Comptehensive Treatise of Electrochemistry Volume 6 Electrodics: Transport

Edited by

Ernest Yeager Case Western Reserve University Cleveland, Ohio

J. O'M. Bockris Texas A&M University College Station, Texas

Brian E. Conway University of Ottawa Ottawa, Ontario, Canada

S. Sarangapani Union Carbide Corporation Parma, Ohio

PLENUM PRESS • NEW YORK AND LONDON

Contents

1. Fundamentals of Transport Phenomena in Electrolytic Systems N.lbl

1. Introduction 1 2. Fundamental Equations of Mass Transport: General Form . . . . 4

2.1. Flux and Velocity of a Species 4 2.2. Driving Forces 4 2.3. Relationship Between Fluxes and Driving Forces 5 2.4. Systems of Reference 7 2.5. Friction Coefficients 9 2.6. Conservation Equations 10 2.7. Recapitulation of Basic Equations and Comparison with Number of

Variables 13 2.8. Scope of Applications of Generalized Equations 15

3. Fundamental Equations of Mass Transport: Approximation of Ideal and Dilute Solutions 16 3.1. Dilute Solutions 16 3.2. Ideal Dilute Solution 17

4. Charge Transport 18 4.1. Electric Current in Solution 18 4.2: Electrode Current 20 4.3. Local and Average Electrode Current Density 22 4.4. Current Efficiency 22 4.5. Electric Mobility; Conductivity 23 4.6. Transport Number 25 4.7. Transport of Charges through the Diffusion Layer 27

5. Elimination of Electric Potential from the Basic Equations . . . . 31 5.1. Ideal Dilute Solution 31 5.2. Nonideal, Concentrated Solutions of a Single Electrolyte . . . 33 5.3. Remarks about the Diffusion Coefficients 35 5.4. Interfacial Flux Densities; Interfacial Velocities 37

xiii

XIV CONTENTS

6. Topics Related to the Electroneutrality Condition 39 6.1. Validity of the Electroneutrality Condition and Application of

Poisson's Law 39 6.2. Concept of Interfacial Quantities 41 6.3. Overlap of the Diffusion Layer and the Double Layer . . . . 42

7. Some General Concepts Related to Mass and Charge Transport . . . 43 7.1. Diffusion Potential 43 7.2. Concentration Overpotential 44 7.3. Mass Transport Control 47

8. Determination of Quantities of Practical Interest: Theoretical and Semi-empirical Methods 50 8.1. Quantities of Practical Interest • 50 8.2. Outline of Theoretical Method 51 8.3. Semiempirical Procedures 54

9. Simplified Approach to Mass Transport in Electrolytic Systems . . . 54 9.1. Approximation of the Ideal Dilute Solution and the Problem of the

Diffusion Coefficient 54 9.2. Problem of the Species 56

10. Historical Note 58 11. Scope of Volume 6 61 References 62

2. Diffusion in the Absence of Convect ion: Steady State and Nonsteady State

S. L. Marchiano and A. J. Arvia

1. Transport Phenomena in Electrochemical Systems 65 2. Migration Flux 68 3. Diffusional Flux 68 4. Convective Flux 72 5. General Expression of the Mass Transfer Equation 72

5.1. Limiting Cases of Equation (27) 73 5.2. Migration Contribution 74

6. Pure Diffusion and the Mathematical Solution of the Diffusion Equation 75 7. Stationary State 77

7.1. Infinite-Plane Interface 78 7.2. Spherical Shell 80 7.3. Cylindrical Interface 81

8. Resolution of the Fick Equation: The Nonstationary State . . . . 82 8.1. Boundary Conditions for the Nonstationary Solutions under a Poten­

tial Step Perturbation 82 8.2. Nonstationary Concentration Distribution Equation: Ideal Semi­

infinite Plane Diffusion 84 8.3. Spherical Diffusion 90 8.4. Expanding Sphere Electrode 92 8.5. Cylindrical Diffusion 100

CONTENTS XV

9. Solution of the Diffusion Equation under a Constant Flux: Galvanostatic Conditions 103 9.1. Reversible Electrochemical Reaction 105 9.2. Irreversible Electrochemical Process 106 9.3. Consecutive Diffusion-Controlled Electrochemical Reactions . 107 9.4. Instantaneous Current Pulse 113

10. Diffusion Equation with Time-Dependent Boundary Conditions . . 114 10.1. Linear Potential/Time Perturbation 115 10.2. Mathematical Procedures 116 10.3. Solution by the Laplace Transform Method 116 10.4. Reversible Reaction 118 10.5. Irreversible and Quasireversible Electrochemical Reactions . 121

11. Time-Dependent Boundary Conditions: Sinusoidal Perturbations . . 125 11.1. Potential Sinusoidal Perturbation 126 11.2. Sinusoidal Current Perturbation 128

References 129

3. Convective Mass Transport N. Ibl and O. Dossenbach

1. Introduction 133 1.1. Convective Mass Transport: Qualitative Considerations . . . . 1 3 7 1.2. Nernst Model for the Diffusion Layer 139 1.3. Mass Transfer Coefficient 140 1.4. Application Example for the Nernst Model 140 1.5. Current-Voltage Curve: Limiting Current 141 1.6. Historical Note 142

2. Theoretical Approach Based on Fundamental Equations 143 2.1. Basic Equations 143 2.2. Prandtl Boundary-Layer Simplifications 146 2.3. Mass Transfer to a Plate in Laminar Flow 147

3. Dimensional Analysis 155 3.1. Principle 155 3.2. Dimensionless Groups 156 3.3. 7r-Theorem 157 3.4. Application Examples 158 3.5. Concluding Remarks 160

