Porous Electrode Model

In electrocatalysis there is great interest in increasing the real surface area of electrodes. In such cases porous electrodes are used. Because modehng of real electrodes is difficult, a simpler model is usually used in which it is assumed that pores have a cylindrical shape with a length / and a radius 

In order to describe the impedance of such electrodes, first a dc solution must be found. Two cases are considered here (1) porous electrodes in the absence of internal diffusion and (2) in the presence of axial diffusion. It is assumed that the electrical potential and concentration of electroactive species depend on the distance from the pore orifice only and there is always an excess of the supporting electrolyte (i.e., migration can be neglected). 

In this case it is assumed that the concentration of the electroactive species is independent of the distance along a pore. In the next section we will see when such an assumption is valid. The axially flowing dc current, /, which enters the pore, flows toward the walls and its value decreases with the distance x from the pore orifice (Fig. 34). This decrease in the current is proportional to the current flowing to the wall  

This equation describes changes in the electrical potential as a function of pore length, de Levie assumed that the impedance of pore walls is independent of the pore distance (Z is not a function of distance), which implies that there is no net dc current. The solution is 

Of course, under dc conditions, when co = 0,, ei= Ret- For the ensemble of n pores and in the presence of the solution resistance outside the pores, the total impedance becomes 

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Porous-Electrode Models. The porous-electrode models are based on the single-pore models above, except that, instead of a single pore, the exact geometric details are not considered. Euler and Nonnenmacher and Newman and Tobias were some of the first to describe porous-electrode theory. Newman and Tiedemann review porous-electrode theory for battery applications, wherein they had only solid and solution phases. The equations for when a gas phase also exists have been reviewed by Bockris and Srinivasan and DeVidts and White,and porous-electrode theory is also discussed by New-man in more detail. 

The next set of models treats the catalyst layers using the complete simple porous-electrode modeling approach described above. Thus, the catalyst layers have a finite thickness, and all of the variables are determined as per Table 1 with a length scale of the catalyst layer. While some of these models assume that the gas-phase reactant concentration is uniform in the catalyst layers,most allow for diffusion to occur in the gas phase. 

The final simple macrohomogeneous porous-electrode models are the ones that are more akin to thin-film models. In these models, the same approach is taken, but instead of gas diffusion in the catalyst layer, the reactant gas dissolves in the electrolyte and moves by diffusion and reaction. The... 

The rest of the comparisons were done for the cathode. The results all showed that the agglomerate model fits the data better than the porous-electrode model. However, it should be noted that the porous-electrode model used was usually a thin-film model and so was not very robust. Furthermore, the agglomerate model has more parameters that can be used to fit experimental data. Finally, some of the agglomerate models compared were actually embedded models that account for both length scales, and therefore, they normally agree better with the experimental data. 

The derivation of the model is similar to that derived for the porous electrode model. We have very similar model equations to Eqs. (115) and (116) for an anodic current, given by... 

The electrochemical behavior of poly(pyrrole) films prepared and cycled in an AICI3 [C2mim][Cl] melt was investigated in detail and improvements in reproducibility and the rate of oxidation and reduction of these films were observed compared to films prepared under similar conditions in acetonitrile [49]. This was postulated to be a result of an increase in the porosity of poly(pyrrole) films deposited from the melt compared to those from acetonitrile, although attempts to describe this porosity using porous electrode models were not totally conclusive. 

Wilemski, G. Simple porous electrode models for molten carbonate fuel cells. J. Electrochem. Soc. 1983, 130 (1), 117-120. 

The porous electrode model described in Eq. (9.7) cannot be represented by a simple connection of R, L, and C elements. However, it can be represented by a semi-infinite series of R-C elements called a transmission fine [410,411], shown in Fig. 9.4. Of course, this representation is equivalent to Eq. (9.7). Some authors tried to use a transmission line to approximate experimental data using a sufficient number of RC elements and verifying whether the number of these parameters was sufficient. This procedure can approximate, then, experimental impedances, but the use of Eq. (9.7) is more appropriate because it allows for the direct estimation of certain parameters and their standard deviations. This model is included in the recent version of the ZView program. 

Figure 4. 45. One-dimensional network distributed homogeneous cylindrical porous electrode model after Gbhr [1997].

