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Diffuse charge and Faradaic reactions in porous electrodes
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Porous electrodes instead of flat electrodes are widely used in electrochemical systems to boost storage capacities for ions and electrons, to improve the transport of mass and charge, and to enhance reaction rates. Existing porous electrode theories make a number of simplifying assumptions: (i) The charge-transfer rate is assumed to depend only on the local electrostatic potential difference between the electrode matrix and the pore solution, without considering the structure of the double layer (DL) formed in between; (ii) the charge-transfer rate is generally equated with the salt-transfer rate not only at the nanoscale of the matrix-pore interface, but also at the macroscopic scale of transport through the electrode pores. In this paper, we extend porous electrode theory by including the generalized Frumkin-Butler-Volmer model of Faradaic reaction kinetics, which postulates charge transfer across the molecular Stern layer located in between the electron-conducting matrix phase and the plane of closest approach for the ions in the diffuse part of the DL. This is an elegant and purely local description of the charge-transfer rate, which self-consistently determines the surface charge and does not require consideration of reference electrodes or comparison with a global equilibrium. For the description of the DLs, we consider the two natural limits: (i) the classical Gouy-Chapman-Stern model for thin DLs compared to the macroscopic pore dimensions, e.g., for high-porosity metallic foams (macropores >50 nm) and (ii) a modified Donnan model for strongly overlapping DLs, e.g., for porous activated carbon particles (micropores
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... In general, electrolyte transport models for porous media, the macroscopic charge density may be nonzero and varies in response to current imbalances. Diffuse electrolyte charge screens internal charged surfaces of porous membrane materials [99][100][101][102] or conducting porous electrodes [103,104] and provides additional pathways for ion transport by electromigration ("surface conduction") and electro-osmotic flows. In addition to Faradaic reactions, porous electrodes can also undergo capacitive charging by purely electrostatic forces, as in electric double layer capacitors and capacitive deionization systems [103], or hybrid pseudo-capacitors [105]. ...
... Diffuse electrolyte charge screens internal charged surfaces of porous membrane materials [99][100][101][102] or conducting porous electrodes [103,104] and provides additional pathways for ion transport by electromigration ("surface conduction") and electro-osmotic flows. In addition to Faradaic reactions, porous electrodes can also undergo capacitive charging by purely electrostatic forces, as in electric double layer capacitors and capacitive deionization systems [103], or hybrid pseudo-capacitors [105]. In batteries, however, such effects are usually neglected [5,49,106], since the focus is on electrochemical, rather than electrostatic, energy storage, using highly concentrated electrolytes. ...
... These changes constitute the Frumkin correction [126,127] to the reaction rate model, and have been recently reviewed [128,129]. Frumkin-corrected Butler-Volmer reaction models have been applied to various electrochemical techniques including steady constant current [127,128,130,131], voltage steps [132], current steps [133], and linear sweep voltammetry [129], as well as nano [134] and porous [103,105] electrodes, with clear indication of departure from models neglecting double layers, especially at low salt concentrations with thick double layers ("Gouy-Chapman limit" [127,128]). To be used consistently with the models developed here, Frumkin reaction kinetics would need to be extended to concentrated electrolyte solutions, including models of individual ionic activities within the double layers, although Frumkin effects are reduced for very thin double layers at high salt concentration ("Helmholtz limit" [127,128]). ...
Porous electrode theory, pioneered by John Newman and collaborators, provides a useful macroscopic description of battery cycling behavior, rooted in microscopic physical models rather than empirical circuit approximations. The theory relies on a separation of length scales to describe transport in the electrode coupled to intercalation within small active material particles. 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. This approach has fundamental limitations, however, and 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 theoretical framework for "multiphase porous electrode theory" implemented in an open-source software package called "MPET", based on electrochemical nonequilibrium thermodynamics. Cahn-Hilliard-type phase field models are used to describe the solid 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 metal electrodes, extended for concentrated solutions. The full equations and numerical algorithms are described, and a variety of example calculations are presented to illustrate the novel features of the software compared to existing battery models.
