Electrode-electrolyte interface electrical model

However, these classical models neglect various aspects of the interface, such as image charges, surface polarization, and interactions between the excess charges and the water dipoles. Therefore, the widths of the electrode/electrolyte interfaces are usually underestimated. In addition, the ion distribution within the interfaces is not fixed, which for short times might lead to much stronger electric helds near the electrodes. 

The electric potential ip is assumed to be continuous throughout the electrodes and electrolyte except at the electrode/electrolyte interfaces. These discontinuities are usually modeled by Nemst s law. The model to calculate the potential jumps at each electrolyte/electrode interface is described in Celik et al. (2005). The source term in Equation (5.24) is non-zero only near the electrode/electrolyte interfaces to account for the potential jumps. 

The properties characteristic for electrochemical nonlinear phenomena are determined by the electrical properties of electrochemical systems, most importantly the potential drop across the electrochemical double layer at the working electrode (WE). Compared to the characteristic length scales of the patterns that develop, the extension of the double layer perpendicular to the electrode can be ignored.2 The potential drop across the double layer can therefore be lumped into one variable, DL, and the temporal evolution law of DL at every position r along the (in general two-dimensional) electrode electrolyte interface is the central equation of any electrochemical model describing pattern formation.3 It results from a local charge bal-... 

Serious efforts have been made to explain the atypical lithium transport behavior using modified diffusion control models. In these models the boundary conditions -that is, "real potentiostatic constraint at the electrode/electrolyte interface and impermeable constraint at the back of the electrode - remain valid, while lithium transport is strongly influenced by, for example (i) the geometry of the electrode surface [53-55] (ii) growth of a new phase in the electrode [56-63] and (iii) the electric field through the electrode [48, 56]. 

The contribution of electric field to lithium transport has been considered by a few authors. Pyun et argued on the basis of the Armand s model for the intercalation electrode that lithium deintercalation from the LiCo02 composite electrode was retarded by the electric field due to the formation of an electron-depleted space charge layer beneath the electrode/electrolyte interface. Nichina et al. estimated the chemical diffusivity of lithium in the LiCo02 film electrode from the current-time relation derived from the Nernst-Planck equation for combined lithium migration and diffusion within the electrode. 

EDLCs store energy within the variation of potential at the electrode/electrolyte interface. This variation of potential at a surface (or interface) is known as the electric double layer or, more traditionally, the Helmholtz layer. The thickness of the double layer depends on the size of the ions and the concentration of the electrolyte. For concentrated electrolytes, the thickness is on the order of 10 A, while the double layer is 1000 A for dilute electrolytes (5). In essence, this double layer is a nanoscale model of a traditional capacitor where ions of opposite charges are stored by electrostatic attraction between charged ions and the electrode surface. EDLCs use high surface area materials as the electrode and therefore can store much more charge (higher capacitance) compared to traditional capacitors. 

A supercapacitor stores energy in electrical double layers at electrode/electrolyte interfaces. In molecular modeling of supercapacitors, the... 

In addition to the solvent contributions, the electrochemical potential can be modeled. Application of an external electric field within a metal/vacuum interface model has been used to investigate the impact of potential alteration on the adsorption process [111, 112]. Although this approach can model the effects of the electrical double layer, it does not consider the adsorbate-solvent, solvent-solvent, and solvent-metal interactions at the electrode-electrolyte interface. In another approach, N0rskov and co-workers model the electrochemical environment by changing the number of electrons and protons in a water bilayer on a Pt(lll) surface [113-115]. Jinnouchi and Anderson used the modified Poisson-Boltzmann theory and DFT to simulate the solute-solvent interaction to integrate a continuum approach to solvation and double layer affects within a DFT system [116-120]. These methods differ in the approximations made to represent the electrochemical interface, as the time and length scales needed for a fiilly quantum mechanical approach are unreachable. 

