Dielectric constant Differential diffusion

Gouy—Chapman Diffuse-Charge Model Jec0kT % ze y 2n Smh 2kT 2 nkT J kT V, = e" Pol 0 er s tial x-> It predicts that differential capacities have the shape of inverted parabolas. Ions are considered as pointcharges. Ion-ion interactions are not considered. The dielectric constant is taken as a constant. 

Here P and y are the thickness of the hydrocarbon tail and of the polar head region of the lipid monolayer, Sp and eY are the corresponding distortional dielectric constants, %e and Xm are the surface dipole potentials due to the electron spillover and to the oriented polar heads, and fa is the potential difference across the diffuse layer. At ion concentrations that are not exceedingly low, fa can be disregarded as a good approximation. Moreover, the orientation of the polar heads of the lipid film is hardly affected by changes in aM. The differential capacity C of the electrode can, therefore, be written ... 

The determination of the real surface area of the electrocatalysts is an important factor for the calculation of the important parameters in the electrochemical reactors. It has been noticed that the real surface area determined by the electrochemical methods depends on the method used and on the experimental conditions. The STM and similar techniques are quite expensive for this single purpose. It is possible to determine the real surface area by means of different electrochemical methods in the aqueous and non-aqueous solutions in the presence of a non-adsorbing electrolyte. The values of the roughness factor using the methods based on the Gouy-Chapman theory are dependent on the diffuse layer thickness via the electrolyte concentration or the solvent dielectric constant. In general, the methods for the determination of the real area are based on either the mass transfer processes under diffusion control, or the adsorption processes at the surface or the measurements of the differential capacitance in the double layer region [56],... 

Equation (2.17) indicates that when (/i = 0, the differential capacitance of the diffuse layer is proportional to the square roots of bofh fhe electrolyte concentration and the dielectric constant. Furthermore, according to the definition of capacifance, if fhe Q,yis expressed as... 

In addition. Equation (2.19) demonstrates that the dielectric constant has the same weight as that of the electrol5de concentration, meaning that the differential capacitance of the diffuse layer is also proportional to the square root of the dielectric constant. Therefore, using different electrolyte solutions such as aqueous, non-aqueous, and ion liquid solutions can produce different capacitances of the double-layer. 

Traditional finite difference methods [55, 81] for solving time-dependent second-degree partial differential equations (such as modified diffusion equation) include forward time-centered space (ETCS), Crank-Nicholson, and so on. For time-independent second-degree partial differential equations such as Poisson-Boltzmann equation, finite difference equations can be written after discretizing the space and approximating derivatives by their finite difference approximations. For space-independent dielectric constant, that is, E(r) = e, a tridiagonal matrix inversion needs to be carried out in order to obtain a solution for tp for a given/. 

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