Gouy-Chapman Theory – double layer capacitance (1/2)

Double layer capacitance
Sähkökemian peruseet
KE-31.4100
Tanja Kallio
[email protected]
C213
CH 5.3 – 5.4
Gouy-Chapman Theory (1/4)
Charge density r(x) is given by Poisson equation
r( x)   r  0
d 2
dx2
Charge density of the solution is obtained
summarizing over all species in the solution
r( x)   zi Fci ( x)
i
Ions are distributed in the solution obeying
Boltzmann distribution
 z F 
ci ( x)  cib exp  i 
 RT 
D.C. Grahame, Chem. Rev. 41
(1947) 441
Gouy-Chapman Theory (2/4)
Previous eqs can be combined to yield Poisson-Boltzmann eq
d 2
dx2

F
r 0

i
 z F 
z i cib exp  i 
 RT 
The above eq is integrated using an auxiliary variable p
d
d 2  dp dp d
dp
p 



p
dx
d
dx2 dx d dx

pdp  
2
F
r 0

1  d 
RT
  
2  dx 
r 0
i

i
 z F 
z i cib exp  i d
 RT 
 z F 
cib exp  i   B
 RT 
Gouy-Chapman Theory (3/4)
Integration constant B is determined using boundary conditions:
i) Symmetry requirement: electrostatic field must vanish at the midplane 
d/dx = 0
ii) electroneutrality: in the bulk charge density must summarize to zero 
=0
Thus
2
1  d 
RT
  
2  dx 
r 0

i
  z F  
cib exp  i   1
  RT  

x =0
Gouy-Chapman Theory (4/4)
Now it is useful to examine a model system containing only a symmetrical
electrolyte
b
2 RTc b 
 d 
 zF 
 zF   8 RTc
 zF 
 exp 
sinh 2 
  
  exp
  2  

 r  0   RT 
 dx 
 RT    r  0
 2 RT 
2
1/ 2

 8 RTc b 
d

 
dx


 r 0 
 zF 
sinh 

 2 RT 
(5.42)
The above eq is integrated giving
tanh(zF / 4 RT )
 exp x 
tanh(zF 0 / 4 RT )
1/ 2
where
potential on the electrode
surface, x = 0
 2c b z 2 F 2 


   RT 
 r 0

Gouy-Chapman Theory – potential profile
The previous eq becomes more pictorial after linearization of tanh
( x)   0 e  x
140
0
2
-20
100
-40
80
-60
 / mV
 / mV
120
60
-100
20
-120
1
x
2
3
-140
0
c
i
1:1 electrolyte
2:1 electrolyte
1:2 electrolyte
linearized
-80
40
0
0
1  d 
RT
  
2  dx 
r 0
1
x
2
3
 zi F  
b
  1
i exp 


RT 

Gouy-Chapman Theory – surface charge
Electrical charge q inside a volume V is given by Gauss law
q   r 0  E  dS
In a one dimensional case electric field strength E penetrating the surface S
is zero and thus E.dS is zero except at the surface of the electrode (x = 0)
where it is (df/dx)0dS. Cosequently, double layer charge density is
 d 
q   r 0  
 dx  0
After inserting eq (5.42) for a symmetric electrolyte in the above eq surface
charge density of an electrode is

 m  q  8RTcb  r  0
1/ 2 sinh zF2RT0 
Gouy-Chapman Theory – double layer
capacitance (1/2)
Capacitance of the diffusion layer is obtained by differentiating the surface
charge eq
1/ 2
 m  2c b z 2 F 2  r  0 
Cd 


 0 
RT

 zF 0 
cosh

 2RT 
10
8
C/C0
6
1:1 electrolyte
2:1 electrolyte
1:2 electrolyte
2:2 electrolyte
4
2
0
-100
-50
0
0 / mV
50
100
Gouy-Chapman Theory – double layer
capacitance (1/2)
1/ 2
 m  2c b z 2 F 2  r  0 
Cd 


 0 
RT

 zF 0 
cosh

 2RT 
Inner layer effect on the capacity (1/2)
If the charge density at the inner layer is zero
potential profile in the inner layer is linear:
 
  
 2 0
 
x2
 x innerlayer
+
+
+
+
+
+
+
x=0
0
(0) = 0
OHL
x = x2
(x2) = 2
(x2) = 2
Inner layer effect on the capacity (2/2)
Surface charge density is obtained from the Gauss law
 m   r  0
 2  0
x2

 8 RTc b rb 0
relative permeability in
the inner layer

1/ 2
 zF2 
sinh 

 2 RT 
relative permeability in the bulk solution
2 is solved from the left hand side eqs and inserted into the right hand side
eq and Cdl is obtained after differentiating
1
1   m 
x
1
1
1
  2 
 


Cdl  0 
 r  0 (2 rb 0 z 2 F 2 cb / RT ) cosh zF2 / 2 RT  C2 Cd
Surface charge density
Cdl
E
m 
 C dl dE
E pzc
Cdl(E)
Cdl,min
m(E)
Epzc
E
Effect of specific adsorption on the
double layer capacitance (1/2)
From electrostatistics, continuation of electric field, for | phase boundary

  q '

 r 0
 2  0
x2

 8 RTc b rb 0
 qd  q´  r  0
 2  0
x2

1/ 2
 zF2 
sinh 
  q'
 2 RT 
 
m
qd
+
Specific adsorbed species are described as point charges
located at point x2. Thus the inner layer is not charged and
its potential profile is linear
+
    
   r  0 

 x
+
  
 r  0 
 x

Effect of specific adsorption on the
double layer capacitance (2/2)
So the total capacitance is
 q
q'
Cdl
  d
  0
  q' 
  
  Cdl  C q'
   0 
H. A. Santos et al.,
ChemPhysChem8(2007)15401547