Window System. 11 1.3 Schematic of the LLI.27 ................................ 12 1.4 Schematic of the SLI.27 ................................ 12 2.1 An Analogy to Static Dipoles - The Absence of Fluctuations. 14 2.2 An Illustration of the Third-Body Effect in Condensed Phases. 15 2.3 Casimir Forces - Accounting for Retardation Effects. 16 3.1 Push and Pull - The Balance of Van der Waals (VdW) and Coulombic Forces. 22 3.2 Effect of Increased Solution pH. 23 3.3 Effect of Electrolyte Screening. 23 3.4 System Geometry. 25 3.5 Divergent Behaviour at Short Separation. 26 3.6 An Illustration of the Size Dependence of the Hamaker Coefficient for Gold Nanoparticles in the Bulk. 27 3.7 Figures taken from the work of Olmon et al.63 ................... 29 3.8 Van der Waals Interaction Profile between Two Nanoparticles of Radius (a) 5 nm, and (b) 10 nm, starting from 5 nm Surface to Surface Separation. 30 3.9 Van der Waals Interaction Profile between Two Nanoparticles of Radius (a) 20 nm, and (b) 40 nm, starting from 5 nm Surface to Surface Separation. 30 3.10 Identical Gold nanoparticles - size comparison using DESY data. 31 3.11 Coulomb Potential Energy as a Function of Surface to Surface Separation for Varying Electrolyte Concentrations. Radius Fixed at 20 nm, 500 Charges per Nanoparticle. 33 3.12 Coulomb Potential Energy as a Function of Surface to Surface Separation for Varying Numbers of Surface Charges per Nanoparticle. Radius Fixed at 20 nm, Electrolyte Concentration of 0.01 M. 33 3.13 Total Interaction Energy as a Function of Surface to Surface Separation for (a) The DESY Hamaker Coefficient and (b) The Johnson and Christy (JC) Hamaker Coefficient. Radius Fixed at 20 nm, 1000 Charges per Nanoparticle. 34 3.14 Total Interaction Energy as a Function of Surface to Surface Separation for (a) The DESY Hamaker Coefficient and (b) The Johnson and Christy (JC) Hamaker Coefficient. Radius Fixed at 20 nm, Electrolyte Concentration of 0.01 M. 35 3.15 Zoomed-in View of the Red Curve from Figure 3.14(a). 35 4.1 System Geometry. 36 4.2 Fits by Parsegian,37 and Roth and Lenhoff68 to the spectral data of Heller et al.69 Figure taken from Roth and Lenhoff.68 .................... 38 6 4.3 Van der Waals Interaction Profile between a Gold Electrode and a Gold Nanopar- ticle of Radius (a) 5 nm, and (b) 10 nm, starting from 5 nm Surface to Surface Separation. 39 4.4 Van der Waals Interaction Profile between a Gold Electrode and a Gold Nanopar- ticle of Radius (a) 20 nm, and (b) 40 nm, starting from 5 nm Surface to Surface Separation. 39 4.5 Fit to the Spectral Data of Konig et al.71 ...................... 40 4.6 Van der Waals Interaction Profile between an ITO Electrode and a Gold Nanopar- ticle of Radius (a) 5 nm, and (b) 10 nm, starting from 5 nm Surface to Surface Separation. 41 4.7 Van der Waals Interaction Profile between an ITO Electrode and a Gold Nanopar- ticle of Radius (a) 20 nm, and (b) 40 nm, starting from 5 nm Surface to Surface Separation. 41 4.8 ITO and Gold Comparison for NP radius = 20 nm. DESY results only for clarity. 41 4.9 Illustration of Method of Images. 42 4.10 Point Particle near the metal/electrolyte interface. 43 4.11 Geometry of the System to accompany Equation 4.6. 44 4.12 Gold varying concentration. 44 4.13 A Comparison with the Classical Image Law. 45 4.14 ITO varying concentration. 45 4.15 Point Particle Result for ITO and Gold compared at 0.01M Concentration. 46 4.16 Nanoparticle near an Interface. 46 4.17 Gold Electrode - Scaled Sphere in the Range of 1-50 Angstroms. 47 4.18 Gold Electrode - Scaled Sphere in the Range of 50-400 Angstroms. 47 4.19 ITO Electrode - Scaled Sphere in the Range of 1-50 Angstroms. 48 4.20 ITO Electrode - Scaled Sphere in Range of 50-400 Angstroms. 48 5.1 Illustration of the Third Body Effect of the Electrode. 49 5.2 Diagram of the Three-Body System considered by Silbey et al.80 ......... 50 5.3 Lateral Geometry. 51 5.4 Vertical Geometry. 51 5.5 An Illustration of the Ratio of the Van der Waals Force between Molecules when compared to the Interaction in Free Space..80 .................... 51 5.6 Diagrammatic of Field Penetration into the Electrode. 53 5.7 Coordinate System. 53 5.8 Gold - Comparison against the Potential in the Bulk at 0.01 M Electrolyte Con- centration. 54 5.9 ITO - Comparison against the Potential in the Bulk at 0.01 M Electrolyte Con- centration. 55 A.1 A point charge near a dielectric-electrolyte interface, where half-space I rep- resents a dielectric with dielectric constant " and half-space II represents the electrolyte medium. The location of the point charge is given by a........ 65 A.2 Point Particle next to Gold with Film. 67 A.3 Point Particle next to ITO with Film. 68 A.4 Comparison of Gold with Film and ITO with Film. 68 A.5 Side-On View of the Characteristic Pancake. 69 A.6 End-On View of the Characteristic Pancake. 69 A.7 General Law over full Range. 71 7 A.8 Short Distance Law. 71 A.9 Long Distance Law. 72 A.10 General Classical.