... Although our theoretical development below does not depend on the specific form of , it is instructive to point out how various existing mathematical models of electrostatic CDI for flat [25] and porous [26,27] blocking electrodes and Faradaic CDI with redox-active porous electrodes [28][29][30] are included as special cases our general thermodynamic formalism. In particular, all Poisson-Boltzmann (PB) models (including the classical dilute-solution approximation as well as various modified PB models [9] for thin or thick double layers) correspond to following form of the free energy functional, which expresses the mean-field and local density approximations for a linear dielectric material (neglecting all chemical or electrostatic correlations between ions): [10] = The free energy functional is constructed so that the physically relevant electrostatic potential, which conserves Maxwell's dielectric displacement field, satisfies the stationarity condition of vanishing first variation. ...
... Existing models of CDI describe electrosorption of ions in quasi-equilibrium diffuse EDLs via the Gouy-Chapman-Stern (GCS) and modified Donnan (mD) models, which correspond to solutions of the PB equation (Eq. (5)) with the Stern "surface capacitor" boundary condition in the limits of thin and thick double layers, respectively, [26,[28][29][30] and are thus included in our thermodynamic framework. More generally, we could relax the quasi-equilibrium assumption in our thermodynamic formulation by defining mass fluxes in terms gradients of the electrochemical potentials defined by Eq. (4) in order to obtain modified Poisson-Nernst-Planck equations for diffuse-charge dynamics out of equilibrium, [9,13,16] which capture additional effects of non-equilibrium charging and tangential surface conduction. ...
... This includes Poisson-Boltzmann (PB) type EDL treatments such as Gouy-Chapman (GC) model as well as modified Poisson-Boltzmann with finite ion size effects. [8,32] EDL descriptions with constant ordependent non-electrostatic potentials such as modified-Donnan (mD) model [27,28] each also respect Eq. (7) (since the integral of the non-electrostatic term on a closed path vanishes). Importantly, as noted above, Eq. (7) is also valid for EDLs with compact layers such as Gouy-Chapman-Stern (GCS) ...
We present a simple, top-down approach for the calculation of minimum energy consumption of electrosorptive ion separation using variational form of the (Gibbs) free energy. We focus and expand on the case of electrostatic capacitive deionization (CDI), and the theoretical framework is independent of details of the double-layer charge distribution and is applicable to any thermodynamically consistent model, such as the Gouy-Chapman-Stern (GCS) and modified Donnan (mD) models. We demonstrate that, under certain assumptions, the minimum required electric work energy is indeed equivalent to the free energy of separation. Using the theory, we define the thermodynamic efficiency of CDI. We explore the thermodynamic efficiency of current experimental CDI systems and show that these are currently very low, less than 1% for most existing systems. We applied this knowledge and constructed and operated a CDI cell to show that judicious selection of the materials, geometry, and process parameters can be used to achieve a 9% thermodynamic efficiency (4.6 kT energy per removed ion). This relatively high value is, to our knowledge, by far the highest thermodynamic efficiency ever demonstrated for CDI. We hypothesize that efficiency can be further improved by further reduction of CDI cell series resistances and optimization of operational parameters.
... This means that the changes promoted by the surface or bulk influence each other. This feature offers the possibility of investigating the effect of the surface through kinetic processes such as charge transfer (Faradaic processes [25][26][27] ) and adsorption-desorption process. 28,29 Thus, the surface will alter the distribution of particles in the system, in contrast to the free diffusion case. ...
This paper investigates several strategies for modeling electrochemical impedance, in particular, exploring the effects of fractional calculus. It focuses on the theoretical approach for describing systems with anomalous diffusion; as a result, these effects can be analytically expressed as functions of frequency when different boundary conditions are considered. Starting with the normal case as a reference scenario, this study discusses how to increase the complexity of mathematical solutions by generalizing fundamental equations. The second strategy extends the continuity equation to include a fractional contribution. Subsequently, Fick’s law is also extended, considering a case that incorporates a fractal derivative. Finally, we utilize electrochemical impedance to determine electric conductivity, analyze mean-square displacement, and connect it to the diffusion process.
... Moreover, the cathode surface in this work was a flat surface, and a different morphology as a porous cathode or a GDE could affect the electrolyte environment near the cathode and the doublelayer characteristics due to both electrical and steric factors. [66,[75][76][77][78][79] Finally, in the case of LiFOB, the FE at the predicted optimum was 47 % � 2.6 %, higher than the one previously predicted for the well-known LiBF 4 . ...
