Electrochemical devices often charge both through Faradaic reactions and electric double layer formation. Here, we study these coupled processes in a model system of a long electrolyte-filled pore subject to a small suddenly applied potential, close to the equilibrium potential Ψeq at which there is no net Faradaic charge transfer. Specifically, we solve the coupled Poisson-Nernst-Planck and Frumkin-Butler-Volmer equations by asymptotic approximations, using the pore's small inverse aspect ratio as the small parameter. In the early time limit, the reaction-diffusion equations yield an extended Faradaic transmission line model that includes a voltage source, Ψeq, biasing the Faradaic reactions, captured by the resistance RF. In the long-time limit, the pore exhibits a nontrivial potential of zero total charge, Ψpztc=Ψeq1-Ẑ(0)/RF, where Ẑ(0) is the experimentally accessible zero-frequency impedance of the system. This expression provides a new means to measure the Faradaic contribution to Ψpztc experimentally.

Charge transport in porous electrodes is foundational for modern energy storage technologies like supercapacitors, fuel cells, and batteries. Supercapacitors in particular rely solely on storing energy in charged pores. Here, we simulate the charging of a single electrolyte-filled pore using the modified Poisson-Nernst-Planck and Navier-Stokes equations. We find that electroconvection can substantially speed up the charging dynamics. We uncover the fundamental mechanism of electroconvection during pore charging through an analytical model that predicts the induced flow field and the electric current arising due to convection. Our findings suggest that convection is especially important in the limit of slender pores with thin electric double layers, and becomes significant beyond a certain threshold voltage that is an inherent electrolyte property.

Electric double layer (EDL) formation underlies the functioning of supercapacitors and several other electrochemical technologies. Here we study how the EDL formation near two flat blocking electrodes separated by 2L is affected by beyond-mean-field Coulombic interactions, which can be substantial for electrolytes of high salt concentration or with multivalent ions. Our model combines the Nernst-Planck and Bazant-Storey-Kornyshev (BSK) equations; the latter is a modified Poisson equation with a correlation length ℓ_{c}. In response to a voltage step, the system charges exponentially with a characteristic timescale τ that depends nonmonotonically on ℓ_{c}. For small ℓ_{c}, τ is given by the BSK capacitance times a dilute electrolyte's resistance, in line with Zhao [Phys. Rev. E 84, 051504 (2011)1539-375510.1103/PhysRevE.84.051504]; here τ decreases with increasing ℓ_{c}. Increasing the correlation length beyond ℓ_{c}≈L^{2/3}λ_{D}^{1/3}, with λ_{D} the Debye length, τ reaches a minimum, rises as τ∝λ_{D}ℓ_{c}/D, and plateaus at τ=4L^{2}/(π^{2}D). Our results imply that strongly correlated, strongly confined electrolytes-such as ionic liquids in the surface force balance apparatus-move slower than predicted so far.

Membrane tubes are essential structural features in cells that facilitate biomaterial transport and inter- and intracellular signaling. The shape of these tubes can be regulated by the proteins that surround and adhere to them. We study the stability of a biomembrane tube coated with proteins by combining linear stability analysis, out-of-equilibrium hydrodynamic calculations, and numerical solutions of a Helfrich -like membrane model. Our analysis demonstrates that both long- and short-wavelength perturbations can destabilize the tubes. Numerical simulations confirm the derived linear stability criteria and yield the nonlinearly perturbed vesicle shapes. Our study highlights the interplay between membrane shape and protein density, where the shape instability concurs with a redistribution of proteins into a banded pattern.

A temperature difference between two electrolyte-immersed electrodes often yields a voltage Δψ between them. This electrolyte Seebeck effect is usually explained by cations and anions flowing differently in thermal gradients. However, using molecular simulations, we found almost the same Δψ for cells filled with pure water as with aqueous alkali halides. Water layering and orientation near polarizable electrodes cause a large temperature-dependent potential drop χ there. The difference in χ of hot and cold electrodes captures most of the thermovoltage, Δψ≈χ_{hot}-χ_{cold}.

