41
Fundamentals of Electrochemical Double-Layer Supercapacitors
the outer potential difference (Δψ M/S ), and is simplified as (Δψ) in Figure 2.3.
This outer potential difference is also called the Volta potential (also Volta
potential difference, contact potential difference, or outer potential difference). It represents the potential difference between one point close to the
electrode outer surface and the end point of the diffuse layer in the solution.
The relationship between the outer and inner potentials can be expressed as:
ϕ = ψ + χ
(2.1)
where ϕ is the inner potential and χ is the surface potential determined by
short range effects of adsorbed ions and oriented water molecules. In general, the outer potential can be measured directly, but the surface potential
cannot. Therefore, this inner potential drop is not experimentally measurable. Regarding the potential distribution within the double-layer, Figure 2.3
shows how the double-layer potential drop can be expressed as either Δϕ M/S
or Δψ M/S , that is, Δϕ M/S = Δψ M/S . If we separate the double-layer into several
phases as shown in Figure 2.4, this relationship can be derived. According
to Equation (2.1), the overall double-layer potential drop in Figure 2.4 can be
expressed as:
Δϕ M/S = Δχ O/M + Δχ M/O + Δψ M/H + Δχ O/H + Δχ H/O + Δψ H/S + Δχ O/S + Δχ S/O (2.2)
Due to Δχ O/M = –Δχ M/O , Δχ O/H = –Δχ H/O , and Δχ O/S = –Δχ S/O , Equation 2.2 becomes:
Δϕ M/S = Δψ M/H + Δψ H/S = Δψ M/S .
The potential drop across the diffuse layer is normally expressed as the
outer potential drop rather than the inner potential drop. The outer potential drop (Δψ H/S ) is expressed as ψ 1 . As shown in Figure 2.3, Δψ M/S and ψ 1 are
the potential drops across the entire double-layer and the potential drop
across the diffuse layer, respectively. Therefore the potential drop across the
Helmholtz layer should be (Δψ M/S – ψ 1 ), and the potential across the entire
double-layer can be expressed as:
Δψ = (Δψ M/S – ψ 1 ) + ψ 1
(2.3)
If the unit charge quantity accumulated on the electrode side or in the electrolyte solution side can be expressed as q on a unit area
⎛
⎜
⎝
q
=
Q
A
⎞
⎟
⎠
,
A is the real rather than geometric planar electrode surface area touched with
the electrolyte solution. For instance, with a unit of μC.cm –2 , the reciprocal of
the overall double-layer differential capacitance with a unit of μF.cm –2 can be
expressed as:
Fundamentals of Electrochemical Double-Layer Supercapacitors
the outer potential difference (Δψ M/S ), and is simplified as (Δψ) in Figure 2.3.
This outer potential difference is also called the Volta potential (also Volta
potential difference, contact potential difference, or outer potential difference). It represents the potential difference between one point close to the
electrode outer surface and the end point of the diffuse layer in the solution.
The relationship between the outer and inner potentials can be expressed as:
ϕ = ψ + χ
(2.1)
where ϕ is the inner potential and χ is the surface potential determined by
short range effects of adsorbed ions and oriented water molecules. In general, the outer potential can be measured directly, but the surface potential
cannot. Therefore, this inner potential drop is not experimentally measurable. Regarding the potential distribution within the double-layer, Figure 2.3
shows how the double-layer potential drop can be expressed as either Δϕ M/S
or Δψ M/S , that is, Δϕ M/S = Δψ M/S . If we separate the double-layer into several
phases as shown in Figure 2.4, this relationship can be derived. According
to Equation (2.1), the overall double-layer potential drop in Figure 2.4 can be
expressed as:
Δϕ M/S = Δχ O/M + Δχ M/O + Δψ M/H + Δχ O/H + Δχ H/O + Δψ H/S + Δχ O/S + Δχ S/O (2.2)
Due to Δχ O/M = –Δχ M/O , Δχ O/H = –Δχ H/O , and Δχ O/S = –Δχ S/O , Equation 2.2 becomes:
Δϕ M/S = Δψ M/H + Δψ H/S = Δψ M/S .
The potential drop across the diffuse layer is normally expressed as the
outer potential drop rather than the inner potential drop. The outer potential drop (Δψ H/S ) is expressed as ψ 1 . As shown in Figure 2.3, Δψ M/S and ψ 1 are
the potential drops across the entire double-layer and the potential drop
across the diffuse layer, respectively. Therefore the potential drop across the
Helmholtz layer should be (Δψ M/S – ψ 1 ), and the potential across the entire
double-layer can be expressed as:
Δψ = (Δψ M/S – ψ 1 ) + ψ 1
(2.3)
If the unit charge quantity accumulated on the electrode side or in the electrolyte solution side can be expressed as q on a unit area
⎛
⎜
⎝
q
=
Q
A
⎞
⎟
⎠
,
A is the real rather than geometric planar electrode surface area touched with
the electrolyte solution. For instance, with a unit of μC.cm –2 , the reciprocal of
the overall double-layer differential capacitance with a unit of μF.cm –2 can be
expressed as: