Porous electrode theory Potential

Porous electrode theory assumes that medium is a superposition of continuous solid and electrolyte phases with a known vo-Inme fraction. The solid phase potential of the positive electrode is because of electronic conduction ... 

Consider a linear electrochemical reaction inside a porous electrode. [15] [16] The dimensionless solid phase potential (d>i) and electrolyte potential (O2) are governed by the macroscopic porous electrode theory ... 

The porous electrode theory was developed by several authors for dc conditions [185-188], bnt the theory is usually applied in the ac regime [92,100,101,189-199], where mainly small signal frequency-resolved techniques are used, the best example of which are ac theory and impedance spectra representation, introdnced in the previons section. The porous theory was first described by de Levi [92], who assumed that the interfacial impedance is independent of the distance within the pores to obtain an analytical solution. Becanse the dc potential decreases as a fnnction of depth, this corresponds to the assnmption that the faradaic impedance is independent of potential or that the porons model may only be applied in the absence of dc cnrrent. In snch a context, the effect of the transport and reaction phenomena and the capacitance effects on the pores of nanostructured electrodes are equally important, i.e., the effects associated with the capacitance of the ionic donble layer at the electrode/electrolyte-solntion interface. For instance, with regard to energy storage devices, the desirable specifications for energy density and power density, etc., are related to capacitance effects. It is a known fact that energy density decreases as the power density increases. This is true for EDLC or supercapacitors as well as for secondary batteries and fnel cells, particnlarly due to the distributed nature of the pores... 

Going from planar to porous electrode introduces another length scale, the electrode thickness. In the case of a PEM fuel cell catalyst layer, the thickness lies in the range of IcL — 5-10 pm. The objective of porous electrode theory is to describe distributions of electrostatic potentials, concentrations of reactant and product species, and rates of electrochemical reactions at this scale. An accurate description of a potential distribution that accounts explicitly for the potential drop at the metal/electrolyte interface would require spatial resolution in the order of 1 A. This resolution is hardly feasible (and in most cases not necessary) in electrode modeling because of the huge disparity of length scales. The simplified description of a porous electrode as an effective medium with two continuous potential distributions for the metal and electrolyte phases appears to be a consistent and practicable option for modeling these stmctures. 

Equation 25.14 can be used to determine the electrical potential in the electrolyte phase. Each term in this equation has been volume averaged, where the last term is a source term describing the volumetric electrochemical reaction. Here, a is the specific interfacial area described as the interfacial area per unit of electrode volume. The effective conductivities and x are used to account for the actual tortuous path length of charge transport and local porosity. Two empirical relations are often used in the porous electrode theory, namely ... 

In this approach, originally developed by Fuller et al. [18, 53] based on the porous electrode theory [42], the active material is assumed to consist of spherical particles with a specific size, and solid phase diffusion in the radial direction is assumed to be the predominant mode of transport. The electrolyte phase concentration (Cg) and the potentials (4>s,4>e) are assumed to vary along the principal (i.e., thickness) direction only, and are henceforth referred to as the x direction. In other words, this model implicitly considers two length scales (1D + 1D), that is, the r direction inside the spherical particle and the x direction along the thickness. All other equations described earlier continue to remain valid except the solid phase diffusion. Equation 25.19 and the corresponding boundary/initial conditions. The solid phase diffusion equation now takes the following form ... 

Figure 1 shows a schematic of a hthium battery. Lithium battery electrodes are usually made by coating a slurry of the active material, conductive filler, and binder onto a foil current collector. This porous configuration provides a high surface area for reaction and reduces the distance between reactants and the surfaces where reactions occur. In these porous electrodes, the electrochemical reaction is distributed over the surface of the particles of active material, and will vary across the depth of the electrode due to the interaction of potential drop and concentration changes in both the solution and solid phases. Porous electrode theory is used to understand these interactions. 

In this section we describe the equations required to simulate the electrochemical performance of porous electrodes with concentrated electrolytes. Extensions to this basic model are presented in Section 4. The basis of porous electrode theory and concentrated solution theory has been reviewed by Newman and Tiedemann [1]. In porous electrode theory, the exact positions and shapes of aU the particles and pores in the electrode are not specified. Instead, properties are averaged over a volume small with respect to the overall dimensions of the electrode but large with respect to the pore structure. The electrode is viewed as a superposition of active material, filler, and electrolyte, and these phases coexist at every point in the model. Particles of the active material generally can be treated as spheres. The electrode phase is coupled to the electrolyte phase via mass balances and via the reaction rate, which depends on the potential difference between the phases. AU phases are considered to be electrically neutral, which assumes that the volume of the double layer is smaU relative to the pore volume. Where pUcable, we also indicate boundary conditions that would be used if a Uthium foil electrode were used in place of a negative insertion electrode. 

