Electrochemistry

Lecture 5
The Electrical Double Layer

Hochun Lee
Energy Science & Engineering

October 2024
 Outline

Faradaic / Non-Faradaic Currents

Ideal polarizable / Non-polarizable electrodes

The Electrical Double Layer (EDL)

Experimental Investigation of EDL

- Surface tension / Electrocapillary curve
- Differential double layer capacitance

Models for EDL

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Faradaic and Non-faradaic Currents

Structure at the metal-solution interface

W. Schmickler & E. Santos, Interfacial Electrochemistry, 2nd ed., 1996, Springer 2/33
 Faradaic and Non-faradaic Currents

Electrode reactions involve both Faradaic and non-Faradaic processes.

Faradic Current
. Transfer of e- to/from electrode by redox reactions
. Governed by Faraday’s law
- the amount of current (charge) is proportional
to the amount of species oxidized or reduced

Non-Faradic Current
. Due to processes other than redox reactions at electrodes
(adsorption, desorption)
. Example: double layer charging/discharging current
- when first apply potential to an electrode,
get redistribution of ions near its surface
to counter the charge on the electrode
- movement of ions = current
- As a system approaches equilibrium, the current decreases

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Ideal polarizable and non-polarizable electrodes

Ideal polarizable electrodes Ideal non-polarizable electrodes

Current is independent of potential Potential is independent of current

: Ideal working electrode : Ideal reference electrode

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Ideal polarizable and non-polarizable electrodes

Modern Electrochemistry 2A: Fundamentals of Electrodics, 2nd Ed., p814 5/33

Electrical Double Layer (EDL)

qM
(-)

qs
0

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 Electrical Double Layer (EDL)

Tightly bound inner layer (IHP & OHP)

Loosely bound outer layer: diffuse layer
(caution: not diffusion layer)

Electrical double layer is the structure of

charge accumulation and charge separation
that always occurs at the interface,
when an electrode is immersed into an electrolyte solution

IHPOHP Note that the electrical double layer

consists of two sides:
the metal side & the solution side

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 Structure of Solution Side of EDL

Solution Side of
Terms Electrical Double Layer

Inner Helmholtz plane (IHP):

The center of specifically adsorbed anions (s.a.a),
contains 1-2 solvent layers & s.a.a

Outer Helmholtz plane (OHP):

The center of the nearest
solvated ions (non-specifically adsorbed ions)

Helmholtz layer (Compact layer or Stern layer):

“IHP + OHP”,
The layer btw. electrode surface & OHP (0 – x2)

Diffuse layer (Gouy-Chapman layer ):

The layer beyond OHP ( > x2),
Solvated ions are distributed by thermal agitation

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 ELD = Space Charge Region of the electrolyte

1. Double layers are characteristics of all phase boundaries.

2. When one phase is charged (q1 ≠0),

the same amount charge with opposite sign is induced at the other phase (q1 = − q2).

3. The charging results in the redistribution of charge carries in both phases around the interface.

4. The charge redistribution is confined within the certain thickness, space charge region (scr).

5. The scr in metal is very thin (< a few Angstrom)

because the charge carrier density of metal is so high.

6. The scr of semiconductor is in the order of μm

depending on the doping level (recall p-n junction !).

7. The scr of the electrolyte is in the range of 1-100 nm,

depending on the electrolyte concentration.

8. The electrical double layer is nothing but the scr of the electrolyte.

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 Experimental Investigation of EDL

Two common experimental methods:

Measuring the variation of

(1) Surface tension (γ) or surface charge (qM)
(2) Differential capacitance (Cd)

Lippman equation:
qM : surface charge density on metal (C/m2)
 ∂γ  γ : surface tension (N/m) or
q M = −  surface energy (J/m2)
 ∂E T ,P , µ E : potential (V)

Differential double layer capacitance:

 ∂ 2γ   ∂q M 
−  2  =   = Cd
 ∂E T , P , µ  ∂E T , P , µ

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 Surface Tention Measurement

Dropping mercury electrode (DME):