4. Analogy between Mass, Heat, and Momentum Transport . . . . 161 4.1. General Aspects 161 4.2. Application Example for a Theoretical Approach 164 4.3. Remarks on the Boundary Layers for Mass, Heat, and Momentum

Transport 167 4.4. Advantage of Using Dimensionless Groups in Analogy Consider­

ations 168 4.5. Considerations on Analogy with Momentum Transport . . . . 170

5. Mass Transport in Turbulent Flow 172 5.1. Fluctuating and Time-Averaged Quantities 172

xvi CONTENTS

5.2. Mass Transport Correlations 176 5.3. General Remarks 180

6. Influence of Migration on Limiting Currents 182 6.1. Introduction 182 6.2. Theoretical Approach 184 6.3. Approximate Method 190

7. Selected Systems of Interest to the Electrochemist 192 7.1. Introduction 192 7.2. Industrial Processes 192 7.3. Natural Convection 192 7.4. Channel Flow 200

8. Mass Transport Coupled with Chemical Reactions 205 9. Examples of Practical Calculations 208

9.1. Introduction \ . 208 9.2. Copper Deposition in a Channel Cell 210 9.3. Copper Deposition Under Natural Convection Conditions . . . 214 9.4. Concluding Remarks 216

10. Evaluation of Interfacial Concentrations 218 11. Applications of Convective Mass Transport Theory 220

11.1. Generalities 220 11.2. Electrochemical Systems as Models for Transport Measurements . 220 11.3. Limitation of Reaction Rate by Mass Transport: Optimization . 223 11.4. Scaling-Up Effects 230

References 234

4. Current Distribution

N. Ibl

1. Introduction: Practical Importance of Current and Potential Distribution 239 2. Experimental Methods 241 3. Main Types of Current and Potential Distributions 242 4. Outline of Theory of Primary and Secondary Distribution . . . . 243 5. Primary Distribution 245

5.1. Boundary Conditions 245 5.2. Kasper Method 246 5.3. Further Examples of Primary Distribution: Analytical Solutions for

Plane Parallel and Disk Electrodes 247 5.4. Numerical Integration of the Laplace Equation 251 5.5. Some General Aspects of Primary Distribution 254

6. Secondary Distribution 255 6.1. Qualitative Considerations 255 6.2. Semiquantitative Concepts and Tests Used in Electroplating . . 258 6.3. Quantitative Treatment 260

7. Distribution with Finite Electrode Conductivity, Three-Dimensional Elec­trodes 277 7.1. Current Distribution 277 7.2. Variation of Potential 287

CONTENTS XVÜ

8. Tertiary Distribution 289 8.1. General Remarks 289 8.2. Distribution Over a Microprofile—Microscopic Throwing Power 290 8.3. Distribution Over a Macroprofile—Macroscopic Throwing Power . 291 8.4. Case of Three-Dimensional Electrodes 296 8.5. Particular Case of Limiting Current-Mass Transport Control . . 299

9. Influence of a Few Other Kinds of Overpotentials 303 10. Current Distribution for the Case of Simultaneous Electrode Reactions . 304 11. Historical Note 308 References 310

5. Porous Electrodes

Yu. A. Chizmadzhev and Yu. G. Chirkov

1. Introduction 317 1.1. Porous Electrode 318 1.2. Types of Porous Electrodes 320

2. Porous Media 321 2.1. Properties of Porous Media 321 2.2. Experimental Methods for Determining Characteristics of Porous

Media 325 3. Capillary Equilibrium 330

3.1. Branching Processes 333 3.2. Lattice Models 335 3.3. Experimental Facts 342

4. Transport Processes 345 4.1. Convective Diffusion 345 4.2. Diffusion in Gases 353 4.3. Effective Electrical Conductivity 357

5. Electrochemical Activity of Electrodes 362 5.1. Two-Phase Electrodes 362 5.2. Three-Phase Hydrophilic Electrodes 368 5.3. Hydrophobie Electrodes 378

6. Conclusion 384 References 385

6. Porous Flow-Through and Fluidized-Bed Electrodes

F. Goodridge and A. R. Wright

1. Introduction 393 1.1. Aims and Treatment of the Chapter 393 1.2. General Considerations 394

2. Hydrodynamic and Mass Transfer Aspects of Three-Dimensional Elec­trodes 398 2.1. Hydrodynamic Aspects 398 2.2. Mass Transfer Aspects 404

XVIII CONTENTS

3. Monopolar Three-Dimensional Electrodes 407 3.1. Mathematical Models 407 3.2. Industrial Aspects 425

4. Bipolar Three-Dimensional Electrodes 429 4.1. Mathematical Models 430 4.2. Experimental Aspects 436 4.3. Industrial Aspects 438

References 440

7. Gas-Evolving Electrodes

Helmut Vogt

1. Characterization of Gas-Evolving Electrodes 445 2. Regimes in Gas Evolution 446 3. Analogy with Boiling 447 4. Nucleate Gas Evolution 447

4.1. Nucleation 447 4.2. Bubble Growth 449 4.3. Bubble Departure 453

5. Mass Transfer 455 5.1. Empirical Correlations 456 5.2. Theoretical Approaches 456 5.3. Mass Transfer with Superposition of Liquid Bulk Flow . . . . 464

6. Heat Transfer 465 7. Film Gas Evolution 466 8. Bubble-Filled Electrolytes 471

8.1. Effective Conductivity 471 8.2. Rising Velocity of Gas Bubbles 473 8.3. Current Distribution and Ohmic Resistance 476 8.4. Influence of Temperature and Pressure 480 8.5. Electrode Geometry and Flow Conditions 481

References 483

Annotated Author Index 491 Subject Index 493