The Electrochemistry Module User’s . - COMSOL Multiphysics

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Electrochemistry ModuleUser’s Guide

Electrochemistry Module User’s Guide 1998–2018 COMSOLProtected by patents listed on www.comsol.com/patents, and U.S. Patents 7,519,518; 7,596,474;7,623,991; 8,457,932; 8,954,302; 9,098,106; 9,146,652; 9,323,503; 9,372,673; and 9,454,625. Patentspending.This Documentation and the Programs described herein are furnished under the COMSOL Software LicenseAgreement (www.comsol.com/comsol-license-agreement) and may be used or copied only under the termsof the license agreement.COMSOL, the COMSOL logo, COMSOL Multiphysics, COMSOL Desktop, COMSOL Server, andLiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are theproperty of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliatedwith, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademarkowners, see www.comsol.com/trademarks.Version: COMSOL 5.4Contact InformationVisit the Contact COMSOL page at www.comsol.com/contact to submit generalinquiries, contact Technical Support, or search for an address and phone number. You canalso visit the Worldwide Sales Offices page at www.comsol.com/contact/offices foraddress and contact information.If you need to contact Support, an online request form is located at the COMSOL Accesspage at www.comsol.com/support/case. Other useful links include: Support Center: www.comsol.com/support Product Download: www.comsol.com/product-download Product Updates: www.comsol.com/support/updates COMSOL Blog: www.comsol.com/blogs Discussion Forum: www.comsol.com/community Events: www.comsol.com/events COMSOL Video Gallery: www.comsol.com/video Support Knowledge Base: www.comsol.com/support/knowledgebasePart number: CM024301

C o n t e n t sChapter 1: IntroductionAbout the Electrochemistry Module14What Can the Electrochemistry Module Do? . . . . . . . . . . . . 14The Electrochemistry Module Physics Interface Guide . . . . . . . . . 15Common Physics Interface and Feature Settings and Nodes. . . . . . 18Electrochemistry Module Study Availability . . . . . . . . . . . . . 18Where Do I Access the Documentation and Application Libraries? . . . . 19Overview of the User’s Guide24Chapter 2: Modeling with ElectrochemistryIntroduction to Electrochemistry Modeling28What is Electrochemistry? . . . . . . . . . . . . . . . . . . . 28Electrochemical Applications . . . . . . . . . . . . . . . . . . 29Fundamentals of Electrochemistry Modeling. . . . . . . . . . . . 29Current Distribution Cases and Choosing the Right Interface toModel an Electrochemical Cell . . . . . . . . . . . . . . . . 31Understanding the Different Approximations for Conservation ofCharge in Electrolytes . . . . . . . . . . . . . . . . . . . 32Modeling Electrochemical Reactions . . . . . . . . . . . . . . . 36Double Layer Capacitance . . . . . . . . . . . . . . . . . . . 43Porous Electrodes . . . . . . . . . . . . . . . . . . . . . . 44Boundary Conditions for Running and Controlling ElectrochemicalCells . . . . . . . . . . . . . . . . . . . . . . . . . . 45Modeling Cyclic Voltammetry . . . . . . . . . . . . . . . . . . 46Common Simplifications when Modeling Electrochemical Cells . . . . . 46Before You Start Building Your Model . . . . . . . . . . . . . . . 48Meshing Advice . . . . . . . . . . . . . . . . . . . . . . . 50Solving Electrochemical Models . . . . . . . . . . . . . . . . . 50Postprocessing Your Solution . . . . . . . . . . . . . . . . . . 55CONTENTS 3

