Modelling and Numerical Simulations II (eBook)

Preface 6
Contributors 19
Chapter 1 Numerical Modeling of Certain Electrochemical Processes 22
I. Elementary Aspects of Electrochemical Reaction 22
II. A Simple Mathematical Model 24
1. Boundary Conditions 27
III. Application in Cathodic Protection 29
1. Iteration Process 31
IV. Analytical Solution to Two Benchmark Problems 32
1. Corrosion Cell 1 33
2. Corrosion Cell 2 34
V. Application in Electrodeposition 35
VI. Analytical Solution to a One-Dimensional Electrodeposition Problem 37
VII. General Framework of Numerical Approximation 40
1. Finite-Difference Method 40
2. Finite-Element Method 41
3. Boundary-Element Method 43
VIII. Implementation of the Finite-Difference Method in Cathodic Protection 44
1. Choosing a Lattice 44
2. Discretization 45
3. Mesh Equations 46
4. Solving the Mesh Equations 47
IX. Implementation of the Boundary-Element Method in Cathodic Protection 52
X. Numerical Implementation of the Boundary-Element Method 54
1. Iteration Procedures 55
XI. Concluding Remarks 58
Chapter 2 Near-Field Optics for Heat-Assisted Magnetic Recording (Experiment, Theory, and Modeling) 73
I. Near-Field Transducers for Heat-Assisted Magnetic Recording 73
II. Modeling Techniques 78
III. Near Field Compared With Far Field 79
IV. Figures of Merit 80
1. Far-Field Transmittance 81
2. Peak Field or Field Intensity 82
3. Percent Dissipated Power in the RecordingMedium 83
4. Temperature Rise in the Recording Medium 84
V. Mechanisms for Enhancement of the Figureof Merit 84
1. Localized Surface Plasmon Resonance 84
2. Lightning Rod Effect 90
3. Dual-Dipole Resonance 92
VI. Comparison Of Near-Field Transducers 93
1. Circular Aperture 95
2. Tapered Rectangular Aperture 96
3. Bow-Tie Aperture 99
4. C Aperture or Ridge Waveguide 102
5. Triangle Antenna 103
6. Bow-Tie Antenna 108
VII. Antenna and Aperture Relationship 111
VIII. Near-Field and Far-Field Relationship 112
1. Radiation from Antennas 116
2. Radiation from Apertures 117
3. Numerical Modeling 119
IX. Photonic Nanojets 124
X. Conclusion 129
Chapter 3 Symmetry Considerations in the Modelling of Light--Matter Interactions in Nanoelectrochemistry 132
I. Introduction 133
II. Optical Properties of Metal Nanoparticles 134
1. Macroscopic Theories 134
2. Discrete-Dipole Approximation 135
3. Optical Properties of Nanoparticles on a Surface 136
4. Towards an Optical Method of Surface Electrochemistry 137
III. Optical Manipulation of Semiconductor Quantum Dots 139
1. Atomic Model of Semiconductor Quantum Dots 139
2. Pseudospectral Method 143
3. Finite-Difference Method 144
IV. Summary 147
Chapter 4 Applications of Computer Simulations and Statistical Mechanics in Surface Electrochemistry 150
I. Introduction 151
II. Molecular Dynamics Simulations of Ion Intercalation in Lithium Batteries 152
1. Molecular Dynamics and Model System 152
2. Simulations and Results 153
III. Lattice-Gas Models of Chemisorbed Systems 155
IV. Calculation of Lattice-Gas Parameters by Density Functional Theory 156
V. Monte Carlo Simulations 161
1. Equilibrium Monte Carlo 161
2. Kinetic Monte Carlo 162
VI. Electrochemical First-Order Reversal CurveSimulations 163
VII. Conclusion 165
Chapter 5 AC-Electrogravimetry Investigation in Electroactive Thin Films 169
I. Introduction 169
II. General Considerations 171
1. Thermodynamics 171
2. Swelling 173
3. Conductivity 174
III. Electrochemical Approach of Electroactive Materials 175
1. Models of the Charge Transport Throughthe Electroactive Film 176
(i) Compact Model (Diffusion--MigrationModel) 176
(ii) Porous Model (Transmission Line Model) 179
2. Calculation of the Impedance 181
(i) Two-Species Problems 181