Ammonia electrosynthesis through the lithium‐mediated approach has recently reached promising results towards high activity and selectivity in aprotic media, reaching high Faradaic efficiency (FE) values and NH3 production rates. To fasten the comprehension and optimization of the complex lithium‐mediated nitrogen reduction system, for the first time a multivariate approach is proposed as a powerful tool to reduce the number of experiments in comparison with the classical one‐factor‐at‐a‐time approach. Doehlert design and surface response methodology are employed to optimize the electrolyte composition for a batch autoclaved cell. The method is validated with the common LiBF4 salt, and the correlations between the FE and the amount of lithium salt and ethanol as proton donor are elucidated, also discussing their impact on the solid electrolyte interphase (SEI) layer. Moreover, a new fluorinated salt is proposed (i.e., lithium difluoro(oxalate) borate (LiFOB)), taking inspiration from lithium batteries. This salt is chosen to tailor the SEI layer, with the aim of obtaining a bifunctional interfacial layer, both stable and permeable to N2, the latter being an essential characteristic for batch systems. The SEI layer composition is confirmed strategic and its tailoring with LiFOB boosts FE values.
Ternary metal oxides, known for their superior electrical and optical properties compared to binary or conventional oxides, hold significant promise for catalysis and energy storage applications. This study investigates the electrochemical performance of Ni1–xMnxCo2O4 nanoparticles for detecting acetaminophen in aqueous phosphate buffer solution. The cobaltite nanoparticles were obtained through a simple gel-combustion synthesis, and the sensors were characterized using cyclic voltammetry, chronoamperometry, and differential pulse voltammetry. The anodic peak currents associated with acetaminophen oxidation were assessed by varying the scan rate of current–voltage cycles. Among the sensors tested, the one fabricated with Ni0.5Mn0.5Co2O4 nanoparticles as an active material exhibited the highest sensitivity of 38.2 μA cm–2 mM–1 and a detection limit of approximately 2 μM, demonstrating its potential for sensitive and efficient acetaminophen detection. Moreover, the sensors fabricated using these ternary oxide nanostructures demonstrate a rapid chronoamperometric response time of 35.4 s and a decay lifetime of 0.31 s, highlighting the fast detection capabilities of acetaminophen. The electrochemical oxidation mechanism of acetaminophen and the charge transfer characteristics at the electrode–electrolyte interface have been discussed.
This study details the development of a computational adsorption model for predicting thermodynamic adsorption parameters for capacitive deionization (CDI) processes. To do this, multiple starting concentrations and temperatures are needed to predict the best fit value. This is first demonstrated experimentally using an in-house CDI cell with custom heaters, and determining maximum adsorption capabilities for a selected range of conditions. This has been done previously for CDI in the published literature, but here, experimental results are incorporated to provide the best fit to a computational model, which runs transient CDI tests in batch mode over multiple concentrations and temperatures to determine adsorption parameters. This saves the eventual challenge of having to run many different experiments independently to determine such adsorption parameters, the accuracy of which may be questionable subject to different experimental errors. With the model, many parameters can be quickly scanned at once, and adsorption parameters can be determined based on the concentration and temperature values selected, as well as other operating conditions, such as voltage and cell resistance. The computational isotherms are generated using the Gouy–Chapman–Stern (GCS) model, which is common for the lower concentration values used for CDI. The model also considers fixed and mobile chemical charges for enhanced CDI (ECDI) and Faradaic CDI (FaCDI), respectively, which have been examined as alternatives to improve CDI performance. While primarily proof-of-concept, the results obtained here demonstrate the benefits in adsorption capabilities, and energy savings obtained here demonstrate benefits in adsorption capabilities and energy savings for FaCDI, coinciding with higher enthalpies and entropies of adsorption. The model also serves as a benchmark in the future for how the results can be further explored and better fits can be obtained experimentally to confirm stability in the thermodynamic values.