Chapter 3: Electrochemistry InterfacesThe Primary and Secondary Current Distribution Interfaces58The Primary Current Distribution and Secondary Current DistributionInterfaces . . . . . . . . . . . . . . . . . . . . . . . . 58Electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . 61Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 61Porous Electrode. . . . . . . . . . . . . . . . . . . . . . . 62Periodic Condition . . . . . . . . . . . . . . . . . . . . . . 62Edge Electrode. . . . . . . . . . . . . . . . . . . . . . . . 63Electrode Line Current Source . . . . . . . . . . . . . . . . . 64Electrolyte Line Current Source . . . . . . . . . . . . . . . . . 64Electrode Symmetry Axis Current Source . . . . . . . . . . . . . 64Electrolyte Symmetry Axis Current Source . . . . . . . . . . . . . 65Electrode Point Current Source . . . . . . . . . . . . . . . . . 65Electrolyte Point Current Source. . . . . . . . . . . . . . . . . 65The Tertiary Current Distribution, Nernst-Planck InterfaceThe Tertiary Current Distribution, Nernst-Planck Interface66. . . . . . 66Electrolyte . . . . . . . . . . . . . . . . . . . . . . . . . 70Porous Electrode. . . . . . . . . . . . . . . . . . . . . . . 70Separator. . . . . . . . . . . . . . . . . . . . . . . . . . 71Reactions. . . . . . . . . . . . . . . . . . . . . . . . . . 71Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 72Ion Exchange Membrane . . . . . . . . . . . . . . . . . . . . 72Ion Exchange Membrane Boundary . . . . . . . . . . . . . . . . 73Shared Physics Features in the Current Distribution Interfaces75Domain, Boundary, Pair, Edge, and Point Nodes for theElectrochemistry Interfaces . . . . . . . . . . . . . . . . . 75Electrode. . . . . . . . . . . . . . . . . . . . . . . . . . 76Electrode Current Source . . . . . . . . . . . . . . . . . . . 77Electrolyte Current Source . . . . . . . . . . . . . . . . . . . 77Porous Electrode Reaction . . . . . . . . . . . . . . . . . . . 78Porous Matrix Double Layer Capacitance . . . . . . . . . . . . . 78Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . 794 CONTENTS

Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 79Electrode Surface. . . . . . . . . . . . . . . . . . . . . . . 79Electrode Reaction . . . . . . . . . . . . . . . . . . . . . . 81Double Layer Capacitance . . . . . . . . . . . . . . . . . . . 85Internal Electrode Surface . . . . . . . . . . . . . . . . . . . 85Electrolyte Potential. . . . . . . . . . . . . . . . . . . . . 86Electrolyte Current . . . . . . . . . . . . . . . . . . . . . . 86Electrolyte Current Density. . . . . . . . . . . . . . . . . . . 87Electrode-Electrolyte Boundary Interface. . . . . . . . . . . . . . 87Electric Ground . . . . . . . . . . . . . . . . . . . . . . . 88Electric Potential . . . . . . . . . . . . . . . . . . . . . . . 88Electrode Current Density . . . . . . . . . . . . . . . . . . . 88Electrode Current . . . . . . . . . . . . . . . . . . . . . . 89Electrode Power . . . . . . . . . . . . . . . . . . . . . . . 89Harmonic Perturbation . . . . . . . . . . . . . . . . . . . . 90Electrode Potential . . . . . . . . . . . . . . . . . . . . . . 90External Short . . . . . . . . . . . . . . . . . . . . . . . . 90Reference Electrode . . . . . . . . . . . . . . . . . . . . . 91Electric Reference Potential . . . . . . . . . . . . . . . . . . . 91Circuit Terminal . . . . . . . . . . . . . . . . . . . . . . . 92The Electrode, Shell Interface93Boundary, Edge, Point, and Pair Nodes for the Electrode, ShellInterface. . . . . . . . . . . . . . . . . . . . . . . . . 94Electrode. . . . . . . . . . . . . . . . . . . . . . . . . . 95Initial Values. . . . . . . . . . . . . . . . . . . . . . . . 96External Current Density. . . . . . . . . . . . . . . . . . . 96Current Source . . . . . . . . . . . . . . . . . . . . . . . 96Normal Current Density . . . . . . . . . . . . . . . . . . . . 97Electric Insulation. . . . . . . . . . . . . . . . . . . . . . 97Boundary Current Source . . . . . . . . . . . . . . . . . . . 97Ground . . . . . . . . . . . . . . . . . . . . . . . . . . 97Electric Potential . . . . . . . . . . . . . . . . . . . . . . . 98The Electroanalysis Interface99Domain, Boundary, and Pair Nodes for the Electroanalysis Interface . .101Transport Properties . . . . . . . . . . . . . . . . . . . .102Initial Values103. . . . . . . . . . . . . . . . . . . . . . .CONTENTS 5

