Many battery electrodes contain ensembles of nanoparticles that phase-separate upon (de)intercalation. In such electrodes, the fraction of actively-intercalating particles directly impacts cycle life: a vanishing population concentrates the current in a small number of particles, leading to current hotspots. Reports on the active particle population in the phase-separating electrode lithium iron phosphate (LFP) vary widely, ranging from around 0% (particle-byparticle) to 100% (concurrent intercalation). Using synchrotron-based X-ray microscopy, we probed the individual state-of-charge for over 3,000 LFP particles. We observed that the active population depends strongly on the cycling current, exhibiting particle-by-particle-like behaviour at low rates and increasingly concurrent behaviour at high rates, consistent with our phase-field porous electrode simulations. Contrary to intuition, the current density, or current per active internal surface area, is nearly invariant with the global electrode cycling rate. Rather, the electrode accommodates higher current by increasing the active particle population. This behaviour results from thermodynamic transformation barriers in LFP, and such a phenomenon likely extends to other phase-separating battery materials. We propose that modifying the transformation barrier and exchange current density can increase the active population and thus the current homogeneity. This could introduce new paradigms to enhance the cycle life of phaseseparating battery electrodes. 3Electrochemical systems can provide clean and efficient routes for energy conversion and storage. Many electrochemical devices such as batteries, fuel cells, and supercapacitors consist of porous electrodes containing ensembles of nanoparticles 1 . For typical microstructures, the particle density can reach as high as 10 15 cm -3 . To further increase complexity, many intercalation battery electrodes, such as graphite 2 , lithium iron phosphate 3,4 , lithium titanate 5 , and spinel lithium nickel manganese oxide 6 , phase-separate upon (de)intercalation. Such electrodes are physically and chemically heterogeneous on the nanoscale, and likely exhibit inhomogeneous current distributions.In phase-separating electrodes, the active particle population is a crucial factor in determining the overall electrode current and the degree of current homogeneity. The electrode current is given by:where is the reaction area of the th actively-intercalating particle, and is the current density of that particle. Under the approximation of similar particle size, we obtain the final expression in equation 1, where ̅ is the average current density of all actively-intercalating particles, is the total internal surface area of all particles (rather than the projected electrode area), and is the so-called active population. When approaches 0%, the electrode intercalates particle-by-particle with a heterogeneous current distribution; when approaches 100%, the electrode intercalates concurrently with a more homogeneous current ...

Porous electrode theory, pioneered by John Newman and collaborators, provides a macroscopic description of battery cycling behavior, rooted in microscopic physical models. Typically, the active materials are described as solid solution particles with transport and surface reactions driven by concentration fields, and the thermodynamics are incorporated through fitting of the open circuit potential. However, this approach does not apply to phase separating materials, for which the voltage is an emergent property of inhomogeneous concentration profiles, even in equilibrium. Here, we present a general framework, "multiphase porous electrode theory", based on nonequilibrium thermodynamics and implemented in an open-source software package called "MPET". Cahn-Hilliard-type phase field models are used to describe the active materials with suitably generalized models of interfacial reaction kinetics. Classical concentrated solution theory is implemented for the electrolyte phase, and Newman's porous electrode theory is recovered in the limit of solid solution active materials with Butler-Volmer kinetics. More general, quantum-mechanical models of faradaic reactions are also included, such as Marcus-Hush-Chidsey kinetics for electron transfer at electrodes, extended for concentrated solutions. The full model and implementation are described, and a variety of example calculations are presented to illustrate the novel features of the software compared to existing battery models. Lithium-based batteries have growing importance in global society 1 as a result of increased prevalence of portable electronic devices, 2 and their enabling role in the transition toward renewable energy sources.3 For example, lithium batteries can help mitigate intermittency of renewable energy sources such as solar power, and lithium battery powered electric vehicles are facilitating movement away from liquid fossil fuels for transportation. Each of these growing areas demands high performance batteries, with requirements specific to the particular needs of the application driving specialized battery design for sub-markets. Thus, it is critical that battery models be based on the underlying physics, enabling them to greatly facilitate cell design to take best advantage of the existing battery technologies.Lithium-ion batteries are generally constructed using two porous electrodes and a porous separator between them. The porous electrodes consist of various interpenetrating phases including electrolyte, active material, binder, and conductive additive. A schematic is shown in Figure 1. In a charged state, most of the lithium in the cell is contained in the active material within the negative electrode. During discharge, the lithium undergoes transport to the surface of the active material, electrochemical reaction to move from the active material to the electrolyte, transport through the electrolyte to the positive electrode, and reaction and transport to move into the active material of the positive electrode.4,5 Physical models must capture eac...

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The Marcus-Hush-Chidsey (MHC) model is well known in electro-analytical chemistry as a successful microscopic theory of outer-sphere electron transfer at metal electrodes, but it is unfamiliar and rarely used in electrochemical engineering. One reason may be the difficulty of evaluating the MHC reaction rate, which is defined as an improper integral of the Marcus rate over the Fermi distribution of electron energies. Here, we report a simple analytical approximation of the MHC integral that interpolates between exact asymptotic limits for large overpotentials, as well as for large or small reorganization energies, and exhibits less than 5% relative error for all reasonable parameter values. This result enables the MHC model to be considered as a practical alternative to the ubiquitous Butler-Volmer equation for improved understanding and engineering of electrochemical systems.

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