Diffuse-charge Dynamics Research Articles

Frequency Response of the Diffuse-Charge Dynamics in Electrochemical Systems with a Graphene Electrode

We investigate the frequency response of diffuse-charge dynamics related to a 1:1 symmetric electrolyte containing a graphene electrode by solving the governing Poisson-Nernst-Planck equations subjected to appropriate boundary conditions in the asymptotic limit ε = λ D /H → 0, where λ D is the Debye screening length and H is the half-thickness of the electrolyte. Using the method of matched asymptotic expansion, we first solve the leading order non-linear problem for equilibrium state at a nonzero applied DC voltage in the presence of Stern layer(s). Then, we extend the leading order asymptotic analysis to derive an analytic expression for the impedance of the graphene-based electrochemical cell when a small AC voltage perturbation is added to the applied DC voltage. Finally, we use a suitable scaling of the impedance parameters to expose the impacts of the ion concentration and the DC bias voltage on the frequency response for possible applications involving the graphene-electrolyte interface.

Developing an Active Microfluidic Pump and Mixer Driven by AC Field-Effect-Mediated Induced-Charge Electro-Osmosis of Metal–Dielectric Janus Micropillars: Physical Perspective and Simulation Analysis

We propose herein a novel microfluidic approach for the simultaneous active pumping and mixing of analytes in a straight microchannel via the AC field-effect control of induced-charge electro-osmosis (ICEO) around metal–dielectric solid Janus cylinders of inherent inhomogeneous electrical polarizability immersed in an electrolyte solution. We coin the term “Janus AC flow field-effect transistor (Janus AC-FFET)” to describe this interesting physical phenomenon. The proposed technique utilizes a simple device geometry, in which one or a series of Janus microcylinders are arranged in parallel along the centerline of the channel’s bottom surface, embedding a pair of 3D sidewall driving electrodes. By combining symmetry breaking in both surface polarizability and the AC powering scheme, it is possible, on demand, to adjust the degree of asymmetry of the ICEO flow profile in two orthogonal directions, which includes the horizontal pump and transversal rotating motion. A comprehensive mathematical model was developed under the Debye–Hückel limit to elucidate the physical mechanism underlying the field-effect-reconfigurable diffuse-charge dynamics on both the dielectric and the metal-phase surfaces of the Janus micropillar. For innovation in applied science, an advanced microdevice design integrating an array of discrete Janus cylinders subjected to two oppositely polarized gate terminals is recommended for constructing an active microfluidic pump and mixer, even without external moving parts. Supported by a simulation analysis, our physical demonstration of Janus AC-FFET provides a brand-new approach to muti-directional electro-convective manipulation in modern microfluidic systems.

Ionic Layering and Overcharging in Electrical Double Layers in a Poisson-Boltzmann Model.

Electrical double layers (EDLs) play a significant role in a broad range of physical phenomena related to colloidal stability, diffuse-charge dynamics, electrokinetics, and energy storage applications. Recently, it has been suggested that for large ion sizes or multivalent electrolytes, ions can arrange in a layered structure inside the EDLs. However, the widely used Poisson-Boltzmann models for EDLs are unable to capture the details of ion concentration oscillations and the effect of electrolyte valence on such oscillations. Here, by treating a pair of ions as hard spheres below the distance of closest approach and as point charges otherwise, we are able to predict ionic layering without any additional parameters or boundary conditions while still being compatible with the Poisson-Boltzmann framework. Depending on the combination of ion valence, size, and concentration, our model reveals a structured EDL with spatially oscillating ion concentrations. We report the dependence of critical ion concentration, i.e., the ion concentration above which the oscillations are observed, on the counter-ion valence and the ion size. More importantly, our model displays quantitative agreement with the results of computationally intensive models of the EDL. Finally, we analyze the nonequilibrium problem of EDL charging and demonstrate that ionic layering increases the total charge storage capacity and the charging timescale.

On ion transport regulation with field‐effect nonlinear electroosmosis control in microfluidics embedding an ion‐selective medium

We study herein numerically the use of induced-charge electrokinetic phenomena to enable a flexible control of ion transport of dilute electrolyte in a straight ion concentration polarization system. The effect of three convection modes of induced-charge electrokinetic phenomena, including induced-charge electroosmosis, flow-field effect transistor, and alternating-current electroosmosis (ACEO), on convective arrestment of diffusive wave-front propagation is investigated by developing a cross-scale and fully coupled transient numerical simulation model, wherein multiple frequency electrochemical polarization and nonlinear diffuse charge dynamics in spatiotemporally varying solution conductivity are taken into account. We demonstrate by detailed comparative simulation studies that ACEO vortex flow field above a metal strip array arranged along the anodic chamber's bottom surface serves as the most efficient way for adjusting the salt density distribution at micrometer and even millimeter dimension, due to its high flexibility in controlling the stirring flow state with the introduction of two extra electrical parameters. The specific operating status is determined by whether the electrode array is floating in potential (induced-charge electroosmosis) or biased to ground (flow-field effect transistor) or forced to oscillate at another Fourier mode (ACEO). These results prove useful for on-chip electric current control with electroconvective stirring.