Ammonia electrosynthesis through the lithium‐mediated approach has recently reached promising results towards high activity and selectivity in aprotic media, reaching high Faradaic efficiency (FE) values and NH3 production rates. To fasten the comprehension and optimization of the complex lithium‐mediated nitrogen reduction system, for the first time a multivariate approach is proposed as a powerful tool to reduce the number of experiments in comparison with the classical one‐factor‐at‐a‐time approach. Doehlert design and surface response methodology are employed to optimize the electrolyte composition for a batch autoclaved cell. The method is validated with the common LiBF4 salt, and the correlations between the FE and the amount of lithium salt and ethanol as proton donor are elucidated, also discussing their impact on the solid electrolyte interphase (SEI) layer. Moreover, a new fluorinated salt is proposed (i.e., lithium difluoro(oxalate) borate (LiFOB)), taking inspiration from lithium batteries. This salt is chosen to tailor the SEI layer, with the aim of obtaining a bifunctional interfacial layer, both stable and permeable to N2, the latter being an essential characteristic for batch systems. The SEI layer composition is confirmed strategic and its tailoring with LiFOB boosts FE values.
Capacitive deionization (CDI) is a technology in which water is desalinated by ion electrosorption into the electric double layers (EDLs) of charging porous electrodes. In recent years significant advances have been made in modeling the charge and salt dynamics in a CDI cell, but the possible effect of surface transport within diffuse EDLs on these dynamics has not been investigated. We here present theory which includes surface transport in describing the dynamics of a charging CDI cell. Through our numerical solution to the presented models, the possible effect of surface transport on the CDI process is elucidated. While at some model conditions surface transport enhances the rate of CDI cell charging, counter-intuitively this additional transport pathway is found to slow down cell charging at other model conditions.
The first model for the distribution of ions near the surface of a metal electrode was devised by Helmholtz in 1874. He envisaged two parallel sheets of charges of opposite sign located one on the metal surface and the other on the solution side, a few nanometers away, exactly as in the case of a parallel plate capacitor. The rigidity of such a model was allowed for by Gouy and Chapman inde pendently, by considering that ions in solution are subject to thermal motion so that their distribution from the metal surface turns out diffuse. Stern recognized that ions in solution do not behave as point charges as in the Gouy-Chapman treatment, and let the center of the ion charges reside at some distance from the metal surface while the distribution was still governed by the Gouy-Chapman view. Finally, in 1947, D. C. Grahame transferred the knowledge of the struc ture of electrolyte solutions into the model of a metal/solution interface, by en visaging different planes of closest approach to the electrode surface depending on whether an ion is solvated or interacts directly with the solid wall. Thus, the Gouy-Chapman-Stern-Grahame model of the so-called electrical double layer was born, a model that is still qualitatively accepted, although theoreti cians have introduced a number of new parameters of which people were not aware 50 years ago.
Storage and conversion are critical components of important energy-related technologies. Advanced Batteries: Materials Science Aspects employs materials science concepts and tools to describe the critical features that control the behavior of advanced electrochemical storage systems.his volume focuses on the basic phenomena that determine the properties of the components, i.e. electrodes and electrolytes, of advanced systems, as well as experimental methods used to study their critical parameters. This unique materials science approach utilizes concepts and methodologies different from those typical in electrochemical texts, offering a fresh, fundamental and tutorial perspective of advanced battery systems. Graduate students, scientists and engineers interested in electrochemical energy storage and conversion will find Advanced Batteries: Materials Science Aspects a valuable reference. © Springer Science+Business Media, LLC 2009. All rights reserved.
We present a model for oxygen reduction in water-filled, cylindrical nanopores with platinum walls. At one end, the pores are in contact with a polymer electrolyte membrane. The electrostatic interaction of the protons with the charged pore walls drives proton migration into the ionomer-free channels. We employ the Stern model to relate the surface charge density at the pore walls to the electrode potential. Proton and potential distributions within the pores are governed by the Poisson-Nernst-Planck theory and the oxygen distribution by Fick's law. Assuming a small local current density from oxygen reduction, we found an approximate analytical solution to the transport equations. The metal surface charge density and the corresponding proton conductivity of the pores are tuned by the deviation of the electrode potential from the potential of zero charge of the metal phase, which is the key determinant of the effectiveness of platinum utilization. Other determinants of pore performance are the Helmholtz capacitance, electrokinetic parameters, and pore size and length. Upon upscaling, the model is consistent with polarization data for ionomer-free, ultrathin catalyst layers in polymer electrolyte fuel cells (PEFCs). We discuss the implications of the model for the materials selection and nanostructural design of such catalyst layers.