Electrode Surface in the Electroanalysis Interface. . . . . . . . . .104Electrode Reaction . . . . . . . . . . . . . . . . . . . . .106Theory for the Current Distribution Interfaces108The Nernst-Planck Equations . . . . . . . . . . . . . . . . .108Domain Equations for Primary and Secondary Current Distributions . .109Electrochemical Reactions and the Difference Between a Primary anda Secondary Current Distribution . . . . . . . . . . . . . .110Domain Equations for Tertiary Current Distributions Using the6 CONTENTSNernst-Planck Equations and Electroneutrality . . . . . . . . .112Mass Fluxes and Sources Due to Electrochemical Reactions . . . . .114Stochiometric Coefficients for Double Layer Capacitive Charging . . .115Film Resistance . . . . . . . . . . . . . . . . . . . . . .116Electrode Kinetics Expressions . . . . . . . . . . . . . . . .116Theory for Specific Current Distribution Feature Nodes . . . . . . .118Theory for Electrochemical Heat Sources127Joule Heating Due to Charge Transport . . . . . . . . . . . . .128Heating Due to Electrochemical Reactions . . . . . . . . . . . .128Heating Due to Heat of Mixing . . . . . . . . . . . . . . . .129Theory for the Electrode, Shell Interface130Governing Equations . . . . . . . . . . . . . . . . . . . .130Coupling to Other Physics Interfaces . . . . . . . . . . . . . .130Theory for the Electroanalysis Interface132Electroanalytical Methods . . . . . . . . . . . . . . . . . .132Supporting Electrolyte . . . . . . . . . . . . . . . . . . . .133Domain Equations for the Electroanalysis Interface . . . . . . . . .134Electrodes in the Electroanalysis Interface . . . . . . . . . . . .135The Electroanalytical Butler–Volmer Equation . . . . . . . . . . .137Counter Electrodes and Overall Charge Balance . . . . . . . . . .138Electrode Potentials and Reference Electrodes139Reference Electrodes . . . . . . . . . . . . . . . . . . . .139Boundary Conditions Using Reference Electrode Potentials. . . . . .140Nodes for Handling Electrode Potentials and Reference Electrodes. . .140

Solving Electrochemical Models142General Current Distribution Problems . . . . . . . . . . . . .142Electrochemistry Coupled to Mass Transport . . . . . . . . . . .143Setting up a Study Sequence for Multiphysics Problems . . . . . . .143Time-dependent Problems with Load Steps . . . . . . . . . . . .144Solver Settings . . . . . . . . . . . . . . . . . . . . . . .145Chapter 4: Chemical Species Transport InterfacesThe Transport of Diluted Species Interface148The Transport of Diluted Species in Porous Media Interface . . . . .152Domain, Boundary, and Pair Nodes for the Transport of DilutedSpecies Interface. . . . . . . . . . . . . . . . . . . . .153Transport Properties . . . . . . . . . . . . . . . . . . . .155Turbulent Mixing . . . . . . . . . . . . . . . . . . . . . .157Initial Values. . . . . . . . . . . . . . . . . . . . . . .158Mass-Based Concentrations . . . . . . . . . . . . . . . . . .158Reactions. . . . . . . . . . . . . . . . . . . . . . . . .158No Flux . . . . . . . . . . . . . . . . . . . . . . . . .160Inflow . . . . . . . . . . . . . . . . . . . . . . . . . .160Outflow . . . . . . . . . . . . . . . . . . . . . . . . .161Concentration . . . . . . . . . . . . . . . . . . . . . . .161Flux . . . . . . . . . . . . . . . . . . . . . . . . . . .161Symmetry . . . . . . . . . . . . . . . . . . . . . . . .162Flux Discontinuity . . . . . . . . . . . . . . . . . . . . .162Partition Condition . . . . . . . . . . . . . . . . . . . . .163Periodic Condition . . . . . . . . . . . . . . . . . . . . .164Line Mass Source . . . . . . . . . . . . . . . . . . . . . .164Point Mass Source . . . . . . . . . . . . . . . . . . . . .165Open Boundary . . . . . . . . . . . . . . . . . . . . . .166Thin Diffusion Barrier . . . . . . . . . . . . . . . . . . . .166Thin Impermeable Barrier . . . . . . . . . . . . . . . . . .166Equilibrium Reaction . . . . . . . . . . . . . . . . . . . .167Surface Reactions. . . . . . . . . . . . . . . . . . . . .168Surface Equilibrium Reaction . . . . . . . . . . . . . . . . .168Fast Irreversible Surface Reaction . . . . . . . . . . . . . . .169CONTENTS 7

