45
Fundamentals of Electrochemical Double-Layer Supercapacitors
potential in two different electrolyte solutions [6]. It can be seen that the
differential capacitance is strongly dependent on the electrode potential
as well as the type of electrolyte used. Therefore, the theoretical treatment
of the diffuse layer seems complicated. In the next subsection, we will
discuss how to theoretically calculate the differential capacitance of the
double-layer.
2.2.2 Double-Layer Net Charge Density by Gouy–
Chapman–Stern (GCS) Modeling
To provide a theoretical description of the diffuse layer, the potential distribution shown in Figure 2.3 is used. According to the Gouy–Chapman–Stern
(GCS) model [7], the ion concentration (C i(x) ) within the diffuse layer at point
x can be expressed as:
⎛
i x
( ) = C
0
z Fψ ⎞
C
exp ⎜ −
i
x ⎟
(2.7)
⎝ RT ⎠
This equation is called the Boltzmann distribution, where C 0 is the bulk concentration of ion i, F is Faraday’s constant, ψ x is the potential at x point, R is
the universal gas constant, and T is the temperature. Thus the net charge in
a unit volume, ρ (μC.cm –3 ), can be expressed as:
⎛ Fψ ⎞
ρ = ∑ z F
o
i C i( )
x = ∑ z F
i C exp ⎜ −
x ⎟
(2.8)
⎝ RT ⎠
The following Poisson formula can be used to connect volume charge with
the potential:
d
2
ψ
ρ
x
2
= −
(2.9)
dx
ε ε
r o
where ε r is the relative dielectric constant of the electrolyte solution and ε o is
the dielectric constant of the vacuum. Then Equation (2.8) can be written as:
d
2
ψ
1
o
⎛ z F ⎞
x
= −
∑ z FC exp ⎜ −
i ψ x
2
⎟
(2.10)
dx
ε ε
i
r o
⎝ RT ⎠
According to