(ii) Three-Species Problems 185
(iii) Applications of Impedance Analysis 190
IV. Coupled Electrochemical and Gravimetric Approach for Electroactive Materials 192
1. Cyclic Voltammetry and Quartz CrystalMicrobalance 192
2. AC Electrogravimetry 201
(i) Steady State 208
(ii) Dynamic Regime 208
(iii) Electrochemical Impedance 209
(iv) Mass/Potential (Electrogravimetric) Transfer Function 213
(v) Diagnostic Criterion 214
(vi) Simulations 216
V. Experimental 223
1. Basic Microbalance Concepts 223
2. AC-Electrogravimetry Aspects 225
3. Dynamic Characterization of the Frequency/Voltage Converter 227
VI. Examples of Applications of AC Electrogravimetry 228
1. Prussian Blue 228
(i) Film Preparation 230
(ii) Voltammetric and Mass/Potential Curves 230
(iii) Electrogravimetric Transfer Function and Electrochemical Impedance 231
2. Polypyrrole 236
(i) Film Preparation 236
(ii) Voltammetric and Mass/Potential Curves 237
(iii) Electrogravimetric Transfer Function and Electrochemical Impedance 237
3. Complex Polymeric Structures 244
(i) Electrode Preparation 244
(ii) Electrogravimetric Transfer Function and Electrochemical Impedance 245
VII. Conclusion 249
Chapter 6 Monte Carlo Simulations of the Underpotential Deposition of Metal Layers on Metallic Substrates: Phase Transitions and Critical Phenomena 257
I. Introduction: Some Basic Aspects of the Underpotential Deposition Phenomenon 258
II. Some Thermodynamics on the UPD Phenomenon 264
III. Description of the Monte Carlo Simulation Method and the Model for Metal Deposition 269
1. The Lattice Model 269
2. The Grand Canonical Monte Carlo Method 270
(i) Change of Occupation 271
(ii) Diffusion 272
(iii) Calculation of the Coverage 273
3. Interatomic Potential: The Embedded-AtomMethod 273
4. Surface Defects 275
5. Energy Tables 276
IV. Adsorption Isotherms 277
1. Systems Studied and Adsorption Energies 277
2. Evaluation of Adsorption Isothermsfor Defect-Free Surfaces 280
(i) UPD Compared with OPD: First-Order Phase Transitions 280
(ii) The Influence of Temperatureon the Isotherms 283
3. Study of the Influence of Surface Defects 284
(i) Isotherms Corresponding to UPD Systems: The Effect of Kinks and Steps as Compared with the Complete Monolayer 284
(ii) Isotherms Corresponding to OPD Systems: The Formation of Surface Alloys 287
4. Comparison with Experiments 290
V. Dynamic Response of AG Monolayers Adsorbed on AU(100) Upon an Oscillatory Variation of the Chemical Potential 291
1. Dynamic Phase Transitions: Basic Concepts 291
2. Simulation Method 292
3. Dynamic Response of the Coverage Degree 293
4. Dynamic Phase Transitions 295
VI. Conclusions 301
Chapter 7 Topics in the Mathematical Modeling of Localized Corrosion 306
I. General Introduction 306
II. Pitting Corrosion 307
1. Introduction 307
2. General Structure of Pitting Models 309
3. Review of Recent Models 310
(i) Models of Pit Growth 310
(ii) Models of Pit Growth and Repassivation 315
4. Transport in Concentrated ElectrolyteSolutions 322
5. Concluding Remarks 328
III. Galvanic Coupling at the Interface BetweenTwo Metals 329
1. Theoretical Description of the Currents and Potentials at the Interface of Two Metals 329
2. Application to the Al--Cu Coupling 331
(i) Mathematical Model 331
(ii) Experimental 335
(iii) Experimental Results and Discussion 335
(iv) Comparison Between Theoretical Calculations and Experimental Observations 337
3. Conclusions 340
IV. Impedance in a Confined Medium 340
1. Introduction 340
2. Experimental 342
3. Theory 344
4. Results and Discussion 350
5. Conclusions 354
Chapter 8 Density-Functional Theory in External Electric and Magnetic Fields 358
I. Scope of this Chapter 358
II. Elements of the Quantum Mechanicsof Many-Electron Systems 360