Porous Electrode Coupling . . . . . . . . . . . . . . . . . .169Reaction Coefficients . . . . . . . . . . . . . . . . . . . .170Electrode Surface Coupling . . . . . . . . . . . . . . . . . .170Porous Media Transport Properties. . . . . . . . . . . . . . .171Adsorption . . . . . . . . . . . . . . . . . . . . . . . .173Partially Saturated Porous Media . . . . . . . . . . . . . . . .174Volatilization . . . . . . . . . . . . . . . . . . . . . . .176Reactive Pellet Bed . . . . . . . . . . . . . . . . . . . . .177Reactions. . . . . . . . . . . . . . . . . . . . . . . . .180Species Source. . . . . . . . . . . . . . . . . . . . . . .181Hygroscopic Swelling . . . . . . . . . . . . . . . . . . . .182Fracture . . . . . . . . . . . . . . . . . . . . . . . . .183The Chemistry Interface184Feature Nodes Available for the Chemistry Interface . . . . . . . .187Reaction . . . . . . . . . . . . . . . . . . . . . . . . .187Species. . . . . . . . . . . . . . . . . . . . . . . . .191Reversible Reaction Group . . . . . . . . . . . . . . . . . .193Equilibrium Reaction Group. . . . . . . . . . . . . . . . . .194Species Group . . . . . . . . . . . . . . . . . . . . . . .196Reaction Thermodynamics . . . . . . . . . . . . . . . . . .196Species Activity . . . . . . . . . . . . . . . . . . . . . .196Species Thermodynamics. . . . . . . . . . . . . . . . . . .196The Nernst-Planck-Poisson Equations Interface198The Electrophoretic Transport Interface200Common Settings for the Species nodes in the Electrophoretic. . . . . . . . . . . . . . . . . . .204Diffusion and Migration Settings . . . . . . . . . . . . . . . .Transport Interface205Domain, Boundary, and Pair Nodes for the Electrophoretic TransportInterface. . . . . . . . . . . . . . . . . . . . . . . .Solvent. . . . . . . . . . . . . . . . . . . . . . . . .Porous Matrix Properties8 CONTENTS206207. . . . . . . . . . . . . . . . . .207Fully Dissociated Species . . . . . . . . . . . . . . . . . . .208Uncharged Species . . . . . . . . . . . . . . . . . . . . .208Weak Acid . . . . . . . . . . . . . . . . . . . . . . . .208Weak Base . . . . . . . . . . . . . . . . . . . . . . . .208

Ampholyte . . . . . . . . . . . . . . . . . . . . . . . .209Protein. . . . . . . . . . . . . . . . . . . . . . . . .209Current Source . . . . . . . . . . . . . . . . . . . . . .209Initial Potential. . . . . . . . . . . . . . . . . . . . . . .209Current . . . . . . . . . . . . . . . . . . . . . . . . .210Current Density . . . . . . . . . . . . . . . . . . . . . .210Insulation . . . . . . . . . . . . . . . . . . . . . . . . .210Potential . . . . . . . . . . . . . . . . . . . . . . . . .210Species Source. . . . . . . . . . . . . . . . . . . . . . .211Initial Concentration . . . . . . . . . . . . . . . . . . . .211Concentration . . . . . . . . . . . . . . . . . . . . . . .211No Flux . . . . . . . . . . . . . . . . . . . . . . . . .211Flux . . . . . . . . . . . . . . . . . . . . . . . . . . .212Inflow . . . . . . . . . . . . . . . . . . . . . . . . . .212Outflow . . . . . . . . . . . . . . . . . . . . . . . . .213The Surface Reactions Interface214Boundary, Edge, Point, and Pair Nodes for the Surface ReactionsInterface. . . . . . . . . . . . . . . . . . . . . . . .215Surface Properties . . . . . . . . . . . . . . . . . . . . .216Initial Values217. . . . . . . . . . . . . . . . . . . . . . .Reactions. . . . . . . . . . . . . . . . . . . . . . . . .217Surface Concentration . . . . . . . . . . . . . . . . . . . .218Theory for the Transport of Diluted Species Interface219Mass Balance Equation . . . . . . . . . . . . . . . . . . . .220Equilibrium Reaction Theory . . . . . . . . . . . . . . . . .221Convective Term Formulation. . . . . . . . . . . . . . . . .223Solving a Diffusion Equation Only. . . . . . . . . . . . . . .Mass Sources for Species Transport. . . . . . . . . . . . . .223224Adding Transport Through Migration . . . . . . . . . . . . . .225Supporting Electrolytes . . . . . . . . . . . . . . . . . . .227Crosswind Diffusion . . . . . . . . . . . . . . . . . . . .228Danckwerts Inflow Boundary Condition . . . . . . . . . . . . .229Transport of Diluted Species in Porous Media . . . . . . . . . . .229Convection . . . . . . . . . . . . . . . . . . . . . . . .231Convective Term Formulation. . . . . . . . . . . . . . . . .232Diffusion . . . . . . . . . . . . . . . . . . . . . . . . .233CONTENTS 9

