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As the design of single-component battery electrodes has matured, the battery industry has turned to hybrid electrodes with blends of two or more active materials to enhance battery performance. Leveraging the best properties of each material while mitigating their drawbacks, multi-component hybrid electrodes open a vast new design space that could be most efficiently explored through simulations. In this article, we introduce a mathematical modeling framework and open-source battery simulation software package for Hybrid Multiphase Porous Electrode Theory (Hybrid-MPET), capable of accounting for the parallel reactions, phase transformations and multiscale heterogeneities in hybrid porous electrodes. Hybrid-MPET models can simulate both solid solution and multiphase active materials in hybrid electrodes at intra-particle and inter-particle scales. Its modular design also allows the combination of different active materials at any capacity fraction. To illustrate the novel features of Hybrid-MPET, we present experimentally validated models of silicon-graphite (Si-Gr) anodes used in electric vehicle batteries and carbon monofluoride (CFx) - silver vanadium oxide (SVO) cathodes used in implantable medical device batteries. The results demonstrate the potential of Hybrid-MPET models to accelerate the development of hybrid electrode batteries by providing fast predictions of their performance over a wide range of design parameters and operating protocols., For code and sample usage, please visit: https://github.com/HarryQL/Hybrid-MPET

Electrosorption is a promising technique for brackish water deionization and waste water remediation. Faradaic materials with redox activity have recently been shown to enhance both the adsorption capacity and the selectivity of electrosorption processes. Development of the theory of electrosorption with redox active materials can provide a fundamental understanding of the electrosorption mechanism and a means to extract material properties from small-scale experiments for process optimization and scale-up. Here, we present an intuitive, physics-based equivalent circuit model to describe the electrosorption performance of redox active materials, which is able to accurately fit experimental cyclic voltammetry measurements. The model can serve as an efficient and easy-to-implement tool to evaluate properties of redox active materials and help to distinguish between the transport-limited and reaction-limited regimes in electrosorption processes. And the extracted intrinsic material properties can be further incorporated into process models under lower supporting electrolyte concentrations for realistic electrosorption applications., 26 pages, 8 figures

Experiments on single crystal Li$_X$FePO$_4$ (LFP) nanoparticles indicate rich nonequilibrium phase behavior, such as suppression of phase separation at high lithiation rates, striped patterns of coherent phase boundaries, nucleation by binarysolid surface wetting and intercalation waves. These observations have been successfully predicted (prior to the experiments) by 1D depth-averaged phase-field models, which neglect any subsurface phase separation. In this paper, using an electro-chemo-mechanical phase-field model, we investigate the coherent non-equilibrium subsurface phase morphologies that develop in the $ab$- plane of platelet-like single-crystal platelet-like Li$_X$FePO$_4$ nanoparticles. Finite element simulations are performed for 2D plane-stress conditions in the $ab$- plane, and validated by 3D simulations, showing similar results. We show that the anisotropy of the interfacial tension tensor, coupled with electroautocatalytic surface intercalation reactions, plays a crucial role in determining the subsurface phase morphology. With isotropic interfacial tension, subsurface phase separation is observed, independent of the reaction kinetics, but for strong anisotropy, phase separation is controlled by surface reactions, as assumed in 1D models. Moreover, the driven intercalation reaction suppresses phase separation during lithiation, while enhancing it during delithiation, by electro-autocatalysis, in quantitative agreement with {\it in operando} imaging experiments in single-crystalline nanoparticles, given measured reaction rate constants.