Dielectrophoretic separation with a floating-electrode array embedded in microfabricated fluidic networks

In this study, we proposed a high-throughput separation strategy of the binary colloid mixture by dielectrophoresis (DEP) induced around large-scale bipolar electrode arrays embedded in microfabricated fluidic networks via a thorough numerical investigation. The usage of a floating electrode (FE) eliminates the need of external Ohmic connection to individual array units, therefore potentially steering the faddish design of new microdevice structures. Diffuse charge dynamics within the induced double layer at opposite ends of every FE permit a sinusoidal electric field to penetrate throughout the whole device, as long as the imposed field frequency is beyond the reciprocal resistor-capacitor time constant at the electrode/electrolyte interface. In this special device configuration, FEs interconnect multiple microchannels arranged in parallel. Pockets embedded on the sidewalls of fluidic channels help create strong field gradients at the tip of FEs and sharp pocket/channel junctions, improving the trapping performance of incoming bioparticles subjected to positive-DEP (pDEP) force, while latex beads experiencing negative-DEP (nDEP) stress are electrically squeezed to the midchannel and finally exit as a series of co-flowing thin streams with unequal translatory velocity. Taking the synergy of DEP force, induced-charge electro-osmosis, alternating-current electrothermal streaming, pressure-driven flow, and buoyancy effect into consideration, a numerical model is established to account for motion trajectories of micro-entities in full-scale three-dimensional space using the Lagrange particle track algorithm, as well as testing the feasibility of the device design in separation of the binary mixture containing yeast cells and polystyrene beads. Applying suitable voltage parameters of frequency O(1) MHz and electric field strength O(10) V/mm, highly efficient DEP separation is theoretically achievable under inlet flow velocity on the order of O(1) mm/s, where most of incoming yeasts are captured by pDEP within these five parallel branching channels, while polystyrene spheres are repelled by nDEP away from the FE array to form slim beams co-flowing into the outlet according to the calculation results. The microfluidic separation device exploiting the FE array offers great potential to build up scalable electrokinetic platforms for high-throughput on-chip sample treatment.

Electrical Double Layers: Effects of Asymmetry in Electrolyte Valence on Steric Effects, Dielectric Decrement, and Ion–Ion Correlations

We study the effects of asymmetry in electrolyte valence (i.e., non z: z electrolytes) on mean field theory of the electrical double layer. Specifically, we study the effect of valence asymmetry on finite ion-size effects, the dielectric decrement, and ion-ion correlations. For a model configuration of an electrolyte near a charged surface in equilibrium, we present comprehensive analytical and numerical results for the potential distribution, electrode charge density, capacitance, and dimensionless salt uptake. We emphasize that the asymmetry in electrolyte valence significantly influences the diffuse-charge relations, and prior results reported in the literature are readily extended to non z: z electrolytes. We develop scaling relations and invoke physical arguments to examine the importance of asymmetry in electrolyte valence on the aforementioned effects. We conclude by providing implications of our findings on diffuse-charge dynamics and other electrokinetic phenomena.

On the Bipolar DC Flow Field-Effect-Transistor for Multifunctional Sample Handing in Microfluidics: A Theoretical Analysis under the Debye⁻Huckel Limit.