Porous carbons find various applications, including as adsorbents for clean water production and as electrode materials in energy storage devices such as supercapacitors. While supercapacitors reach higher power densities than batteries, they are less widely used, as their energy density is lower. We present a low-temperature wet ultrasonochemical synthesis technique to modify the surface of activated carbon with 1 wt% Cu nanoparticles. We analyzed the modified carbon using X-ray diffraction, Raman spectroscopy, scanning electron microscopy, and transmission electron microscopy and confirmed the composite formation by N2 adsorption-desorption isotherms at 77 K. For comparison, we did the same tests on pristine carbon. We used the modified carbon as an electrode material in a home-built supercapacitor filled with gel polymer electrolyte and as an absorbent of Malachite green dye. In both applications, the modified carbon performed substantially better than its pristine counterpart. The modified-carbon supercapacitor exhibited a single electrode-specific capacitance of approximately 68.9 F g-1. It also demonstrated an energy density of 9.8 W h kg-1 and a power density of 1.4 kW kg-1. These values represent improvements over the pristine-carbon supercapacitor, with increases of 25.7 F g-1 in capacitance, 3.8 W h kg-1 in energy density, and 0.5 kW kg-1 in power density. After 10 000 charging-discharging cycles, the capacitance of the modified-carbon supercapacitor decreased by approximately 10%, indicating good durability of the material. We found that the modified carbon's absorbance capacity for Malachite dye is more than that of the pristine carbon; the adsorption capacity value was similar to 153.16 mg g-1 for modified carbon with pseudo-second kinetic order, in accordance with the Redlich-Peterson adsorption model. Carbon framework modification to improve energy storage and dye adsorption.

The flow of electrolytes through pores is important in various fields of science and technology. Examples range from axons in the brain and plasmodesmata in plant cells to energy storage in batteries and supercapacitors. In the porous carbon electrodes of supercapacitors, ions move over micrometers through macropores until they arrive at nanopores, forming so-called electrical double layers. In this presentation, I discuss several models for this multi-scale charging process. First, I present analytical [1] and finite-element solutions [2,3] to the Poisson-Nernst-Planck (PNP) equation for the charging of cylindrical electrolyte-filled pores [Figure (a)]. With these methods, I will delineate the validity of the famous “transmission line” circuit model developed by de Levie in the 1960s [4,5] [Figure (b)]. The TL model assumes equipotential lines in a pore to be straight, which is not the case at a pore’s entrance and end, see Figure (a). As a result, the TL model does not accurately describe short-pore charging. Related, the impedance of short pores deviates from the impedance of the TL circuit (the “Warburg open” impedance W o ), especially at high frequencies, see Figure (c). Figure Caption: (a) The electrostatic potential in a cylindrical pore next to an electrolyte reservoir of resistance R b shortly after applying a potential step. The data represent numerical solutions to the Poisson-Nernst-Planck (PNP) equations, which we solved with finite elements using a Newton solver from the FEniCS library. (b) The transmission line circuit is an equivalent circuit that describes the charging of pores such as the one shown in panel (a). In the circuit, the resistance R and capacitance C of the pore are cut into small pieces r and c . (c) The impedance of the TL circuit is given by R b + W o (blue dotted line), where W o is the Warburg open impedance. Also shown is the impedance of a pore whose length is five times its diameter, as obtained from numerical PNP simulations (black). T. Aslyamov and M. Janssen, Electrochim. Acta 424, (2022) J. Yang, M. Janssen, C. Lian, R. van Roij, J. Chem. Phys. 156, (2022) C. Pedersen, T. Aslyamov, M. Janssen, in preparation R. de Levie, Electrochim. Acta 8, (1963) M. Janssen, Phys. Rev. Lett. 126, (2021) Figure 1

Macromolecular crowding affects biophysical processes as diverse as diffusion, gene expression, cell growth, and senescence. Yet, there is no comprehensive understanding of how crowding affects reactions, particularly multivalent binding. Herein, we use scaled particle theory and develop a molecular simulation method to investigate the binding of monovalent to divalent biomolecules. We find that crowding can increase or reduce cooperativity-the extent to which the binding of a second molecule is enhanced after binding a first molecule-by orders of magnitude, depending on the sizes of the involved molecular complexes. Cooperativity generally increases when a divalent molecule swells and then shrinks upon binding two ligands. Our calculations also reveal that, in some cases, crowding enables binding that does not occur otherwise. As an immunological example, we consider immunoglobulin G-antigen binding and show that crowding enhances its cooperativity in bulk but reduces it when an immunoglobulin G binds antigens on a surface.

Innovative ways of harnessing sustainable energy are needed to meet the world's ever-increasing energy demands. Supercapacitors may contribute, as they can convert waste heat to electricity through cyclic charging and discharging at different temperatures. Herein, we use an analytically-solvable model of a cylindrical pore filled with a single file of ions to identify optimal conditions for heat-to-electricity conversion with supercapacitors. We consider Stirling and Ericsson-like charging cycles and show that the former or latter yields more work when a supercapacitor operates under charge or voltage limitations, respectively. Both cycles yield the most work for pores almost as narrow as the size of the ions they contain, as is the case for energy storage with supercapacitors. In contrast to energy storage, which can be maximised by ionophobic pores, such pores do not yield the best heat-to-electricity conversion, independently of the applied potential. Instead, we find that for a given pore size, a moderately ionophilic pore harvests more work than ionophobic and strongly ionophilic pores.