Computational modeling is vital for the fundamental understanding of processes in Li-ion batteries. However, capturing nanoscopic to mesoscopic phase thermodynamics and kinetics in the solid electrode particles embedded in realistic electrode morphologies is challenging. In particular for electrode materials displaying a first order phase transition, such as LiFePO$_4$, graphite and spinel Li$_4$Ti$_5$O$_{12}$ (LTO), predicting the macroscopic electrochemical behavior requires an accurate physical model. Herein, we present a thermodynamic phase field model for Li-ion insertion in LTO which captures the performance limitations presented in literature as a function of all relevant electrode parameters. The phase stability in the model is based on ab-initio DFT calculations and the Li-ion diffusion parameters on nanoscopic NMR measurements of Li-ion mobility, resulting in a parameter free model. The direct comparison with prepared electrodes shows good agreement over three orders of magnitude in the discharge current. Overpotentials associated with the various charge transport processes, as well as the active particle fraction relevant for local hotspots in batteries, are analyzed. It is demonstrated which process limits the electrode performance under a variety of realistic conditions, providing comprehensive understanding of the nanoscopic to microscopic properties. These results provide concrete directions towards the design of optimally performing LTO electrodes.

The Debye-H\" uckel formula for ionic activity coefficients is extended for concentrated solutions by solving a simple model of many-body Coulomb correlations and adding the Born solvation energy. Given the bulk permittivity, our formula is able to fit activity data for diverse electrolytes with only one parameter to adjust the correlation length, which interpolates between the Bjerrum length and the mean ion spacing. The results show that ionic activity in most electrolytes is dominated by three types of electrostatic forces: (i) mean-field charge screening, (ii) solvation, and (iii) Coulomb correlations, both "over-screening" (charge oscillations) and "under-screening" (extending beyond the Debye screening length)., Comment: 5 pages, 3 figures

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.

Multicomponent gas transport is investigated with unprecedented precision by AC impedance analysis of porous YSZ anode-supported solid oxide fuel cells. A fuel gas mixture of H2-H2O-N2 is fed to the anode, and impedance data are measured across the range of hydrogen partial pressure (10-100%) for open circuit conditions at three temperatures (800C, 850C and 900C) and for 300mA applied current at 800C. For the first time, analytical formulae for the diffusion resistance (Rb) of three standard models of multicomponent gas transport (Fick, Stefan-Maxwell, and Dusty Gas) are derived and tested against the impedance data. The tortuosity is the only fitting parameter since all the diffusion coefficients are known. Only the Dusty Gas model leads to a remarkable data collapse for over twenty experimental conditions, using a constant tortuosity consistent with permeability measurements and the Bruggeman relation. These results establish the accuracy of the Dusty Gas model for multicomponent gas diffusion in porous media and confirm the efficacy of electrochemical impedance analysis to precisely determine transport mechanisms.

The basic physics of nucleation in solid \hl{single-crystal} nanoparticles is revealed by a phase-field theory that includes surface energy, chemical reactions and coherency strain. In contrast to binary fluids, which form arbitrary contact angles at surfaces, complete "wetting" by one phase is favored at binary solid surfaces. Nucleation occurs when surface wetting becomes unstable, as the chemical energy gain (scaling with area) overcomes the elastic energy penalty (scaling with volume). The nucleation barrier thus decreases with the area-to-volume ratio and vanishes below a critical size, and nanoparticles tend to transform in order of increasing size, leaving the smallest particles homogeneous (in the phase of lowest surface energy). The model is used to simulate phase separation in realistic nanoparticle geometries for \ce{Li_XFePO4}, a popular cathode material for Li-ion batteries, and collapses disparate experimental data for the nucleation barrier, with no adjustable parameters. Beyond energy storage, the theory generally shows how to tailor the elastic and surface properties of a solid nanostructure to achieve desired phase behavior., 7 pages, 4 figs

We consider the effects of anisotropic coherency strain (due to lattice mismatch) on phase separation in intercalation materials, motivated by the high-rate Li-ion battery material \fepo. Using a phase-field model coupled to elastic stresses, we analyze spinodal decomposition (linear instability of the homogeneous state) as well as nonlinear evolution of the phase pattern at constant mean filling. We consider fully anisotropic coherency strain and focus on the novel case of simultaneous expansion and contraction along different crystal axis, as in the case of \fepo, which leads to tilted, striped phase boundaries in equilibrium.