We present herein a novel method of bipolar field-effect control on DC electroosmosis (DCEO) from a physical point of view, in the context of an intelligent and robust operation tool for stratified laminar streams in microscale systems. In this unique design of the DC flow field-effect-transistor (DC-FFET), a pair of face-to-face external gate terminals are imposed with opposite gate-voltage polarities. Diffuse-charge dynamics induces heteropolar Debye screening charge within the diffuse double layer adjacent to the face-to-face oppositely-polarized gates, respectively. A background electric field is applied across the source-drain terminal and forces the face-to-face counterionic charge of reversed polarities into induced-charge electroosmotic (ICEO) vortex flow in the lateral direction. The chaotic turbulence of the transverse ICEO whirlpool interacts actively with the conventional plug flow of DCEO, giving rise to twisted streamlines for simultaneous DCEO pumping and ICEO mixing of fluid samples along the channel length direction. A mathematical model in thin-layer approximation and the low-voltage limit is subsequently established to test the feasibility of the bipolar DC-FFET configuration in electrokinetic manipulation of fluids at the micrometer dimension. According to our simulation analysis, an integrated device design with two sets of side-by-side, but upside-down gate electrode pair exhibits outstanding performance in electroconvective pumping and mixing even without any externally-applied pressure difference. Moreover, a paradigm of a microdevice for fully electrokinetics-driven analyte treatment is established with an array of reversed bipolar gate-terminal pairs arranged on top of the dielectric membrane along the channel length direction, from which we can obtain almost a perfect liquid mixture by using a smaller magnitude of gate voltages for causing less detrimental effects at a small Dukhin number. Sustained by theoretical analysis, our physical demonstration on bipolar field-effect flow control for the microfluidic device of dual functionalities in simultaneous electroconvective pumping and mixing holds great potential in the development of fully-automated liquid-phase actuators in modern microfluidic systems.

Diffuse charge dynamics in ionic thermoelectrochemical systems.

Thermoelectrics are increasingly being studied as promising electrical generators in the ongoing search for alternative energy sources. In particular, recent experimental work has examined thermoelectric materials containing ionic charge carriers; however, the majority of mathematical modeling has been focused on their steady-state behavior. Here, we determine the time scales over which the diffuse charge dynamics in ionic thermoelectrochemical systems occur by analyzing the simplest model thermoelectric cell: a binary electrolyte between two parallel, blocking electrodes. We consider the application of a temperature gradient across the device while the electrodes remain electrically isolated from each other. This results in a net voltage, called the thermovoltage, via the Seebeck effect. At the same time, the Soret effect results in migration of the ions toward the cold electrode. The charge dynamics are described mathematically by the Poisson-Nernst-Planck equations for dilute solutions, in which the ion flux is driven by electromigration, Brownian diffusion, and thermal diffusion under a temperature gradient. The temperature evolves according to the heat equation. This nonlinear set of equations is linearized in the (experimentally relevant) limit of a "weak" temperature gradient. From this, we show that the time scale on which the thermovoltage develops is the Debye time, 1/Dκ^{2}, where D is the Brownian diffusion coefficient of both ion species, and κ^{-1} is the Debye length. However, the concentration gradient due to the Soret effect develops on the bulk diffusion time, L^{2}/D, where L is the distance between the electrodes. For thin diffuse layers, which is the condition under which most real devices operate, the Debye time is orders of magnitude less than the diffusion time. Therefore, rather surprisingly, the majority of ion motion occurs after the steady thermovoltage has developed. Moreover, the dynamics are independent of the thermal diffusion coefficients, which simply set the magnitude of the steady-state thermovoltage.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes.

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Simulation of diffuse-charge capacitance in electric double layer capacitors

We use a Lattice Boltzmann Model (LBM) in order to simulate diffuse-charge dynamics in Electric Double Layer Capacitors (EDLCs). Simulations are carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). The steric effect of concentrated solutions is considered by using a Modified Poisson–Nernst–Planck (MPNP) equations and compared with regular Poisson–Nernst–Planck (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. Our studies show how electrode morphology can be used to tailor the properties of supercapacitors.

Moderately nonlinear diffuse-charge dynamics under an ac voltage.

The response of a symmetric binary electrolyte between two parallel, blocking electrodes to a moderate amplitude ac voltage is quantified. The diffuse charge dynamics are modeled via the Poisson-Nernst-Planck equations for a dilute solution of point-like ions. The solution to these equations is expressed as a Fourier series with a voltage perturbation expansion for arbitrary Debye layer thickness and ac frequency. Here, the perturbation expansion in voltage proceeds in powers of V_{o}/(k_{B}T/e), where V_{o} is the amplitude of the driving voltage and k_{B}T/e is the thermal voltage with k_{B} as Boltzmann's constant, T as the temperature, and e as the fundamental charge. We show that the response of the electrolyte remains essentially linear in voltage amplitude at frequencies greater than the RC frequency of Debye layer charging, D/λ_{D}L, where D is the ion diffusivity, λ_{D} is the Debye layer thickness, and L is half the cell width. In contrast, nonlinear response is predicted at frequencies below the RC frequency. We find that the ion densities exhibit symmetric deviations from the (uniform) equilibrium density at even orders of the voltage amplitude. This leads to the voltage dependence of the current in the external circuit arising from the odd orders of voltage. For instance, the first nonlinear contribution to the current is O(V_{o}^{3}) which contains the expected third harmonic but also a component oscillating at the applied frequency. We use this to compute a generalized impedance for moderate voltages, the first nonlinear contribution to which is quadratic in V_{o}. This contribution predicts a decrease in the imaginary part of the impedance at low frequency, which is due to the increase in Debye layer capacitance with increasing V_{o}. In contrast, the real part of the impedance increases at low frequency, due to adsorption of neutral salt from the bulk to the Debye layer.

