Continuum modeling of solid oxide fuel cell electrodes: introducing the minimum dissipation principle

A simple continuum one-dimensional model of porous mixed-conducting electrode applied onto solid oxide electrolyte is proposed and discussed. Assuming linear relationships between the reaction overpotential and current, the model makes it possible to derive analytical solutions for the electrode thickness dependence of the overall electrode resistance. The least dissipation principle is used to determine the distribution of the ionic and electronic currents throughout the mixed-conducting layer. The area-specific resistance is expressed in terms of the electron and ion resistivities of the electrode material, its “reaction resistivity” as a slope of current–overpotential dependence, and geometric parameters. The solution is expanded to describe the electrode impedance and gas transport resistance under DC conditions.     

玻碳电极界面的阻抗谱数学表达及定量分析

玻碳电极（GCE）是各类电化学传感器常用的基础电极,其界面特征直接影响检测性能。本文详细考察了电极体系的电化学过程,针对GCE传感界面,探讨了一个等效电路中电解质电阻、电荷输运电阻、扩散阻抗、电化学（氧化/还原）反应阻抗、表面吸附阻抗和双电层电容等电学元件的物理意义,并给出了对应的数学模型。通过改变模型中5个参数值,模拟了不同状态下的阻抗谱,分析了电极系统各参数对GCE阻抗谱的贡献规律。最后,采用该数学模型对裸GCE和修饰GCE在铁氰化钾溶液中的阻抗谱进行分析,拟合结果与实验数据吻合度高;基于拟合获得参数,定量对比分析了修饰前后电极表面的特征变化。     

A contribution to the voltammetric study of cystine and cysteine at Pt electrodes in 0.5 M H2SO4

Both cysteine and cystine adsorb at the Pt electrode according the Frumkin—Temkin isotherm with the heterogeneity factor f = 51 for cysteine and 21 for cystine. Both the adsorbed cysteine and cystine give in a solution without any dissolved cystine or cysteine almost identical first cyclic voltammetric curves. Each substance dissolved in the electrolyte gives two oxidation peaks which differ when the oxidation is carried out at a “reduced” or an “oxidized” Pt electrode. On the basis of the dependence of the height and potential of the peaks on polarization rate and concentration (in the case of oxidation of dissolved substances) and of coulometric measurements the following conclusions have been made concerning the kinetics and mechanism:(i) Neither cysteine nor cystine change their oxidation state on adsorption at the electrode.(ii) The final oxidation product of both adsorbed cysteine and cystine may be the cysteic acid.(iii) For cysteine there are two adsorbed species, one strongly adsorbed, the other one weakly adsorbed.(iv) The oxidation of dissolved cysteine takes place via the weakly adsorbed species, the surface concentration of which is influenced by the coverage of the strongly adsorbed species. This process is described by an electrode reaction rate equation.(v) In the overall oxidation of cysteine one electron is transferred while the detailed mechanism requires an oxidation by splitting-off two electrons with a subsequent ion—substrate dimerization reaction.     

Effect of ion solvation on the diffuse layer structure in mixed electrolytes

A “solvionic” model of a multicomponent electrochemical system (mixed electrolyte) is considered. An ion in the solution is considered as a point charge rigidly fixed inside its solvation shell. The corresponding equations for the diffuse layer on an ideally polarizable electrode are derived, and an effective method of their numerical solution is formulated. The calculations are performed in order to follow the changes in the diffuse layer structure with variations in the electrode charge and electrolyte composition. Far from the zerocharge potential of solution, the dependences of distributions of solution components over the diffuse layer on the electrode charge radically differ from those within the classic Gouy-Chapman theory. Analytical equations (asymptotics at large electrode charges) for concentrations of solvated ions in the plane of their maximum approach and for their “surface excesses” (diffuse adsorption) are determined. Results of numerical calculations for a 0.2 M LiCl + 0.05 M BaCl₂ solution are plotted.     

Turbulent Electrochemical Noise: A Theoretical Analysis in the Frequency–Potential Coordinates

Spectral density of random alternating current, which is induced by turbulent pulsations of electrolyte velocity in a cell, is theoretically analyzed within the Ershler–Randles charge transfer model in the Nernst diffusion layer approximation. The effect of electrode potential on the spectral current density is analyzed at various frequencies of hydrodynamic pulsations.     

Specific features of the behavior of an electrochemical system in the case of the Hopf instability for a spherical electrode

Model approximations are developed that allow establishing a quantitative relationship between the geometrical parameters of a spherical electrode, the faradaic impedance, and instabilities of the electrochemical system for an electrode reaction under potentiostatic conditions for the adsorption of species preceding their discharge. It is shown that the control parameter ω_H in the Hopf bifurcation point depends on the electrode size. The effect of the Nernst diffusion layer is observed at low frequencies in the range of negative faradaic impedance values.     

Electrochemical behavior of tungsten-containing nanocomposite with siloxane matrix

Electrochemical properties of a thin-film nanocomposite “silicon-carbon matrix-tungsten carbide” deposited onto pyroceramics (“sitall”) substrate are studied by potentiodynamic curves and electrochemical impedance spectroscopy. Transfer coefficients in model redox system [Fe(CN)₆]^3?/4? are measured. With the decreasing of the films’ electrical resistance, their experiment behavior gradually changed from that of “poor conductor” till nearly metal-like one. In particular, the electrode differential capacitance increases, which is explained by the increase in the number of conducting metal-containing clusters in the film bulk and at the film/electrolyte solution interface. Some specific features of the complex-plane plots of impedance spectra are tentatively explained by the adsorption at the nanocomposite surface elements.     

Couple axial gradients of potential and concentration in a cylindrical pore electrode: an impedance model

The impedance of a cylindrical pore electrode in the case where the potential gradient due to the electrolyte resistivity is coupled to the axial concentration gradient of reacting species has been calculated semi-analytically from the approximate solution reported previously for the steady-state concentration and current profiles in the pore. Complex plane impedance plots, computed by an iteration technique for the transmission line, indicate: (i) a quasi-semi-circular diffusion loop at low frequencies due to diffusion control; and (ii) a high frequency loop in which the frequency dispersion is strongly dependent on the electrode parameters (electrolyte resistivity, diffusion coefficient of the reacting species, pore depth, Tafel coefficient of the electrochemical reaction and overall current flowing through the pore).     

Kinetics of Adsorption of Neutral Molecules from a Thin Layer on a Rough Electrode

A theoretical model for the generalized finite adsorption impedance of a rough electrode is offered. A complex function of the roughness of the electrode surface is introduced and its frequency dependence is studied. The frequency characteristics of constituents of the adsorption impedance are shown to depend on the electrode roughness and the diffusion layer thickness. The frequency dependences of the phase angle and impedance components are found to considerably differ from the Frumkin–Melik-Gaikazyan result for an ideal smooth electrode.     

Characterization of Electrode/Electrolyte Interface of ECIS Devices

Electric cell‐substrate impedance sensing requires low electrode/electrolyte interface impedance for effective biomedical and biophysical applications. Thus a complete understanding of physical processes involved in the formation of an electric double layer is required to design a low interface impedance device. This paper presents the numerical simulation of the impedance for the electrode/electrolyte interface of three‐electrode devices along with the practical realization for the effective workout of impedance sensing devices. The three‐electrode based impedance sensing devices along with phosphate buffered saline as electrolyte is simulated using COMSOL Multiphysics to evaluate the impedance of the electrode/electrolyte interface. Microfabrication technology is used to realize three‐electrode impedance sensing devices with diverse configuration which are used to measure the electrode/electrolyte interface impedance. The measured impedance data were then compared with the COMSOL simulated results and it is found that both the data sets fitted well with less than 5 % RSE. The results obtained from simulation and experiments indicate that the impedance due to double layer diffusion dominates in the low frequency region up to few kHz whereas electrolytic bulk resistance plays a major role in the higher frequency range. The experimental impedance data were further interpreted by electrochemical impedance spectroscopy analysis software to model the equivalent circuit of the electrochemical system.     

