Early work on modeling lithium batteries, performed prior to the ready availability of high-speed digital computers, used simplified models neglecting kinetic or cencentration effects, assuming constant properties, or neglecting the separator, in order to obtain a close approximation to battery behavior within the limits of computational power available at the time. Today's computers can easily simulate the entire cell sandwich model, and simply present the model in the best form developed to date in section 3, followed by considerations of special situations which are not essential to the basic modeling framework. Simplifying cases which have contributed to our understanding of the lithium battery are presented in section 5. Finally, we discuss applications of modeling, such as interpreting experimental data and optimizing geometric parameters.

Multicomponent mass-transport in cation-exchange membranes involves the movement of multiple species whose motion is coupled one to another. This phenomenon mediates the performance of numerous electrochemical and water purification technologies. This work presents and validates against experiment a mathematical model for multicomponent mass transport in phase-separated cation-exchange membranes (e.g., perfluorinated sulfonic-acid ionomers). Stefan-Maxwell-Onsager theory describes concentrated-solution transport. Hydrodynamic theory provides constitutive relations for the solute/solvent, solute/membrane, and solvent/membrane friction coefficients. Classical porous-medium theories scale membrane tortuosity. Electrostatic relaxation creates friction between ions. The model uses calculated ion and solvent partitioning between the external solution and the membrane from Part I of this series and incorporates the corresponding ion speciation into the transport coefficients. The proposed transport model compares favorably to properties (e.g., membrane conductivity, transference numbers, electroosmosis, and permeability) measured in dilute and concentrated aqueous binary and ternary electrolytes. The results reveal that the concentration and type of ions in the external solution alter the solvent volume fraction and viscosity in the hydrophilic pathways of the membrane, changing macroscale ionomer conductivity, permeability, and transference numbers. This work provides a physicochemical framework to predict ion-exchange-membrane performance in multicomponent systems exhibiting coupled transport.

We present and validate a mathematical model for multicomponent thermodynamic activity in phase-separated cation-exchange membranes (e.g., perfluorinated sulfonic-acid ionomers). The model consists of an expression for the free energy of the membrane and of the surrounding electrolyte solution. A modified Stokes-Robinson ionic solvation framework treats the solution-like non-idealities resulting from hydration, electrostatics, ion association, and physical interactions in bulk solution and in ionomer hydrophilic domains. Inside the membrane, a mechanics-based composite approach accounts for the swelling of the hydrophobic matrix. Treating the membrane microstructure as a disordered system of domains calculates steric exclusion of ions. Electroneutrality guarantees that the charge of mobile ions in the membrane is equal to the charge on polymer groups. Osmotic coefficients for electrolytes from literature parameterize solution-like interactions while mechanical and X-ray scattering characterization gives most membrane-specific parameters. Model predictions compare favorably to measured membrane thermodynamics (i.e., water and ion uptake) in dilute and concentrated binary and ternary salt electrolytes and in water vapor. Interactions between ions in the membrane are similar to those present in bulk electrolytes. Our results reveal that water and ion uptake is dictated by a balance between solution-like energetics and membrane swelling.

Shunt currents occur in electrochemical reactors like flow batteries, electrolyzers, and fuel cells where many bipolar cells that are connected in series electrically contact a mobile electrolyte through one or more common fluid distribution manifolds. Shunt currents reduce energy efficiency, and can cause unwanted side reactions including corrosion and gas generation. Equivalent-circuit models have been widely used to examine shunt currents in multi-cell electrochemical reactors. However, a detailed investigation of the interesting electrochemical processes occurring at the edges of the active areas has not been presented. In this work, the generation of shunt currents and their tendency to drive corrosion at the edges of positive electrodes in the most positive cells in a reactor stack are investigated with a comprehensive numerical model. An analytical model based on the penetration of current into a semi-infinite electrode, that can be used in conjunction with traditional equivalent-circuit models to assess the tendency for shunt currents to drive corrosion, is developed and compared to the numerical model. The models provided here can be used to set requirements on maximum allowable port currents in order to achieve a particular durability goal.