Range of Validity of a Simplified Model for Diffuse Charge Dynamics

AbstractThe modelling of electrochemical processes often requires the solution of the Poisson‐Nernst‐Planck (PNP) equations. In complex geometries, such as porous electrodes, that is challenging due to the presence of disparate length scales, ranging from the Debye screening length (∼nm) to the device length scale (∼cm). To overcome this difficulty, one often assumes that the electric double layer (EDL) is at quasi‐equilibrium to construct a simplified model that accounts for ion diffusion in the electro‐neutral bulk of the electrolyte while replacing the EDLs with appropriate boundary conditions. Various researchers have demonstrated that such an approach is valid in the asymptotic limit of a thin EDL and moderate electrode potentials. In this note, we explore the range of validity of this approximation by considering a one‐dimensional electrolytic cell with blocking electrodes subjected to a step change and time‐periodic alternations in the electrodes’ potentials by calculating the errors associated with the approximate approach as functions of the EDL thickness and electric field frequency and intensity. Additionally, we delineate numerical instabilities associated with the numerical solutions of the bulk equations with the nonlinear boundary condition peculiar to this problem.

Kelvin probe force microscopy in liquid using electrochemical force microscopy.

Conventional closed loop-Kelvin probe force microscopy (KPFM) has emerged as a powerful technique for probing electric and transport phenomena at the solid–gas interface. The extension of KPFM capabilities to probe electrostatic and electrochemical phenomena at the solid–liquid interface is of interest for a broad range of applications from energy storage to biological systems. However, the operation of KPFM implicitly relies on the presence of a linear lossless dielectric in the probe–sample gap, a condition which is violated for ionically-active liquids (e.g., when diffuse charge dynamics are present). Here, electrostatic and electrochemical measurements are demonstrated in ionically-active (polar isopropanol, milli-Q water and aqueous NaCl) and ionically-inactive (non-polar decane) liquids by electrochemical force microscopy (EcFM), a multidimensional (i.e., bias- and time-resolved) spectroscopy method. In the absence of mobile charges (ambient and non-polar liquids), KPFM and EcFM are both feasible, yielding comparable contact potential difference (CPD) values. In ionically-active liquids, KPFM is not possible and EcFM can be used to measure the dynamic CPD and a rich spectrum of information pertaining to charge screening, ion diffusion, and electrochemical processes (e.g., Faradaic reactions). EcFM measurements conducted in isopropanol and milli-Q water over Au and highly ordered pyrolytic graphite electrodes demonstrate both sample- and solvent-dependent features. Finally, the feasibility of using EcFM as a local force-based mapping technique of material-dependent electrostatic and electrochemical response is investigated. The resultant high dimensional dataset is visualized using a purely statistical approach that does not require a priori physical models, allowing for qualitative mapping of electrostatic and electrochemical material properties at the solid–liquid interface.

Influence of nonelectrostatic ion-ion interactions on double-layer capacitance

Recently a Poisson-Helmholtz-Boltzmann (PHB) model [Bohinc et al., Phys. Rev. E 85, 031130 (2012)] was developed by accounting for solvent-mediated nonelectrostatic ion-ion interactions. Nonelectrostatic interactions are described by a Yukawa-like pair potential. In the present work, we modify the PHB model by adding steric effects (finite ion size) into the free energy to derive governing equations. The modified PHB model is capable of capturing both ion specificity and ion crowding. This modified model is then employed to study the capacitance of the double layer. More specifically, we focus on the influence of nonelectrostatic ion-ion interactions on charging a double layer near a flat surface in the presence of steric effects. We numerically compute the differential capacitance as a function of the voltage under various conditions. At small voltages and low salt concentrations (dilute solution), we find out that the predictions from the modified PHB model are the same as those from the classical Poisson-Boltzmann theory, indicating that nonelectrostatic ion-ion interactions and steric effects are negligible. At moderate voltages, nonelectrostatic ion-ion interactions play an important role in determining the differential capacitance. Generally speaking, nonelectrostatic interactions decrease the capacitance because of additional nonelectrostatic repulsion among excess counterions inside the double layer. However, increasing the voltage gradually favors steric effects, which induce a condensed layer with crowding of counterions near the electrode. Accordingly, the predictions from the modified PHB model collapse onto those computed by the modified Poisson-Boltzmann theory considering steric effects alone. Finally, theoretical predictions are compared and favorably agree with experimental data, in particular, in concentrated solutions, leading one to conclude that the modified PHB model adequately predicts the diffuse-charge dynamics of the double layer with ion specificity and steric effects.