The sorption-desorption hysteresis observed in many nanoporous solids, at vapor pressures low enough for the the liquid (capillary) phase of the adsorbate to be absent, has long been vaguely attributed to changes in the nanopore structure, but no mathematically consistent explanation has been presented. The present work takes an analytical approach to account for discrete molecular forces in the nanopore fluid and proposes two related mechanisms that can explain the hysteresis at low vapor pressure without assuming any change in the nanopore structure. The first mechanism, presented in Part I, consists of a series of snap-through instabilities during the filling or emptying of non-uniform nanopores or nanoscale asperities. The instabilities are caused by non-uniqueness in the misfit disjoining pressures engendered by a difference between the nanopore width and an integer multiple of the thickness of a monomolecular adsorption layer. The second mechanism, presented in Part II, consists of molecular coalescence within a partially filled surface, nanopore or nanopore network. This general thermodynamic instability is driven by attractive intermolecular forces within the adsorbate and forms the basis to develop a unified theory of both mechanisms. The ultimate goals of the theory are to predict the fluid transport in nanoporous solids from microscopic first principles, and to determine the pore size distribution and internal surface area from sorption tests., Infrastructure Science (Northwestern University)

Salt transport in bulk electrolytes is limited by diffusion and convection, but in microstructures with charged surfaces (e.g. microfluidic devices, porous media, soils, or biological tissues) surface conduction and electro-osmotic flow also contribute to ionic fluxes. For small applied voltages, these effects lead to well known linear electrokinetic phenomena. In this paper, we predict some surprising nonlinear dynamics that can result from the competition between bulk and interfacial transport at higher voltages. When counter-ions are selectively removed by a membrane or electrode, a "desalination shock" can propagate through the microstructure, leaving in its wake an ultrapure solution, nearly devoid of co-ions and colloidal impurities. We elucidate the basic physics of desalination shocks and develop a mathematical theory of their existence, structure, and stability, allowing for slow variations in surface charge or channel geometry. Via asymptotic approximations and similarity solutions, we show that desalination shocks accelerate and sharpen in narrowing channels, while they decelerate and weaken, and sometimes disappear, in widening channels. These phenomena may find applications in separations (desalination, decontamination, biological assays) and energy storage (batteries, supercapacitors) involving electrolytes in microstructures., 14 pages, 7 figures

Conformal mapping has been applied mostly to harmonic functions, i.e. solutions of Laplace's equation. In this paper, it is noted that some other equations are also conformally invariant and thus equally well suited for conformal mapping in two dimensions. In physics, these include steady states of various nonlinear diffusion equations, the advection-diffusion equations for potential flows, and the Nernst-Planck equations for bulk electrochemical transport. Exact solutions for complicated geometries are obtained by conformal mapping to simple geometries in the usual way. Novel examples include nonlinear advection-diffusion layers around absorbing objects and concentration polarizations in electrochemical cells. Although some of these results could be obtained by other methods, such as Boussinesq's streamline coordinates, the present approach is based on a simple unifying principle of more general applicability. It reveals a basic geometrical equivalence of similarity solutions for a broad class of transport processes and paves the way for new applications of conformal mapping, e.g. to non-Laplacian fractal growth., Comment: 20 pages, 4 figures (11 color images), v3: more on ADLA fractal growth

A key factor preventing the market penetration of renewable, intermittent energy sources, such as solar, wind and wave, is the lack of cost-effective energy storage options to counteract intermittency. Here, we propose and demonstrate a novel flow battery architecture that replaces traditional ion-exchange membranes with less expensive heterogeneous flow-through porous media. We present an experimentally-validated model which demonstrates that our architecture promises reduced crossover of reactive species compared to typical membraneless systems employing co-laminar flows in open channels. In addition, our prototype battery exhibits significantly improved power density (0.925 W/cm2) and maximum current density (3 A/cm2) compared to previous membraneless systems.