The conventional electrical model of an electrochemical cell that represents the electrode-electrolyte interface (EEI) includes the association of resistances with capacitance as shown in Fig. 1. The parallel elements are related to the total current through the working electrode that is the sum of distinct contributions from the faradaic process and double-layer charging. The double layer capacitance resembles a pure capacitance, represented in the equivalent circuit by the element C and the faradaic process represented by a resistance, R2. The parameters E and Ri represent the equilibrium potential and the electrolyte resistance, respectively. 

Various models have been proposed for the electric double layer at an electrode-electrolyte interface. Briefly explain the structure of the electric double layer starting from the Helmholtz model to the triple-layer model and then identify the key features of each model. 

Fig. 1 The electrode/electrolyte interface, iUustiatmg Faradaic chaige transfer (top) and capacitive redistribution of chaige (bottom) as the electrode is driven negative, (a) Physical representation (b) Two-element electrical circuit model for mechanisms of charge transfer at the interface. The capacitive process involves reversible redistribution of chaige. The Faradtiic process involves transfer of electrons from the metal electrode, reducing hydrated cations in solution (symbolically 0 + e R, where the cation O is the oxidized form of the redox couple O/R). An example reaction is the reduction of silver ions in solution to form a silver plating on the electrode, reaction (8a). Faradaic charge injection may or may not be reversible...

If only non-Faradaic redistribution of charge occurs, the electrode/electrolyte interface may be modeled as a simple electrical capacitor called the double-layer capacitor Cdi. This capacitor is formed due to several physical phenomena [2, 3, 4, 5, 6]. First, when a metal electrode is placed in an electrolyte, charge redistribution occurs as metal ions in the electrolyte combine with the electrode. This involves a transient transfer of electrons between the two phases, resulting in a plane of charge at the surface of the metal electrode, opposed by a plane of opposite charge, as counterions, in the electrolyte. The excess charge on the electrode surface, symbolized... 

Faradak Charge Transfer and the Electrical Model of the Electrode/Electrolyte Interface... 

Fig. 3 Electrical Circuit Models (a) Single-Electrode/Electrolyte Interface (b) Three-Electrode System External access to the system is at three points labeled WE , CE , and RE . If the counter electrode has a large surface area, it may be considered as strictly a capacitance as shown. A reference electrode with very low valued Faradaic resistance will maintain the interfacial potential VRE-soiution constant...

For lower frequencies and slower signal waveforms, electrode model is more complicated than the Randles model. It has nonlinear characteristics which cannot be modeled by standard electrical elements having fixed values, like resistors and capacitors. Also incorporating non-linear elements like diodes may model some behaviors of the electrode-electrolyte interface like the dependence of the faradaic resistor on the interface voltage drop, but it cannot completely represent the reality. 

As stated before, electrodes are not standard linear electrical elements. Electrode properties depend on the electrode potential. Especially, in stimulation electrodes, the electrode potential fluctuates over a relatively wide range, thus enhancing the nonlinear characteristics of the electrode-electrolyte interface. However, an approximate electrical model can be helpful in designing the interface circuits to the electrodes, like signal recording or driver circuits. As explained in Chap. 5, impedance spectroscopy is one of the methods to extract the electrode model. In the following this and other methods are explained through practical examples. 

Full evaluation of equation (2.4) thus requires knowledge of the charge distribution at the electrode - electrolyte interface, a problem that has been explored in various works.For example, Dickinson and Compton recently used numerical modelling to solve the Poisson - Boltzmann equation, which describes the electric field in an electrolyte solution under thermodynamic equilibrium, for hemispherical electrodes. The simulations revealed a transition between two classical limits a planar double layer as predicted by the Gouy - Chapman model and the spherical double layer associated with a point charge (Coulomb s Law). This is illustrated in Fig. 2.2, in which the dimensionless charge density, Q ( FrqjRTEQEg) is plotted as a function of the dimensionless hemispherical electrode radius,... 

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