Figure 4.5.53. Equivalent circuit (porous electrode model) for the evaluation of the impedance spectra of the silver GDE measured during ORR at different times of operation at 100mAcm. ...

Figure 4.5.60. (a) Equivalent circuit (EC) of the PEFC with O2/H2 gas supply, for low current densities, (b) Equivalent circuit of the PEEC with O2/H2 gas supply, for high current densities with an additional diffusion step (Nernst-impedance), (c) Equivalent circuit of the PEFC with O2/H2 gas supply with Nernst-impedance and porous electrode model. 

EIS Measured at SOFC at Different Experimental Conditions—At Different Current Densities. EIS was applied by van Heuveln [1993] to determine the electrical losses in the SOFC, further refinements have been made by Richter [1997] to represent the porous structure of the electrode using the porous electrode model proposed by Gohr [1997]. 

Yuh, C.Y. and Selman, ).R. (1992) Porous-electrode modeling of the molten-carbonate fuel-cell electrodes. 

Hence we note that the simple semiinfinite transient current response obtained from analyzing a porous electrode model is similar in form to the Cottrell equation derived via a diffusion-based approach in both cases i varies as In the transmission line model, we examine... 

FIGURE 1.55. Predicted dimensionless plots for analyzing chronoamperometric transients from transmission line or porous electrode model using Eqn. 286 (fidl curve). Note that the short-time limit given by Eqn. 283 dashed curve), and the long-time limit given by Eqn. 287 are also displayed. 

The porous electrode model can also be applied to current step chronopotentiometry this has been done by Martin and coworkers. The expression for the variation of potential with time following application of a constant current i is obtained via the following expression ... 

Fletcher proposes adopting a porous electrode model, considering the conductive polymer film in contact with an aqueous electrolyte solution as consisting of a large number of identical, noninterconnected pores. The electrolyte solution is contained within the pores. The analysis then considers a single pore of uniform cross section. Three general impedance elements are considered the solution impedance x within the pore the interfacial impedance y between the solution within the pore and the pore wall and z, the internal impedance of the polymer. The latter quantities are assumed not to vary with distance inside the pore. 

FIGURE 1.82. Schematic representation of the equivalent circuit ladder network corresponding to Fletcher porous electrode model for electronically conducting polymers (see Refs. 68, 69). The specific equivalent circuit representation of the interfacial impedance element is also illustrated. 

FIGURE 15.8 Calculation results of ID dynamic porous electrode model for an electrode with a selective cation exchange membrane layer in front (located atx = 0) that is perfectly blocking for co-ions, (a) Macropore salt concentration profiles as function of time (direction of arrows) during application of -770 mV voltage to the electrode relative to the bulk (spacer channel) outside the membrane, and (b). After subsequent increase of the potential to +770 mV. 

Detailed derivations of the porous electrode model and the volume averaging approach for battery modeling can be found in Refs [14,42, 52], respectively. 

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. 

Intercalation-induced stresses have been modeled extensively in the Hterature. A one-dimensional model was proposed to estimate stress generation in the lithium insertion process in the spherical particles of a carbon anode [24] and an LiMn204 cathode [23]. In this model, displacement inside a particle is related to species flux by lattice velocity, and total concentration of species is related to the trace of the stress tensor by compressibihty. Species conservation equations and elasticity equations are also included. A two-dimensional porous electrode model was also proposed to predict electrochemicaUy induced stresses [30]. Following the model approach of diffusion-induced stress in metal oxidation and semiconductor doping [31-33], a model based on thermal stress analogy was proposed to simulate intercalation-induced stresses inside three-dimensional eUipsoidal particles [1]. This model was later extended to include the electrochemical kinetics at electrode particle surfaces [2]. This thermal stress analogy model was later adapted to include the effect of surface stress [34]. 

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. 

To obtain accurate information about microstructural characteristics ofSOFC electrodes, a set of experimental i-r) curves for a given electrode may be fitted, similar to Figure 11.3, against predictions of a more complex porous electrode model, as discussed in Section 11.8. 

One-Dimensional Porous Electrode Models Based on Complete Concentration, Potential, and Current Distributions... 

S. Al-Hallaj and J. R. Selman, Porous-electrode model for SOFC electrodes, in Proceedings NETL Workshop on Fuel Cell Modeling, U.S, Department of Energy, National Energy Technology Laboratory, Morgantown, PA., 2000. 

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