Batteries, supercapacitors, and several other electrochemical devices charge by accumulating ions in the pores of electrolyte-immersed porous electrodes. The charging of such devices has long been interpreted using equivalent circuits and the partial differential equations these give rise to. Here, we discuss the validity of the transmission line (TL) circuit and equation for modeling a single electrolyte-filled pore in contact with a reservoir of resistance R_r. The textbook derivation of the pore-reservoir impedance R_r+Z_p from the TL equation does not correctly account for ionic current conservation at the pore-reservoir interface. However, correcting this shortcoming leads to the same impedance. We also show that the pore impedance Z_p can be derived directly from the TL circuit, bypassing the TL equation completely. The TL circuit assumes equipotential lines in an electrolyte-filled pore to be straight, which is not the case near the pore entrance and end. To determine the importance of these regions, we numerically simulated the charging of pores of different lengths ℓ_p and radii ϱ_p through the Poisson-Nernst-Planck equations. We find that pores with aspect ratios beyond ℓ_p/ϱ_p⪆5 have impedances in good agreement with Z_p.

We consider the charging of a model capacitor comprised of two planar electrodes and an electrolyte. Upon switching on a voltage difference, electric double layers build up in this setup, which we characterize with a classical dynamic density functional theory (DDFT) that accounts for electrostatic correlations and for molecular excluded volume of finite-sized ions and solvent molecules. Our DDFT predicts the electrode charge Q(t) to form exponentially with two timescales: at early times, the system relaxes on the RC time, namely, λDL/[D(2 + σ/λD)], with λD being the Debye length, L being the electrode separation, σ being the ion diameter, and D being the ionic diffusivity. Contrasting an earlier DDFT study, this early-time response does not depend on the applied potential. At late times, the capacitor relaxes with a relaxation time proportional to the diffusion time L2/D.

We study the charging dynamics of a long electrolyte-filled slit pore in response to a suddenly applied potential. In particular, we analytically solve the Poisson-Nernst-Planck (PNP) equations for a pore for which lambda(D)<< H << L, with lambda(D) the Debye length and H and L the pore's width and length. For small applied potentials, we find the time-dependent potential drop between the pore's surface and its center to be in complete agreement with a prediction of the celebrated transmission line model. For moderate to high applied potentials, prior numerical work showed that charging slows down at late times; Our analytical model reproduces and explains such biexponential charge buildup.

Jie Yang, Mathijs Janssen, ∗ Cheng Lian, † and René van Roij School of Chemistry and Molecular Engineering, East China University of Science and Technology, Shanghai 200237, China Department of Mathematics, Mechanics Division, University of Oslo, N-0851 Oslo, Norway Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht, The Netherlands (Dated: April 5, 2022)

Constant potential methods (CPMs) enable computationally efficient simulations of the solid-liquid interface at conducting electrodes in molecular dynamics. They have been successfully used, for example, to realistically model the behavior of ionic liquids or water-in-salt electrolytes in supercapacitors and batteries. CPMs model conductive electrodes by updating charges of individual electrode atoms according to the applied electric potential and the (time-dependent) local electrolyte structure. Here, we present a feature-rich CPM implementation, called ELECTRODE, for the Large-scale Atomic/Molecular Massively Parallel Simulator, which includes a constrained charge method and a thermo-potentiostat. The ELECTRODE package also contains a finite-field approach, multiple corrections for nonperiodic boundary conditions of the particle-particle particle-mesh solver, and a Thomas-Fermi model for using nonideal metals as electrodes. We demonstrate the capabilities of this implementation for a parallel-plate electrical double-layer capacitor, for which we have investigated the charging times with the different implemented methods and found an interesting relationship between water and ionic dipole relaxations. To prove the validity of the one-dimensional correction for the long-range electrostatics, we estimated the vacuum capacitance of two coaxial carbon nanotubes and compared it to structureless cylinders, for which an analytical expression exists. In summary, the ELECTRODE package enables efficient electrochemical simulations using state-of-the-art methods, allowing one to simulate even heterogeneous electrodes. Moreover, it allows unveiling more rigorously how electrode curvature affects the capacitance with the one-dimensional correction.