The principle is based on the relationship between the tension
and the time required for the mercury to form a drop to fall down
tmax : life time of Hg drop (sec)
2πrc
t max = γ m: mass flow rate of mercury (g/sec)
mg rc :
g:
the diameter of the drop
gravitational acceleration

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 Surface Tention Measurement

(1) qM can be derived from the slope:

 ∂γ 
q M = −
 mg  ∂t max  
 ∂E T ,P , µ
q = −
M
  2πrc
 2πrc  ∂E T , P , µ t max =
mg
γ

(2) γ(tmax) is maximized at zero qM,

the point of zero charge (pzc)  why?

tmax Electrocapillary curve

(sec)
qM < 0 𝝏𝝏𝝏𝝏 𝝏𝝏𝝏𝝏 𝝏𝝏𝝏𝝏
qM > 0 < 𝟎𝟎 = 𝟎𝟎 > 𝟎𝟎
3.0 𝝏𝝏𝑬𝑬 𝝏𝝏𝑬𝑬 𝝏𝝏𝑬𝑬
low γ high γ low γ
qM > 0 qM = 0 qM < 0
2.5
Excess qM repels + ↔ + −↔ −
each other Hg (@pzc) Hg
Hg
1.5
+ E

0 -0.5 -1.0 -1.5 -2.0

pzc (qM = 0) E(V vs. Reference)
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 Electrocapillary Curve

At potentials more negative than the pzc (qM < 0),

γ follows the same pattern, regardless of the solution composition.
However, at potentials more positive (qM > 0) than pzc,
the γ curves of different compositions diverge markedly from on another.
 Due to the specifically adsorbed anions

Why ?

Large anions (I−, Br −, CNS −) can be specifically adsorbed at IHP,

because they are readily dehydrated.
In contrast, small anions and cations
maintain their hydration shell
so that they can approach only up to OHP, not IHP.

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 Electrocapillary Curve

The presence of specifically adsorbed anion (s.a.a)

shifts pzc toward (-) direction and lowers γ at the pzc.

Without s.a.a

electrolyte + − : +/- charge on metal

− − − + + +

Hg + + + − − − + − : cation & anion

in electrolytes
− : specifically
pzc
adsorbed anion
+ E

Surface excess of anion/cation > 0

With s.a.a even at pzc

− − − − − − − + − + +
+ + + + + + + − + − −
s.a.a induces
additional (+) charge pzc
on metal
+ E
(−) shift of pzc
Why lower γ at pzc?
s.a.a. repels each other, and thus lowers γ.
s.a.a. acts like qM rather than qS
because it is tightly bound to metal surface.

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 Electrocapillary Curve Advanced Level

The tendency of dehydration & being specifically adsorbed at IHP: I− > Br − > Cl − > F −

σ : excess quantity
S : actual system
R: reference system

ΓK+ > 0 ΓK+ > 0

ΓK+ > 0

ΓF- < 0
At pzc,
ΓK+ > 0
& ΓBr- > 0
ΓK+ < 0
ΓF- > 0
ΓBr- > 0

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 Electrocapillary Curve Advanced Level

The higher the salt concentration , The presence of s.a.a makes

the more specific adsorption will occur, the potential at IHP more (−),
and the pzc becomes more (−). causing overshooting in potential profile

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 Differential Double Layer Capacitance

Measurements of the double layer capacitance provide valuable insights into

adsorption/desorption processes.

The differential capacitance (Cd):  ∂q  2

Cd =   q = surface charge density C/cm
 ∂E 
Cd is usually measured by ac impedance technique (Lecture 7)

In reality, Cd is affected by
the electrode potential
and the electrolyte composition
(type & concentration)

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 Models for EDL: Helmholtz Model

• Helmholtz (100+ years ago) proposed that surface charge

is balanced by oppositely charged ions at OHP

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Models for EDL: Helmholtz Model

A layer of ions at the OHP constitutes

the entire excess charge in the solution:
qs @OHP = − qM

Equivalent to a parallel plate capacitor:

the differential capacitance (Cd) is
constant, independent of E
εε 0
q= E
d
 ∂q  εε 0  Fails to reflect the reality
Cd =  =
 ∂E  d