Studies of Adsorption Kinetics and Defects of Self‐Assembled Thiol Monolayers on Gold by Capacitance Plane Plot

The capacitive property of an electrode/electrolyte interface can be described by complex capacitance. The capacitance plane plots (CPPs) of ideal polarized and kinetic controlled electrodes are derived based on the concept of complex capacitance. By using CPPs, the capacitance of electrode/electrolyte interface can be conveniently determined. In this work, CPPs obtained in ac impedance experiments are employed for the first time in studying the kinetics of adsorption process of the thiol monolayer. The coverage of octadecanethiol (ODT) monolayer on gold is examined as a function of adsorption time. The adsorption process of ODT molecules on gold exhibits two distinct phases: an initial rapid step followed by a slow one. The simple Langmuir model best explains our experimental data in the initial adsorption stage. CPPs and cyclic voltammetry (CV) indicate that, in the initial adsorption step, the ODT monolayer contains defects whose number decreases with the increasing of adsorption time.     

钒液流电池用石墨毡电极电化学活化机理的交流阻抗研究

研究了不同氧化程度下的石墨毡在钒溶液中的吸附性、润湿性及其交流阻抗图谱(EIS), 结果发现随着氧化程度增加, 吸附性和润湿性增强; 交流阻抗谱包括两个半圆和一条直线, 高频半圆对应离子的吸脱附反应, 低频半圆对应电化学反应, 直线对应离子在溶液中的扩散过程. 随石墨毡氧化程度的增加, 低频半圆显著减小, 通过等效电路拟合及动力学参数计算, 发现电荷传递电阻显著减小.     

Kinetics of electric double layer formation at the blocked spherical or cylindrical electrode/solid electrolyte interface under galvanodynamic and potentiodynamic conditions

The kinetics of charging of the blocked (inert) electrode/solid electrolyte interface is studied for spherical or cylindrical electrodes by the impedance method in two modes (galvanodynamic and potentiodynamic). The case of slow diffusion and adsorption-desorption is analyzed for species of one type, namely, defects of the solid-electrolyte rigid sublattice (minor carriers). The roles that both the slow lattice defects and the fast conduction ions play in the electric double layer formation are taken into account. Calculations involve the use of both the diffusion model of a spherical or cylindrical electrode in electrolyte (proposed by Jacobsen and West) and the ac circuit of an ideally polarizable planar electrode in a solid electrolyte (developed by Grafov, Ukshe, and Bukun).     

贮氢电极电化学阻抗谱及其数学模型

从分析贮氢电极的放电过程着手，建立了具有明显物理意义贮氢电极电化学阻抗谱的数学模型，以该数学模型为基础，讨论了与电极材料性质和电极荷电状态相联系的一些参数。     

Adsorption Kinetics of Normal-Heptanol on the Bismuth Single Crystal Planes

Cyclic voltammetry and impedance methods are employed for a quantitative study of normal-heptanol (n-HepOH) adsorption kinetics at the bismuth single-crystal plane/aqueous Na₂SO₄ solution interface. The results of nonlinear regression analysis show that the Frumkin–Melik-Gaikazyan (FMG) or Frumkin–Melik-Gaikazyan–Randles (FMGR) equivalent circuits can be used for the simulation of experimental impedance data. The dependences of adsorption capacitance (caused by the potential dependence of surface coverage), Warburg diffusion impedance, and adsorption resistance on the electrode potential and organic-compound concentration are established. Analysis of impedance data demonstrates that the adsorption of n-HepOH is mainly limited by the rate of diffusion of organic compound to the electrode surface. Small deviations toward mixed adsorption kinetics are established at very high frequencies. In the region of maximum adsorption in more concentrated n-HepOH solutions, the slow reorganization or two-dimensional association of adsorbed molecules is possible. However, the very low adsorption or partial charge transfer resistance values indicate that then-HepOH adsorption at Bi planes is a practically reversible process and thus there is no noticeable partial charge transfer between adsorbed n-HepOH molecules and Bi surface atoms.     

