电化学界面的电化学阻抗-1

An electrochemical double layer (EDL) can be described by Gouy-Chapman-Stern (GCS) model.
The EDL is composed of a compact layer between the metal surface and the Helmholtz plane (HP), and a diffuse layer stretching toward the solution bulk
The HP layer has the thickness around several $\mathring{A}$.
The diffuse layer has a characteristic thickness, termed the Debye length and expressed as
$${\lambda }_D=\sqrt{{ϵ}_{s}RT/2F^2{c}_0}$$
where,
${\lambda }_D$ is the Debye length,
${ϵ}_s$ Is the dielectric constant of the bulk solution,
$c_0$ Is the bulk concentration,
$R$ Is the ideal gas constant,
$T$ Is temperature,
$F$ Is the Faraday constant.
EDL can be modeled as a series connection of a compact capacitor $C_H$ And a diffuse layer part $C_{GC}$.
The capacitance of the compact capacitor is expressed as
$$C_H=\frac{{ϵ}_H}{{\delta }_H}$$
where,
${ϵ}_H$ is the permittivity of the space between the metal surface and the compact layer; ${\delta }_H$is the thickness of the space between the metal surface and the compact layer.
Therefore, the impedance of the EDL is expressed as
$$Z_{DL}=Z_H+Z_{GC}=\frac{1}{j\omega C_H}+\frac{1}{j\omega Z_{GC}}$$
where $\omega$ Is the angular frequency of the perturbation.
Gouy-Chapman capacitance can be simplified as the impedance of a pure capacitor, even though it shows frequency dispersion which most often described empricially using a s constant phase element (CPE).
$$Z_{GC}=\frac{{ϵ}_s}{{\lambda }_D}cosh\left(\frac{U_M-U_{pzc}}{2}\right)$$
where,
$$U_M=\frac{F{\varphi }_M}{RT}$$
is the electrode potential normalized with respect to thermal voltage,
$U_{pzc}$ is the normalized potential of zero charge.

NOTE

$$coshx=\frac{e^x+e^{-x}}{2}$$