Linear potential profile

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 Models for EDL: Gouy-Chapman (GC) Model

• Assumed Poisson-Boltzmann distribution of ions from surface

• ions are point charges
• ions do not interact with each other
• Assumed that diffuse layer begins from OHP

OHP Diffuse Layer

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 Models for EDL: Gouy-Chapman (GC) Model

Cd has a parabolic shape, & depends on

Excess charge is distributed salt concentration and E.
spatially in the solution.  2 z 2 e 2εε 0 n 0   zeφ0 
Cd =   cosh 
 kT   2 kT 

 But still not enough

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Models for EDL: Gouy-Chapman (GC) Model

Exponentially decaying potential profile, not linear profile.

φ = φ0 exp(−κ x)
1/κ (Debye length) has units of length, is regarded as
the characteristic thickness of diffuse layer.

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 Debye Length: EDL Thickness

The effective thickness of EDL

∝ 1/κ (Debye length) ∝ (1 / ni)

Debye-Hückel parameter (κ)

1/ 2
 2e 2 m
2
κ =  ∑ ni zi 
 εε 0 kT i =1 
1/ 2
 2nz e 2 2

κ =  

for symmetric salt
(e.g., NaCl, CuSO4)
 εε 0 kT 
zi = electrolyte valence
e = electron charge (C)
ni = concentration of species i (#/m3)
m = number of types of ions
εr = dielectric constant of medium
ε0 = permittivity of a vacuum (F/m)
k = Boltzmann constant (J/K)
T = temperature (K)

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Models for EDL: Gouy-Chapman-Stern (GCS) Model

GCS: diffuse layer + Stern layer of adsorbed charge

OHP Shear Plane

Stern Layer Diffuse Layer

(Compact,
Helmholtz Layer)

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Models for EDL: Gouy-Chapman-Stern (GCS) Model

A layer of ions stuck to the electrode

and the remainder scattered in cloud fashion:
Combination of Helmholtz and GC models

Cd is a in-series connection of Helmholtz

capacitance (CH) and GC capacitance (CD)
1 1 1
= +
Cd CH CD
εε 0 εε
CH = (or 0 )
a x2
 2z 2e 2εε 0n 0   zeφ0 
CD =   cosh
 
 kT   2kT 
Smaller capacitance (CH or CD) dominates !
 Much better but agree with only the case
of non-specifically adsorbed ions

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Models for EDL: Gouy-Chapman-Stern (GCS) Model

Combination of
a linear potential profile over the compact layer (0∼x2)
and exponentially decaying profile over the diffuse layer (> x2)

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Comparison of Three EDL Models

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 Electrical
Charge / Discharge Double Layer Capacitor (EDLC)
Profile

Current output

Cd’ Rs Cd’
Step Voltage Input

Cd Rs Rs: solution resistance

∝ ionic conductivity−1

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Charge / Discharge Profile EDLC

Constant Current Input Experimental Voltage Profile

for Constant Current Charge/Discharge

Voltage Output

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 Charge / Discharge Profile EDLC

Voltage Ramp Input

Experimental Cyclic Voltammogram

(=Scan rate)

Current Output

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More Infos.

(1) Chap 3.9

of 오승모, 전기화학, 자유아카데미, 2010

(2) Chap 13
of A. J. Bard and L. R. Faulkner, Electrochemical Methods:
Fundamentals and Applications, 2nd Ed., John Wiley &
Sons, 2001

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Homework-1

For the RC circuit as shown below

1. Derive voltage response for the constant current

charge/discharge (p.28)

2. Derive current response for the voltage sweep (p.29)

Due: 29th Oct.

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Homework-2

Derive the following:

 2 z 2 e 2εε 0 n 0   zeφ0 
Cd =   cosh 
 kT   2 kT 

refer to Chap 13.3.2

of A. J. Bard and L. R. Faulkner, Electrochemical Methods:
Fundamentals and Applications, 2nd Ed., John Wiley & Sons, 2001

Due: 29th Oct.

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