1. Hamiltonians and Wave Functions 360
2. Density Matrices and Density Functionals 364
3. Functionals and Their Derivatives 369
III. The Hohenberg--Kohn Theorem 370
1. Enunciation and Discussionof the Hohenberg--Kohn Theorem 370
2. A Simple Example: Thomas--Fermi Theory 376
IV. The Exchange--Correlation Energy 378
1. Definition of the Exchange--Correlation Energy 378
2. Interpretation of the Exchange--Correlation Energy 381
3. Selected Exact Propertiesof the Exchange--Correlation Energy 382
V. The Kohn--Sham Equations 384
1. Self-Consistent Single-Particle Equations and Ground-State Energies 385
2. Single-Particle Eigenvalues and Excited-State Energies 388
VI. An Overview of Approximate Exchange--Correlation Functionals 392
1. Local Functionals: LDA 393
2. Semilocal Functionals: GEA, GGA and Beyond 396
3. Orbital Functionals and Other Nonlocal Approximations: Hybrids, Meta-GGA, SIC,OEP, etc. 398
4. Performance of Approximate Functionals:A Few Examples 403
VII. External Electric And Magnetic Fields 407
1. Magnetic Fields Coupling to the Spins: SDFT 407
2. Brief Remarks on Relativistic DFT 410
3. Magnetic Fields Coupling to Spins and Currents: CDFT 410
4. Electric Fields 415
5. Polarization and Magnetization 416
VIII. Outlook 418
Chapter 9 Acoustic Microscopy Applied to Nanostructured Thin Film Systems 426
I. Introduction 426
II. Principle of the Scanning Acoustic Microscope 429
1. Imaging Mechanism 429
2. Description of Acoustic Lens 432
(i) Piezoelectric Transducer 432
(ii) Buffer Rod 433
(iii) Lens 433
(iv) Acoustic Antireflection Coating 435
III. Resolution 436
IV. Principle Of Quantitative Data Acquisition 439
1. V(z) Curve 439
2. Phase Change 441
3. Theory of the Surface Acoustic Wave Velocity Measurement 444
4. Optimizing Measurement Precision 445
V. Contrast 446
1. Reflectance Function 446
2. Reflectance Function for Layered Media 447
3. Contrast Enhancement Caused by Discontinuities 453
4. Computer Simulation 457
5. Experimental Result 461
VI. Conclusion 465
Chapter 10 Current Distribution in Electrochemical Cells: Analytical and Numerical Modeling 468
I. Introduction and Overview 468
II. Significance of Modeling the Current Distribution 469
III. Experimental Determination of the CurrentDistribution 470
IV. Analytical Derivation of the Current Distribution 471
1. The Current Density 471
2. Material Balance 472
3. Boundary Conditions 474
4. General Solution Procedure 476
5. Thin Boundary Layer Approximation 477
V. Common Approximations for the Current Distribution 479
1. Primary Distribution: s+c 480
2. Secondary Distribution: +sc 483
3. Mass Transport Controlled Distribution: c+a 487
4. Tertiary Distribution: cs(``Mixed Control'') 489
VI. Scaling Analysis of Electrochemical Cells 490
VII. Transport Effects On Kinetically Controlled Systems 494
VIII. Comparison of Analytical and Numerical Solutions 495
IX. A Simplified Solution Algorithm 496
X. Numerical Procedures for Solving the LaplaceEquation 497
1. The Finite-Difference Method 498
(i) Methods of Solving the Finite-Difference Equation 500
2. The Finite-Element Method 502
3. Boundary-Element Methods 502
4. Orthogonal Collocation 503
XI. Numerically Implemented Solutions for the Current Distribution 504
XII. Determination of the Current Distributionin Special Applications 506
1. Multiple Simultaneous Electrode Reactions, Including Alloy Codeposition and Gas Coevolution 506
2. Moving Boundaries in Deposition and Dissolution Applications 508
3. Electropolishing, Leveling, and Anodizing 509
4. Current Distribution on Resistive Electrodes 509
5. Current Distribution in the Metallization of Through-Holes, Blind Vias, and Trenches 510
Index 519