The recently developed modified Donnan (mD) model provides a simple and useful description of the electrical double layer in microporous carbon electrodes, suitable for incorporation in porous electrode theory. By postulating an attractive excess chemical potential for each ion in the micropores that is inversely proportional to the total ion concentration, we show that experimental data for capacitive deionization (CDI) can be accurately predicted over a wide range of applied voltages and salt concentrations. Since the ion spacing and Bjerrum length are each comparable to the micropore size (few nanometers), we postulate that the attraction results from fluctuating bare Coulomb interactions between individual ions and the metallic pore surfaces (image forces) that are not captured by mean-field theories, such as the Poisson-Boltzmann-Stern model or its mathematical limit for overlapping double layers, the Donnan model. Using reasonable estimates of the micropore permittivity and mean size (and no other fitting parameters), we propose a simple theory that predicts the attractive chemical potential inferred from experiments. As additional evidence for attractive forces, we present data for salt adsorption in uncharged microporous carbons, also predicted by the theory., Denmark. Ministry of Infrastructure and the Environment (Wetsus), Denmark. Ministry for Economic Affairs (Wetsus), European Union (Regional Development Fund), Denmark. Province of Fryslân, Denmark (Northern Netherlands Provinces)

In situ electrochemical dilatometry was used to study, for the first time, the expansion behavior of a porous carbon electrode in a pure ionic liquid, 1-ethyl-3-methyl-imidazolium-tetrafluoroborate. For a single electrode, an applied potential of -2 V and +2 V against the potential of zero charge resulted in maximum strain of 1.8 % and 0.5 %, respectively. During cyclic voltammetry, the characteristic expansion behavior strongly depends on the scan rate, with increased scan rates leading to a decrease of the expansion. Chronoamperometry was used to determine the equilibrium specific capacitance and expansion. The obtained strain versus accumulated charge relationship can be fitted with a simple quadratic function. Cathodic and anodic expansion data collapses on one parabola when normalizing the surface charge by the ratio of ion volume and average pore size. There is also a transient spike in the height change when polarity is switched from positive to negative that is not observed when changing the potential from negative to positive indicating the size and the shape of the ion is influencing the expansion behavior., 10 pages double spaced, 3 figs, Electrochemistry Communications, accepted

Solid-state programmable metallization cells have attracted considerable attention as memristive elements for Redox-based Resistive Random Access Memory (ReRAM) for low-power and low-voltage applications. In principle, liquid-state metallization cells could offer the same advantages for aqueous systems, such as biomedical lab-on-a-chip devices, but robust resistive switching has not yet been achieved in liquid electrolytes, where electrodeposition is notoriously unstable to the formation of fractal dendrites. Here, the recently discovered physics of shock electrodeposition are harnessed to stabilize aqueous copper growth in polycarbonate nanopores, whose surfaces are modified with charged polymers. Stable bipolar resistive switching is demonstrated for 500 cycles with

Lithium-air batteries have been considered as ultimate solutions for the power source of long-range electrified transportation, but state-of-the-art prototypes still suffer from short cycle life, low efficiency and poor power output. Here, a lithium-bromine rechargeable fuel cell using highly concentrated bromine catholytes is demonstrated with comparable specific energy, improved power density, and higher efficiency. The cell is similar in structure to a hybrid-electrolyte Li-air battery, where a lithium metal anode in nonaqueous electrolyte is separated from aqueous bromine catholytes by a lithium-ion conducting ceramic plate. The cell with a flat graphite electrode can discharge at a peak power density around 9 mW cm -2 and in principle could provide a specific energy of 791.8 Wh kg -1 , superior to most existing cathode materials and catholytes. It can also run in the regenerative mode to recover the lithium metal anode and free bromine with 80-90% voltage efficiency, without any catalysts. Degradation of the solid electrolyte and the evaporation of bromine during deep charging are challenges that should be addressed in improved designs to fully exploit the high specific energy of the liquid bromine. The proposed system offers a potential power source for long-range electric vehicles, beyond current Li-ion batteries yet close to envisioned Li-air batteries. Keywords: high specific energy; solid-state electrolytes; lithium-air batteries; flow batteries; electric vehicles