Benzene oxidation at boron-doped diamond anode: An electrochemical-impedance spectroscopy study

Benzene oxidation at boron-doped diamond anode in 1 M KCl aqueous solution is studied by electrochemical-impedance spectroscopy method. The measurements of differential capacitance and anodic current showed that in the ideal polarizability region benzene does not adsorb at the diamond electrode (or its adsorption does not affect the electrode capacitance). At more positive potentials adsorption of an inter-mediate of the benzene oxidation occurs, which partially blocks the electrode surface and results in decrease of anodic current. Analysis of the shape of the complex-plane plots of impedance spectra points out to diffusion of part of the adsorbed intermediate into solution.     

铜在低电导溶液中扩散机理的研究

双水内冷发电机空芯铜导线的腐蚀问题直接影响到火电厂发电机的安全经济运行．其腐蚀体系为纯铜和低电导水溶液（pH＝7．3，空气饱和，电导率＜10μS·cm－1）．我们已经用交流阻抗法对该体系进行了一些研究．由于该体系溶液电导率低，在交流阻抗谱的高频区出现高频半圆；低频区为一条文于实轴Z约45°的直线（不加缓蚀剂时）．如果在该体系中MABTA、MBT等缓蚀剂，低频区直线变为半圆．所以，低频区的阻抗代表了电极表面的特征，而高频区（如10k～100kHz范围）半圆是由于参比回路（包括参比电极、盐桥、鲁金毛细管等）中存在太大的电阻…     

Adsorption kinetics and electrode processes

Prof. H. A. Laitinen (Urbana):We have used double layer capacity measurements to measure adsorption as a function of time during the life of slowly forming mercury drops. For many substances that are not extremely surface active, so that the solution concentration is 1 to 10 mM, the double layer capacity per unit area is essentially constant during the time interval of 5 to 15 sec of drop life. It appears, therefore, that adsorption equilibrium has essentially been attained during the later portions of drop life.Prof. P. Delahay:We have also used a similar method of following the differential capacity during drop life. I would agree that one can achieve conditions for which adsorption equilibrium with respect to bulk concentration is practically reached (high enough concentration and long drop time). But this is quite often not the case (drop time of 3–4 sec in polarography).Mr. G.C. Whitnack (China Lake, U.S.A.);In determination of nitrate esters in presence of phthalate esters we have observed considerable dropping of current height of nitrate ester wave before start of phthalate ester wave. Is this due to an adsorption process occurring at the DME?Prof. P. Delahay:It might well be, but I would hesitate to comment any further on this point because more information would be required. You probably have quite a complicated situation. Perhaps we can discuss the matter later?Prof. Dr. H. Fischer (Ettingen):Ich möchte hinzufügen, class es Fälle geben kann, in denen mcht allem der Blockierungseffekt don Grenzstrom verringert, sondern eine Veränderung des Diffusionsfilmes durch Sekundärreaktion des Inhibitors. Dies ist z.B. bei der Abscheidung von Wasserstoff an einer festen Elektrode (Fe) der Fall, wenn der Inhibitor sich spaltet in eine schwerlösliche Verbindung und ein Proton (RNH⁺ → [RN] + H⁺). Dies beobachtet man z.B, bei den Kationen von heterozyklischen Aminen (Acridin). Offentlich bildet sich eine diffusionshindernde Barriere aus.Prof. P. Delahay:The case of the hydrogen electrode is complicated, and I shall attempt to answer your question only in the case of an ideally smooth electrode. Then, diffusion toward the electrode (partially covered with an adsorbed substance) is little affected by adsorption because the size of “the holes” in the film is very small in comparison with the diffusion layer thickness. Of course, there is no diffusion where there is complete blocking.Prof. B. Breyer (Sydney):The importance of the chemical nature of the film adsorbed at the interface, which has been mentioned by Prof. delahay, seems to me to play a major part in the type of processes discussed. Thus it must be kept in mind that complex formation between the diachargeable ion and the adsorbed film might occur (cf. e.g. heyrovsky? and matyas, 1941). The fact that T1+ ion is little influenced by the presence of an adsorbed film at the electrode solution interface could then be partly explained by the notoriously low co-ordination tendency of that ion.Prof. P. Delahay:The difficulty involved in the “blocking” of the limiting current for thallium is due, I think, to the small size of this ion (large diffusion coefficient of Tl+ in comparison with other ions). Of course, complex ions can be relatively very bulky and this enhances “blocking”.Prof. E. Lange (Erlangen):I agree with prof. delahay that it is very important to investigate the connections between the adsorption and electrode reaction.This is easy for a steady state, e.g., each heterogeneous reaction between two phases that is accompanied by a transfer of ions or electrons, i.e. of electrical charges, from one phase to the other. In such a case, the Galvani tension does not change and the transfer between the two phases must be compensated by a corresponding transfer of charges within the phases.But in the non-steady state, also, an adsorption process may behave as an “electrode reaction” for instance, even an adsorption of a dipole molecule may cause a “transfer of charge” accompanied by a corresponding change of the Galvani tension. In this manner, it seems to me that for the non-steady state it is necessary to define precisely what one means by “electrode reaction”.Prof. P. Delahay:I entirely agree with Prof. i.ange about the necessity of clear definitions. I think that the fact that a steady state with respect to diffusion of the reductible or oxidizable substance has not been reached is not too serious because this scarcely affects the Galvani potential (large excess of supporting electrolyte).Variations in the amount of adsorbed organic substances indeed affect the Galvani potential (dipole orientation), but this effect is included in the dependence of the rate constant ks (at the standard potential) on the electrode coverage.Prof. N. Tanaka (Sendai, Japan).I am very grateful to Prof. delahay for his beautiful work on adsorption kinetics. I should like to make one comment in connection with the rotated dropping mercury electrode. The relation between log i and log t on the current — time curve was found to be 0.5 only in the absence of the surface-active substance. In the presence of surface-active substance, the slope of log i vs. log t changed at a certain point of the current — time curve. This can be explained when the slow adsorption of the surface-active substance on the surface of the electrode is taken into consideration.Prof. P. Delahay:Even for stirred solutions, adsorption equilibrium is not reached very rapidly. A simple calculation based on a model of the Nernst diffusion layer shows that perhaps 1–5 sec are required. Your conclusion is, therefore, quite correct.Prof. W. Kemula (Warsaw):We have recently published that, in several cases, the addition of extremely small concentrations of organic surface-active substances provokes at first a rise of the diffusion current, this current then being suppressed by additional quantities of the substance.     