Local transport losses limit high-current density performance of fuel-cells, especially at low Pt-loadings. Optimizing catalyst-layer structure and material properties to mitigate these losses is critical for wide-scale fuel-cell commercialization. Carbon supports form the primary porous structure of catalyst layers in which Pt particles are deposited and ionomer films are distributed. Unlike non-porous carbon supports, high surface area carbon support is unique in its relative humidity dependent Pt utilization and performance due to presence of Pt particles in the micropores interior of carbon particles. We model water uptake and Pt utilization in catalyst layers with high-surface-area carbon support. Interactions between functional groups on the carbon pore walls and water play a vital role in controlling water uptake from water vapor by catalyst layers. Ionomer is shown to influence proton access in micropores, even though it may not deposit into the micropores.

Transport through vanadium redox-flow-battery membranes strongly influences cell performance. In this work, we use a multicomponent concentrated-solution model of transport and thermodynamics in phase-separated cation-exchange membranes, the most common separator type, to develop structure-performance relationships. The model incorporates species partitioning into the membrane, thermodynamic nonidealities, and Stefan-Maxwell-Onsager frictions between species. Molecular-thermodynamics and -transport theories parameterize the model. We validate the calculations against measured Coulombic and voltage efficiencies of a vanadium flow battery as a function of current density. Our model shows that species transport is the result of collective interactions between all species present in the system. The magnitude of coupling suggests that predictions made using dilute-solution theory for transport in these systems will be misleading in many situations. As a demonstration of the capabilities of the model, we predict cell performance, incorporating these interactions, as a function of electrolyte concentration and composition and membrane equivalent weight and backbone modulus. We find that electrolytes with high sulfuric acid concentrations provide the greatest cell performance (quantified by maximizing power density at a target energy efficiency). In the case of membrane properties, low equivalent-weight polymers perform better; at high equivalent weights, a low membrane modulus is preferred.

Understanding transport phenomena is a critical component of electrochemical engineering in terms of optimizing energy-conversion and -storage devices including fuel cells and flow batteries. In these systems, transport through the ion-conducting electrolytes including both polymer (e.g., perflurosulfonic-acid (PFSA) membranes) and liquid electrolyte control the overall device performance. In this talk, recent work on the multiscale modeling of ionomers will be discussed including thermodynamic and multiscale continuum treatments of multi-ion transport through the ionomer nanodomains and the relevant changes to the chemical potential resulting in selective ion and water uptake. This treatment involves a simple accounting for polymer backbone deformation, donnan equilibrium, solvation, and interactions with external electrolyte. The continuum modeling at the nanodomain uses a mean-field local-density theory that accounts for ion/ion, ion/water, and ion/polymer effects. In addition, upscaling methodologies using resistor-network approaches informed by experiment that explore phenomenon-dependent transport pathways for ions and electrolyte will be introduced and demonstrated to yield nonlinear and nonintuitive results. The overall modeling results will be compared to experimental property and uptake data and used to explain the nature of observed phenomena including electro-osmosis, ion conduction, and ion transference. Acknowledgements This work was funded under the Fuel Cell Performance and Durability Consortium (FC PAD) funded by the Energy Efficiency and Renewable Energy, Fuel Cell Technologies Office, of the U.S. Department of Energy as well as by the ARPA-E IONICS program under contract DE-AC02-05CH11231.