An advection--diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and then collapse to a single point in finite time. The simulations reveal a surprising exact relationship whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton--Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). In summary, the model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems., 20 pages, 11 figures

The development of energy and infrastructure efficient water purification systems are among the most critical engineering challenges facing our society. Water purification is often a multi-step process involving filtration, desalination, and disinfection of a feedstream. Shock electrodialysis (shock ED) is a newly developed technique for water desalination, leveraging the formation of ion concentration polarization (ICP) zones and deionization shock waves in microscale pores near to an ion selective element. While shock ED has been demonstrated as an effective water desalination tool, we here present evidence of other simultaneous functionalities. We show that, unlike electrodialysis, shock ED can thoroughly filter micron-scale particles and aggregates of nanoparticles present in the feedwater. We also demonstrate that shock ED can enable disinfection of feedwaters, as approximately $99\%$ of viable bacteria (here \textit{E. coli}) in the inflow were killed or removed by our prototype. Shock ED also separates positive from negative particles, contrary to claims that ICP acts as a virtual barrier for all charged particles. By combining these functionalities (filtration, separation and disinfection) with deionization, shock ED has the potential to enable more compact and efficient water purification systems.

Porous electrodes composed of multiphase active materials are widely used in Li-ion batteries, but their dynamics are poorly understood. Two-phase models are largely empirical, and no models exist for three or more phases. Using a modified porous electrode theory based on non-equilibrium thermodynamics, we show that experimental phase behavior can be accurately predicted from free energy models, without artificially placing phase boundaries or fitting the open circuit voltage. First, we simulate lithium intercalation in porous iron phosphate, a popular two-phase cathode, and show that the zero-current voltage gap, sloping voltage plateau and under-estimated exchange currents all result from size-dependent nucleation and mosaic instability. Next, we simulate porous graphite, the standard anode with three stable phases, and reproduce experimentally observed fronts of color-changing phase transformations. These results provide a framework for physics-based design and control for electrochemical systems with complex thermodynamics., National Science Foundation (U.S.) (Contract DMS-0948071), Samsung-MIT Alliance

A general physics-based model is developed for heterogeneous electrocatalysis in porous electrodes and used to predict and interpret the impedance of solid oxide fuel cells. This model describes the coupled processes of oxygen gas dissociative adsorption and surface diffusion of the oxygen intermediate to the triple phase boundary, where charge transfer occurs. The model accurately captures the Gerischer-like frequency dependence and the oxygen partial pressure dependence of the impedance of symmetric cathode cells. Digital image analysis of the microstructure of the cathode functional layer in four different cells directly confirms the predicted connection between geometrical properties and the impedance response. As in classical catalysis, the electrocatalytic activity is controlled by an effective Thiele modulus, which is the ratio of the surface diffusion length (mean distance from an adsorption site to the triple phase boundary) to the surface boundary layer length (square root of surface diffusivity divided by the adsorption rate constant). The Thiele modulus must be larger than one in order to maintain high surface coverage of reaction intermediates, but care must be taken in order to guarantee a sufficient triple phase boundary density. The model also predicts the Sabatier volcano plot with the maximum catalytic activity corresponding to the proper equilibrium surface fraction of adsorbed oxygen adatoms. These results provide basic principles and simple analytical tools to optimize porous microstructures for efficient electrocatalysis.