Effect of Adsorption of Neutral Molecules on the EDL Capacitance of a Rough Electrode

The perturbation theory is used to design a theoretical model for the Frumkin–Melik-Gaikazyan impedance of a sine electrode with a perturbation parameter equal to the ratio between the amplitude of surface oscillations and its period for a diffusion-controlled adsorption stage. The unequal diffusion access to the interface increases its capacitance. At higher frequencies, the electrode capacitance is defined by its geometric roughness factor. At lower frequencies, the capacitance depends on the adsorption conditions. The adsorption of electroactive substances results in a constant phase angle; the electrode roughness alters it and the frequency range where it remains constant.     

Impedance Study of GaN and InGaN Semiconductor Anion Selective Electrodes

The response of potentiometric anion selective electrodes employing undoped GaN or In_0.2Ga_0.8N films as sensing element to detect various anions was investigated in solutions of KF, KNO₃, KCl, HOC₆H₄COONa, KSCN, CH₃COOK, KClO₄ and KBr salts. The calibration plots for the GaN and In_0.2Ga_0.8N semiconductor electrodes contained linear regions extending over four decades of activity change in most solutions. The structure of the GaN and In_0.2Ga_0.8N semiconductor electrode/ electrolyte interface was studied through electrochemical impedance spectroscopy. Analogous equivalent circuits modeling the GaN or In_0.2Ga_0.8N electrode/electrolyte interface were proposed and their parameters were calculated. The space charge layer of the GaN and In_0.2Ga_0.8N semiconductors dominated the impedance of the electrochemical system at high frequencies (>10 kHz), whereas at low frequencies (<10 kHz), the impedance was controlled by the diffusion of electroactive species across the layer of adsorbed ions at the surface of the electrode. Results imply a strong dependence of the electrodes performance on the adsorption capacity of tested anions.