The averaging inherent to porous electrode theory introduces some (usually very small) degree of error in calculating the actual potential and concentration at the pore waU. Mass-transfer coefficients can be introduced to try and compensate, such as using an effective mass-transfer equation, j = -km(c-Cwn). to solve to solve for the electrolyte concentration at the pore waU [31], Tsaur and PoUard [8] report that mass transfer within pores has a significant effect on ceU performance only for species present in smaU concentrations. 

The models that consider this approach are largely based on the assumption of effectively homogeneous local relaxation processes related to transport in each of the phases and electrical charge exchange between them. Thus, the complex problem of an uneven distribution of electrical current and potential inside the electrode can be described analytically, and impedances can be calculated. Furthermore the models may be conveniently pictured as a double-channel transmission line (Fig. 3.5). In several papers, the theory of the impedance of porous electrodes has been extended to cover those cases in which a complex frequency response arises in the transport processes [100] or at the inner surface [194,203]. 

Eirst, let us show how to derive the required porous electrode RC theory ( transmission line-theory ) from our results in Section 15.4.3. At high ionic strength conditions, the diffuse (or, Donnan) layer potential will be snfficiently low such that we are in the low-potential limit where we have no net salt (electro-)adsorption in the EDLs (see Equation 15.4). This is the fundamental reason... 

Oscillatory behavior observed as periodic potential transients at constant current or periodic current transients at constant potential is found frequently when more than two parallel electrode reactions are coupled. Usually, an upper and a lower current-potential curve limit the oscillation region. These two curves represent stable states [139] according to the theory of stability of electrode states [140]. Oscillatory phenomena occurring during the oxidation of certain fuels on solid electrodes are discussed in this section. The discussion is not extended to porous electrodes because the theory of the diffusion electrode has not been developed to the point to allow an adequate description of the complex coupling of parallel electrode reactions and mass transport processes in the liquid and gaseous phase. 

Garcia etal. [41] developed a two-dimensional porous electrode model and accounted for potential and charge distributions in the electrolyte. They employed transport equations derived from dilute solution theory, which is generally not adequate for LIB systems. The stress generation effect is built into the 2D DNS modeling framework with a simplified, sphere-packed electrode microstmcture description. 

The traditional approach to understanding both the steady-state and transient behavior of battery systems is based on the porous electrode models of Newman and Tobias (22), and Newman and Tiedermann (23). This is a macroscopic approach, in that no attempt is made to describe the microscopic details of the geometry. Volume-averaged properties are used to describe the electrode kinetics, species concentrations, etc. One-dimensional expressions are written for the fluxes of electroactive species in terms of concentration gradients, preferably using the concentrated solution theory of Newman (24). Expressions are also written for the species continuity conditions, which relate the time dependence of concentrations to interfacial current density and the spatial variation of the flux. These equations are combined with expressions for the interfacial current density (heterogeneous rate equation), electroneutrality condition, potential drop in the electrode, and potential drop in the electrolyte (which includes spatial variation of the electrolyte concentration). These coupled equations are linearized using finite-difference techniques and then solved numerically. 

Polar Cell Systems for Membrane Transport Studies Direct current electrical measurement in epithelia steady-state and transient analysis, 171, 607 impedance analysis in tight epithelia, 171, 628 electrical impedance analysis of leaky epithelia theory, techniques, and leak artifact problems, 171, 642 patch-clamp experiments in epithelia activation by hormones or neurotransmitters, 171, 663 ionic permeation mechanisms in epithelia biionic potentials, dilution potentials, conductances, and streaming potentials, 171, 678 use of ionophores in epithelia characterizing membrane properties, 171, 715 cultures as epithelial models porous-bottom culture dishes for studying transport and differentiation, 171, 736 volume regulation in epithelia experimental approaches, 171, 744 scanning electrode localization of transport pathways in epithelial tissues, 171, 792. 

Another problem in application of the basic theories is associated with surface geometry. Most theories are developed to describe the relationships among the area-averaged quantities such as charge density, current density, and potentials assuming a uniform electrode surface. In fact, the silicon surface may not be uniform at the micrometer, nanometer, or atomic scales. There can be great variations in the distribution of reactions from extremely uniform, for example, in electropolishing, to extremely nonuniform, for example, in the formation of porous silicon. 

Adherents to this theory have different opinions on the potential at which the film forms. Its thickness, the mechanism of formation, and, most Important, the "cause of passivity. In the earlier theories It was postulated that the passivation follows the formation of a "primary layer" of small conductivity, x lth porous character, which Is sometimes due to precipitation of metal salt on and near the electrode.(32) In the pores the current Increases, and by polarization at an "Umschlagspotentlal" (Vj, = V, Figure 1) an actual passive layer is formed. Thus the essential concept of the passivation process Is connected with the change of the properties (chemical or physical) of the primary film at a certain potential. The passive film Is free from pores and presents a barrier between the metal and the environment. It is electronically conductive and slowly corrodes In solution.(6,8,24,37)... 

---