Reducing the loading of expensive Pt-based catalysts in polymer-electrolyte fuel-cell (PEFC) catalyst layers (CLs) is imperative for wide-scale commercialization. However, a local resistance close to the Pt reaction sites becomes significant at low Pt loadings and hinders adequate performance.1,2 In our previous study, we demonstrated that the local resistance is primarily related to the proton-conducting ionomer thin-film within the CL and consists of: (i) reactant transport resistance due to low permeability of ionomer thin-film (compared to bulk ionomer) and (ii) interfacial resistance due to Pt/sulfonic-moiety interactions.3 Thus, there is need to optimize ionomer design to minimize CL resistance. In this talk, we investigate the impact of ionomer chemistry, content, and equivalent weight (EW) on CL resistance. Using H2 limiting-current measurements, the total CL resistance is segregated into transport and interfacial components. CL resistance increases monotonically with ionomer content, primarily due to thicker ionomer films. Low EW ionomers demonstrate higher interfacial resistance due to high density of sulfonic groups, whereas higher EW ionomers exhibit higher transport resistance due to low-reactant permeability associated with low-water uptake. These findings are rationalized through comparison of properties and structure/function relationships elucidated by ionomer thin-film studies on planar substrates, thus bringing together model and in-situ studies. Further, time-permitting, the impact of carbon support (low versus high surface area carbon) on local CL resistance will also be discussed, mainly via modeling water uptake and proton pathways. Overall, the knowledge gained demonstrate pathways towards ameliorating the local resistance and enabling high performing low Pt-loaded PEFCs. Acknowledgements We would like to thank Sarah A. Berlinger and Robert Darling for helpful discussions. This work was funded under the Fuel Cell Performance and Durability Consortium (FC-PAD) funded by the Energy Efficiency and Renewable Energy, Fuel Cell Technologies Office, of the U. S. Department of Energy under contract number DE-AC02-05CH11231. References: 1. T. A. Greszler, D. Caulk and P. Sinha, J Electrochem Soc, 159, F831 (2012). 2. A. Z. Weber and A. Kusoglu, J Mater Chem A, 2, 17207 (2014). 3. T. Schuler, A. Chowdhury, A. T. S. Freiberg, B. T. Sneed, F. B. Spingler, M. C. Tucker, K. More, C. J. Radke and A. Z. Weber, J Electrochem Soc, 166, F3020 (2019).

Solvent-filled membranes, such as Nafion, are an important class of polymer electrolytes because the imbibed solvent imbues the material with high ionic conductivity, while still allowing the membrane to maintain its mechanical stability. Both of these attributes are essential for membrane electrolytes in numerous energy-storage and -conversion devices including redox-flow batteries. The transport phenomena is complicated due to the multiple ions in the electrolytes employed in these applications and the higher concentrations desired for energy density. Furthermore, while there is understanding on the single-ion transport in these materials, there is a need to develop a self-consistent theory for describing the ion transport and ion partitioning and thermodynamics when interacting with the charged polymer membrane. Moreover, there are few quantitative material design guides for polymers that balance the movement of certain species that are desirable, such as current-carrying ions, against transport of contaminants, additives, or redox-active species, which decrease device performance. To address the above gaps and provide material design insights, this work describes the theory of multi-ion transport in polymer-electrolyte membranes. To specify the thermodynamics of the system, we formulate the membrane and electrolyte solution free energies. The model accounts for long-range electrostatic and short-range solvation and physical interactions between ions. Externally validated solution and membrane properties parameterize the model. The model calculations agree with measured water and ion uptake in dilute and concentrated binary and ternary salt electrolytes, with the uptake being a balance between osmotic and elastic pressures that are impacted by ion specificity and membrane pretreatment.1-2 To model the transport phenomena, we use Stefan-Maxwell-Onsager theory. To connect performance to the molecular-level structure of the polymer, hydrodynamic theory provides constitutive relations for the involved ion/solvent and ion/ and solvent/polymer friction coefficients, with classical porous-media theories scaling tortuosity. The model compares favorably to a number of published measured membrane transport properties1-2 (i.e. conductivity, transference numbers, electroosmosis, and water-transport coefficient) in dilute and concentrated binary and ternary electrolytes. Importantly, the model isolates the contribution of each type of molecular-scale to the measured properties. Acknowledgements: This study was funded with support of the Fuel Cell Performance and Durability Consortium (FC-PAD) funded by the Energy Efficiency and Renewable Energy, Fuel Cell Technologies Office, of the U.S. Department of Energy under contract number DE-AC02- 05CH11231 and, in part, by the Advanced Research Projects Agency - Energy (ARPA-E), U.S. Department of Energy (DOE) under Award Number DEAR0000149 . References Okada, T.; Satou, H.; Okuno, M.; Yuasa, M., Ion and Water Transport Characteristics of Perfluorosulfonated Ionomer Membranes with H+ and Alkali Metal Cations. J. Phys. Chem. B 2002, 106 (6), 1267-1273. Tang, Z.; Svoboda, R.; Lawton, J. S.; Aaron, D. S.; Papandrew, A. B.; Zawodzinski, T. A., Composition and Conductivity of Membranes Equilibrated with Solutions of Sulfuric Acid and Vanadyl Sulfate. J. Electrochem. Soc. 2013, 160 (9), F1040-F1047.