Electrical double layer capacitor model

FIGURE 2.54 Normalized capacitance change versus the pore size of the CDC samples prepared at different temperatures normalized capacitance is obtained by dividing the specific capacitance by the SSA. HyperChem models of the structure of EMI and TFSI ions show a size correlation. (Reprinted with permission from Largeot, C. et al., 2008. Relation between the ion size and pore size for an electric double-layer capacitor. Journal of the American Chemical Society 130 2730-2731. Copyright 2008 American Chemical Society.)... 

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. 

In [22], A1 Sakka et al., proposed a thermal model for cylindrical electric double-layer capacitors (EDLC), by performing specific techniques on the layer level. In [23] a methodology is proposed for simulation of the internal temperature of a cylindrical lithium iron phosphate battery cell. 

Nishihara H, Itoi H, Kogure T, Hou P-X, Touhara H, Okino F, Kyotani T (2009) Investigation of the ion storage/transfer behavior in an electrical double-layer capacitor by using ordered microporous carbons as model materials. Chem Eur J 15 5355-5363... 

The electrified interface is generally referred to as the electric double layer (EDL). This name originates from the simple parallel plate capacitor model of the interface attributed to Helmholtz.1,9 In this model, the charge on the surface of the electrode is balanced by a plane of charge (in the form of nonspecifically adsorbed ions) equal in magnitude, but opposite in sign, in the solution. These ions have only a coulombic interaction with the electrode surface, and the plane they form is called the outer Helmholtz plane (OHP). Helmholtz s model assumes a linear variation of potential from the electrode to the OHP. The bulk solution begins immediately beyond the OHP and is constant in potential (see Fig. 1). 

For a long time, the electric double layer was compared to a capacitor with two plates, one of which was the charged metal and the other, the ions in the solution. In the absence of specific adsorption, the two plates were viewed as separated only by a layer of solvent. This model was later modified by Stem, who took into account the existence of the diffuse layer. He combined both concepts, postulating that the double layer consists of a rigid part called the inner—or Helmholtz—layer, and a diffuse layer of ions extending from the outer Helmholtz plane into the bulk of the solution. Accordingly, the potential drop between the metal and the bulk consists of two parts ... 

In the absence of specific adsorption of anions, the GCSG model regards the electrical double layer as two plate capacitors in series that correspond respectively, to two regions of the electrolyte adjacent to the electrode, (a) An inner compact layer of solvent molecules (one or two layers) and immobile ions attracted by Coulombic forces (Helmholtz inner plane in Fig. 2). Specific adsorption of anions at the electrode surface may occur in this region by electronic orbital coupling with the metal, (b) An outer diffuse region of coulombically attracted ions in thermal motion that complete the countercharge of the electrode. 

The interphase between an electrolyte solution and an electrode has become known as the electrical double layer. It was recognized early that the interphase behaves like a capacitor in its ability to store charge. Helmholtz therefore proposed a simple electrostatic model of the interphase based on charge separation across a constant distance as illustrated in Figure 2.12. This parallel-plate capacitor model survives principally in the use of the term double layer to describe a situation that is quite obviously far more complex. Helmholtz was unable to account for the experimentally observed potential dependence and ionic strength dependence of the capacitance. For an ideal capacitor, Q = CV, and the capacitance C is not a function of V. 

Double-Layer Capacitor Electrical Equivalent Model.443... 

DOUBLE-LAYER CAPACITOR ELECTRICAL EQUIVALENT MODEL... 

Figure 17.1 An electrical double-layer model (a) and a double-layer capacitor (b).

Figure 10. Electrical double layer models. Top right (a) typical type of potential vs. composition plot for a charged surface compared to (b) constant capacitance model. Top left Two double-layer models, (a) diffuse double layer, (b) part parallel plate capacitor and part diffuse layer.. Bottom left Stem layer model. Incorporation of adsorbed ions to surface. From Hiemenz and Rajagopalan (1997) Bottom right Comparison of Gouy-Chapman and Stem-Grahame models of the electrical double layer. From Davis and Kent (1990).

Figure 2. Diagram showing the equivalent circuit model for a cell in suspension. The electrical double layer (EDL) on the surface of the electrodes is model as a capacitor C i.

By introducing a local dielectric permittivity for the water part and the air part of a real interface Vogel Mobius (1988a, b) suggested a two layer capacitor model shown in Fig. 2.6, which is in agreement with their experimental findings. This model considered the polarisation effects of ionic head groups as well as the influence of an electrical double layer. 

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. 

The simplest model of the electrical double layer between a metal and an electrolyte is the simple capacitor visualized by Helmholtz as shown in Figure 14. The diffuse ion distribution in the liquid phase was recognized by Gouy and Chapman- to form a space charge region adjacent to the electrode surface. 

Capacitive effects. The presence of a protective layer on the surface of a metallic cluster decreases the capacity of the cluster. For a planar metal electrode, the electrical double layer comprised of the charge on the surface of the electrode and the ions of opposite charge in the solution (or the solvent dipoles) can be modeled as a parallel plate capacitor with capacity Cpp in Farads (F) given by... 

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