The classical Poisson-Boltzmann (PB) theory of electrolytes assumes a dilute solution of point charges with mean-field electrostatic forces. Even for very dilute solutions, however, it predicts absurdly large ion concentrations (exceeding close packing) for surface potentials of only a few tenths of a volt, which are often exceeded, e.g., in microfluidic pumps and electrochemical sensors. Since the 1950s, several modifications of the PB equation have been proposed to account for the finite size of ions in equilibrium, but in this two-part series, we consider steric effects on diffuse charge dynamics (in the absence of electro-osmotic flow). In this first part, we review the literature and analyze two simple models for the charging of a thin double layer, which must form a condensed layer of close-packed ions near the surface at high voltage. A surprising prediction is that the differential capacitance typically varies nonmonotonically with the applied voltage, and thus so does the response time of an electrolytic system. In PB theory, the differential capacitance blows up exponentially with voltage, but steric effects actually cause it to decrease while remaining positive above a threshold voltage where ions become crowded near the surface. Other nonlinear effects in PB theory are also strongly suppressed by steric effects: The net salt adsorption by the double layers in response to the applied voltage is greatly reduced, and so is the tangential “surface conduction” in the diffuse layer, to the point that it can often be neglected compared to bulk conduction (small Dukhin number).
A mathematical analysis of demineralization occurring in a model electrochemical cell resembling practical cells containing porous cation- and anion-responsive electrodes has been carried out. The end result is an equation describing effluent-concentration/time curves in terms of three physically significant system parameters, employed in data analysis as fitting parameters. Good agreement between theory and experiment has been obtained for a wide variety of data. The theory has been applied to the determination of faradaic efficiency, a quantity that resists accurate direct measurement in practical cells.
A phenomenological theory of direct current across the metal–solid electrolyte interface is presented, using the assumption that cations are the only mobile charge carriers in the electrolyte. The current–voltage characteristic of the system is obtained under the assumption that a delayed stage of the electrode reaction is the electron transfer in cation discharge or ionization of metal ions.[Russian text ignored]
Electrochemical double-layer capacitors built from nanoporous electrodes can have such a high ratio of electrode surface area to pore volume that charging the capacitor can deplete the salt from the liquid volume. This can result in increased resistance, resulting in a slow, nonlinear charging rate of which quantitative understanding is limited. In some cases, this effect is masked by an external solution resistance or by the transport of salt into the pore from an external reservoir. However, in forms relevant to a compact energy storage device, the phenomenon can have an important effect on charging time and linearity, and understanding it is important for such design. We have observed salt depletion effects by using dealloyed gold, which has well-defined 10 nm pores and a chemically well-understood surface, and by minimizing the amount of external salt within range of diffusion. Good correspondence is observed with a modified de Levie model that accounts for reduced local conductivity due to salt depletion. The model's assumption that the Stern layer (ions closely bound to the pore wall) makes a low contribution to conductance in the pore is validated by experimental data.
For hybrid-electric vehicle applications, supercapacitors show promise for providing low cost per unit power, high-efficiency energy storage, substantially temperature-invariant operation, and high cycle life (potentially equating to one supercapacitor pack per vehicle lifetime). Here we provide a microstructural and mathematical analysis for the characterization of a relatively high specific energy supercapacitor. The governing equations and the boundary conditions are generalized so that the cell can be modeled using a current, potential, or power excitation source. An Appendix clarifies the role of binary-electrolyte diffusion. The overall analysis allows one to determine the optimal target operating voltage for hybrid vehicle operation, calculate the available energy vs. available power for the system, and clarify the electrode utilization through the examination of the stored charge as a function of position. (c) 2005 The Electrochemical Society.
A model based on dilute-solution porous-electrode theory is proposed to describe electrochemical devices that store energy in the double layer (double-layer capacitors). Various assumptions, such as neglecting concentration polarization, potential-dependent capacitance, and micropore effects, are made in the model. For constant-current discharges, the model reduces to a resistance-capacitance (RC) series circuit model after the initial discharging transients. The RC series circuit model is seen to fit existing experimental data for the discharge times available. The specific energy for constant-current and constant-power discharges is maximized over a range of discharge times by optimizing the electrode thickness, electrode porosity, and the final voltage constrained by the cutoff voltage. This maximization allows predictions of the attainable specific energies and powers for these devices and shows the influence of the various cell properties. Energy efficiencies are found for cycles when the capacitor is discharged in a given time and then charged infinitely slowly, and when the capacitor is discharged and charged in the same amount of time. Limitations to the model are discussed.