Porous capacitor model

Still in Hsu et al. work, electrochemical impedance was used to analyze the reaction kinetics and interfacial characteristics of an anode in DMFC [142]. Several analogy-circuit models are proposed (Fig. 8.11). The new model incorporates CPEs rather than conventional capacitors in the equivalent-circuits taking into account the porous structure of the anode, particularly that in the CL and at the anode/... 

It is often found that the double-layer capacitance or a coating capacitance does not behave like an ideal capacitor, experimentally manifested in the complex plane plot by a depressed semicircle whose center lies below the real axis. This behavior is usually attributed to some distribution (or dispersion) in some physical property of the system (e.g., the porous surface of the metal or the varying thickness or composition of a coating) and is modeled by the use of a constant phase element (CPE) [30]. 

CPE is used in a model in place of a capacitor to compensate for non-homogeneity in the system. A rough or porous surface can cause double-layer capacitance to appear as CPE and Warburg element [116, 117]. 

It is noteworthy that, in all these models, the electrode surface is considered as a plane whereas, practically in an electrical double-layer capacitor (EDLC), it is a porous material - most often carbon - of highly developed SSA, which might be approximately estimated by gas (generally nitrogen) physisorption. Since the gas molecule used to probe the pore volume and the electrolyte ions display different size and interaction with the material surface, it is obvious that the active surface area that takes part in EDL charging is different from the one evaluated by gas adsorption. 

However, it is useful first to illustrate the complexity of such systems by showing a hierarchy of equivalent RC circuits that can be developed, from which an ultimate model of a porous-electrode CR device can be constructed. Note that the behavior of an electrochemical capacitor device is, electrically, far from that of a pure capacitor in its ac response spectrum to AV modulation this is primarily due to the complexity of the distributed, internal connections within the matrix, associated with resistivity of electrolyte channels and the intrinsic resistance of the C microparticles or fibrils and their interparticle contact resistances which usually depend on the pressure applied during fabrication of electrode structures. 

Figure 4.5.27. Five-element ladder RC circuit modeling the impedance behavior of a porous, electrochemical capacitor, designed for memory back-up (from Miller [1972]).

The model is strictly valid only for an identical set of uniform (constant diameter) pores. As will be seen below, however, this simple model provides a good description of the electrical response of carbon electrochemical capacitors. The model is evidently qualitatively descriptive of the average properties of some porous materials. It is expected that the model will apply to a material composed of a set of pores in which r and... 

Fig. 19 Equivalent circuit models for caibon-based porous electrodes RC circuits for a series and b parallel connections, representing an equivalent circuit (simplest) of a capacitor. R resistor, C capacitor. Equivalent circuits of only one capacitor (Cdl or CP) in parallel to a resistor R and in series to resistor RS (c) and considering both Cdl (in parallel to RE) and CP (in parallel to RE ) in series with RS (d) are also shown. The ac responses to the latter two circuits are shown in (e, f) [33] (Reprinted with permission from Ref. [33] Copyright (2012) by John Wiley and Sons)...

Fig. 7. Model of porous plug [6]. - (A) schematic representation of current path through plug. (1) represents current through solution and spheres in series, (2) through spheres in contact with each otlier, (3) current through solution. (B) simplified model for d.c. conductance, representing situation shown in (A). Different zones represent separate resistors of dimensions shown. a + b- -c=l cm. (C) Extension of same model to a.c. properties. Each zone is represented by a parallel circuit of resistor and capacitor. Impedance of each zone is determined by its dimension (equal to those shown in (B)) and conductivity and dielectric constant of material forming the zone.

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