Understanding how electrolyte-filled porous electrodes respond to an applied potential is important to many electrochemical technologies. Here, we consider a model supercapacitor of two blocking cylindrical pores on either side of a cylindrical electrolyte reservoir. A stepwise potential difference 2Φ between the pores drives ionic fluxes in the setup, which we study through the modified Poisson-Nernst-Planck equations, solved with finite elements. We focus our discussion on the dominant timescales with which the pores charge and how these timescales depend on three dimensionless numbers. Next to the dimensionless applied potential Φ, we consider the ratio R/Rb of the pore's resistance R to the bulk reservoir resistance Rb and the ratio rp/λ of the pore radius rp to the Debye length λ. We compare our data to theoretical predictions by Aslyamov and Janssen (Φ), Posey and Morozumi (R/Rb), and Henrique, Zuk, and Gupta (rp/λ). Through our numerical approach, we delineate the validity of these theories and the assumptions on which they were based.

The correlation of different methods of measurement can become an important tool to identify the dominant physical elements that govern the electronic and ionic dynamics in perovskite solar cells. The diverse phenomena underlying the response of halide perovskite materials to different stimuli are reflected in time-domain measurements, where transients appear with time scales spanning orders of magnitude, from nanoseconds to hours. We discuss the connection between different frequency- and time-domain methods to probe the voltage and current response of halide perovskite solar cells to different small perturbations. To solve the frequency-to-time transformation, we start from models of the transfer function of intensity-modulated photocurrent spectroscopy (IMPS) and derive the associated impulse response function, the transient photocurrent (TPC), in response to a short light pulse. Similarly, we determine the transient photovoltage (TPV) starting from the intensity-modulated photovoltage spectroscopy (IMVS) transfer function. We also discuss the open-circuit voltage decays (OCVD). We first show the response of simple equivalent circuit models, and then we treat the full model for generation–diffusion–recombination of electrons that shows a spiraling loop in IMPS. This model gives rise to overshoots in the time domain.

Several modern technologies for energy storage and conversion are based on the screening of electric charge on the surface of porous electrodes by ions in an adjacent electrolyte. This so-called electric double layer (EDL) exhibits an intricate interplay with the electrolyte’s temperature that was the focus of several recent studies. In one of them, Janssen et al. [Phys. Rev. Lett. 119, 166002 (2017)] experimentally determined the ratio Qrev/Wel of reversible heat flowing into a supercapacitor during an isothermal charging process and the electric work applied therein. To rationalize that data, here, we determine Qrev/Wel within different models of the EDL using theoretical approaches such as density functional theory (DFT) as well as molecular dynamics simulations. Applying mainly the restricted primitive model, we find quantitative support for a speculation of Janssen et al. that steric ion interactions are key to the ratio Qrev/Wel. Here, we identified the entropic contribution of certain DFT functionals, which grants direct access to the reversible heat. We further demonstrate how Qrev/Wel changes when calculated in different thermodynamic ensembles and processes. We show that the experiments of Janssen et al. are explained best by a charging process at fixed bulk density or in a “semi-canonical” system. Finally, we find that Qrev/Wel significantly depends on parameters such as pore and ion size, salt concentration, and valencies of the cations and anions of the electrolyte. Our findings can guide further heat production measurements and can be applied in studies on, for instance, nervous conduction, where reversible heat is a key element.

The impedance of diffusion is an important tool to investigate a wide variety of systems, including electrochemical devices such as Li-ion batteries, porous electrodes, and solar cells. The classical impedance model for diffusion in a thin layer with a blocking boundary contains two separate regimes: Warburg diffusion at high frequency and capacitive charging at low frequency. Here, we provide a physical criterion for the transition between these two regimes, as the point of closest approach between early- and late-time approximations of the exact diffusion current. The resulting frequency is (π2/2)ωd with respect to the natural frequency ωd = Dn/L2, with Dn being the diffusion constant and L being the thickness of the layer.

I discuss the strong link between the transmission line (TL) equation and the TL circuit model for the charging of an electrolyte-filled pore of finite length. In particular, I show how Robin and Neumann boundary conditions to the TL equation, proposed by others on physical grounds, also emerge in the TL circuit subject to a stepwise potential. The pore relaxes with a timescale τ, an expression for which consistently follows from the TL circuit, TL equation, and from the pore's known impedance. An approximation to τ explains the numerically determined relaxation time of the stack-electrode model of Lian et al. [Phys. Rev. Lett. 124, 076001 (2020)PRLTAO0031-900710.1103/PhysRevLett.124.076001].