Understanding over-limiting current (faster than diffusion) is a long-standing challenge in electrochemistry with applications in desalination and energy storage. Known mechanisms involve either chemical or hydrodynamic instabilities in unconfined electrolytes. Here, it is shown that over-limiting current can be sustained by surface conduction in nanopores, without any such instabilities, and used to control dendritic growth during electrodeposition. Copper electrodeposits are grown in anodized aluminum oxide membranes with polyelectrolyte coatings to modify the surface charge. At low currents, uniform electroplating occurs, unaffected by surface modification due to thin electric double layers, but the morphology changes dramatically above the limiting current. With negative surface charge, growth is enhanced along the nanopore surfaces, forming surface dendrites and nanotubes behind a deionization shock. With positive surface charge, dendrites avoid the surfaces and are either guided along the nanopore centers or blocked from penetrating the membrane., Korea (South). Ministry of Education, Science and Technology (MEST) (National Research Foundation of Korea. Basic Science Research Program. 2012R1A6A3A03039224), Singapore. Agency for Science, Technology and Research (National Science Scholarship), International Business Machines Corporation (Faculty Award), Saint-Gobain Corporation. Northboro R&D Center

The conductivity of highly charged membranes is nearly constant, due to counterions screening pore surfaces. Weakly charged porous media, or “leaky membranes,” also contain a significant concentration of coions, whose depletion at high current leads to ion concentration polarization and conductivity shock waves. To describe these nonlinear phenomena in the absence of electro-osmotic flow, a simple leaky membrane model is formulated, based on macroscopic electroneutrality and Nernst–Planck ionic fluxes. The model is solved in cases of unsupported binary electrolytes: steady conduction from a reservoir to a cation-selective surface, transient response to a current step, steady conduction to a flow-through porous electrode, and steady conduction between cation-selective surfaces in cross flow. The last problem is motivated by separations in leaky membranes, such as shock electrodialysis. The article begins with a tribute to Neal Amundson, whose pioneering work on shock waves in chromatography involved similar mathematics. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3539–3555, 2013

In nanoparticulate phase-separating electrodes, phase separation inside the particles can be hindered during their charge/discharge cycles even when a thermodynamic driving force for phase separation exists. In such cases, particles may (de)lithiate discretely in a process referred to as mosaic instability. This instability could be the key to elucidating the complex charge/discharge dynamics in nanoparticulate phase-separating electrodes. In this paper, the dynamics of the mosaic instability is studied using Smoothed Boundary Method simulations at the particle level, where the concentration and electrostatic potential fields are spatially resolved around individual particles. Two sets of configurations consisting of spherical particles with an identical radius are employed to study the instability in detail. The effect of an activity-dependent exchange current density on the mosaic instability, which leads to asymmetric charge/discharge, is also studied. While we show that our model reproduces the results of a porous-electrode model for the simple setup studied here, it is a powerful framework with the capability to predict the detailed dynamics in three-dimensional complex electrodes and provides further insights into the complex dynamics that result from the coupling of electrochemistry, thermodynamics, and transport kinetics., National Science Foundation (U.S.) (Contract No. DMS-0842504), National Science Foundation (U.S.) (grant number OCI-1053575, under allocation No. TG-DMR110007), National Science Foundation (U.S.) (Contract No. DMS-0948071), Samsung (Firm) (Samsung-MIT Program for Materials Design and Energy Applications)

Motivated by the puzzle of sorption hysteresis in Portland cement concrete or cement paste, we develop in Part II of this study a general theory of vapor sorption and desorption from nanoporous solids, which attributes hysteresis to hindered molecular condensation with attractive lateral interactions. The classical mean-field theory of van der Waals is applied to predict the dependence of hysteresis on temperature and pore size, using the regular solution model and gradient energy of Cahn and Hilliard. A simple "hierarchical wetting" model for thin nanopores is developed to describe the case of strong wetting by the first monolayer, followed by condensation of nanodroplets and nanobubbles in the bulk. The model predicts a larger hysteresis critical temperature and enhanced hysteresis for molecular condensation across nanopores at high vapor pressure than within monolayers at low vapor pressure. For heterogeneous pores, the theory predicts sorption/desorption sequences similar to those seen in molecular dynamics simulations, where the interfacial energy (or gradient penalty) at nanopore junctions acts as a free energy barrier for snap-through instabilities. The model helps to quantitatively understand recent experimental data for concrete or cement paste wetting and drying cycles and suggests new experiments at different temperatures and humidity sweep rates., Comment: 26 pages, 10 figs