Dispersion . . . . . . . . . . . . . . . . . . . . . . . .234Adsorption . . . . . . . . . . . . . . . . . . . . . . . .235Reactions. . . . . . . . . . . . . . . . . . . . . . . . .236Theory for the Reactive Pellet Bed . . . . . . . . . . . . . . .237References . . . . . . . . . . . . . . . . . . . . . . . .242Theory for the Electrophoretic Transport Interface243Theory for the Surface Reactions Interface249Governing Equations for the Surface Concentrations . . . . . . . .249Governing Equations for the Bulk Concentrations . . . . . . . . .250ODE-Formulations for Surface Concentrations . . . . . . . . . .252Surface Reaction Equations on Deforming Geometries . . . . . . .253Reference for the Surface Reactions Interface . . . . . . . . . . .254Theory for the Coupling of Mass Transport toElectrochemical Reactions255Molar Sources and Sinks . . . . . . . . . . . . . . . . . . .255Mass Sources and Sinks . . . . . . . . . . . . . . . . . . .256Chapter 5: Fluid Flow InterfacesThe Darcy’s Law Interface258Domain, Boundary, Edge, Point, and Pair Nodes for the Darcy’s10 C O N T E N T SLaw Interface . . . . . . . . . . . . . . . . . . . . . .259Fluid and Matrix Properties . . . . . . . . . . . . . . . . . .261Mass Source. . . . . . . . . . . . . . . . . . . . . . .262Initial Values. . . . . . . . . . . . . . . . . . . . . . .262Porous Electrode Coupling . . . . . . . . . . . . . . . . . .262Electrode Surface Coupling . . . . . . . . . . . . . . . . . .263Pressure . . . . . . . . . . . . . . . . . . . . . . . . .263Mass Flux. . . . . . . . . . . . . . . . . . . . . . . . .264Inlet . . . . . . . . . . . . . . . . . . . . . . . . . . .264Symmetry . . . . . . . . . . . . . . . . . . . . . . . .265No Flow . . . . . . . . . . . . . . . . . . . . . . . . .265Flux Discontinuity . . . . . . . . . . . . . . . . . . . . .265

Outlet . . . . . . . . . . . . . . . . . . . . . . . . . .266Cross Section . . . . . . . . . . . . . . . . . . . . . . .266Thickness. . . . . . . . . . . . . . . . . . . . . . . . .266The Free and Porous Media Flow Interface268Domain, Boundary, Point, and Pair Nodes for the Free and PorousMedia Flow Interface . . . . . . . . . . . . . . . . . . .269Fluid Properties . . . . . . . . . . . . . . . . . . . . . .270Fluid and Matrix Properties . . . . . . . . . . . . . . . . . .271Volume Force . . . . . . . . . . . . . . . . . . . . . . .272Forchheimer Drag . . . . . . . . . . . . . . . . . . . . .272Porous Electrode Coupling . . . . . . . . . . . . . . . . . .272Initial Values. . . . . . . . . . . . . . . . . . . . . . .273Electrode-Electrolyte Interface Coupling . . . . . . . . . . . . .273Wall274. . . . . . . . . . . . . . . . . . . . . . . . . .The Brinkman Equations Interface275Domain, Boundary, Point, and Pair Nodes for the BrinkmanEquations Interface . . . . . . . . . . . . . . . . . . . .277Fluid and Matrix Properties . . . . . . . . . . . . . . . . . .278Forchheimer Drag . . . . . . . . . . . . . . . . . . . . .279Mass Source. . . . . . . . . . . . . . . . . . . . . . .279Volume Force . . . . . . . . . . . . . . . . . . . . . . .280Initial Values. . . . . . . . . . . . . . . . . . . . . . .280Fluid Properties . . . . . . . . . . . . . . . . . . . . . .280Theory for the Darcy’s Law Interface282Darcy’s Law — Equation Formulation . . . . . . . . . . . . . .282Theory for the Free and Porous Media Flow Interface284Reference for the Free and Porous Media Flow Interface. . . . . . .284Theory for the Brinkman Equations Interface285About the Brinkman Equations. . . . . . . . . . . . . . . .285Brinkma

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