Diffuse-charge dynamics of ionic liquids in electrochemical systems

We employ a continuum theory of solvent-free ionic liquids accounting for both short-range electrostatic correlations and steric effects (finite ion size) [Bazant et al., Phys. Rev. Lett. 106, 046102 (2011)] to study the response of a model microelectrochemical cell to a step voltage. The model problem consists of a 1-1 symmetric ionic liquid between two parallel blocking electrodes, neglecting any transverse transport phenomena. Matched asymptotic expansions in the limit of thin double layers are applied to analyze the resulting one-dimensional equations and study the overall charge-time relation in the weakly nonlinear regime. One important conclusion is that our simple scaling analysis suggests that the length scale √(λ*(D)l*(c)) accurately characterizes the double-layer structure of ionic liquids with strong electrostatic correlations where l*(c) is the electrostatic correlation length (in contrast, the Debye screening length λ*(D) is the primary double-layer length for electrolytes) and the response time of λ(D)(*3/2)L*/(D*l(c)(1/2)) (not λ*(D)L*/D* that is the primary charging time of electrolytes) is the correct charging time scale of ionic liquids with strong electrostatic correlations where D* is the diffusivity and L* is the separation length of the cell. With these two new scales, data of both electric potential versus distance from the electrode and the total diffuse charge versus time collapse onto each individual master curve in the presence of strong electrostatic correlations. In addition, the dependance of the total diffuse charge on steric effects, short-range correlations, and driving voltages is thoroughly examined. The results from the asymptotic analysis are compared favorably with those from full numerical simulations. Finally, the absorption of excess salt by the double layer creates a depletion region outside the double layer. Such salt depletion may bring a correction to the leading order terms and break down the weakly nonlinear analysis. A criterion which justifies the weakly nonlinear analysis is verified with numerical simulations.

Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging

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).

Diffuse-charge dynamics in electrochemical systems.

The response of a model microelectrochemical system to a time-dependent applied voltage is analyzed. The article begins with a fresh historical review including electrochemistry, colloidal science, and microfluidics. The model problem consists of a symmetric binary electrolyte between parallel-plate blocking electrodes, which suddenly apply a voltage. Compact Stern layers on the electrodes are also taken into account. The Nernst-Planck-Poisson equations are first linearized and solved by Laplace transforms for small voltages, and numerical solutions are obtained for large voltages. The "weakly nonlinear" limit of thin double layers is then analyzed by matched asymptotic expansions in the small parameter epsilon= lambdaD/L, where lambdaD is the screening length and L the electrode separation. At leading order, the system initially behaves like an RC circuit with a response time of lambdaDL/D (not lambdaD2/D), where D is the ionic diffusivity, but nonlinearity violates this common picture and introduces multiple time scales. The charging process slows down, and neutral-salt adsorption by the diffuse part of the double layer couples to bulk diffusion at the time scale, L2/D. In the "strongly nonlinear" regime (controlled by a dimensionless parameter resembling the Dukhin number), this effect produces bulk concentration gradients, and, at very large voltages, transient space charge. The article concludes with an overview of more general situations involving surface conduction, multicomponent electrolytes, and Faradaic processes.

Charge-gap formation upon the Verwey transition inFe3O4

The temperature-dependent electronic structure has been investigated for a single crystal of magnetite $({\mathrm{Fe}}_{3}{\mathrm{O}}_{4})$ by measurements of the optical-conductivity spectrum. Upon the charge-ordering (Verwey) transition ${(T}_{V}=121 \mathrm{K})$, a clear opening of the optical gap (0.14 eV) is observed. Above ${T}_{V},$ the optical conductivity spectra shows notable spectral weight transfer in the $\mathrm{Fe} 3d$ intersite transition region $(0\ensuremath{-}2 \mathrm{eV})$, while preserving the pseudogap or polaronic feature, indicating highly diffuse charge dynamics.