Sähkökemian peruseet
KE-31.4100
Tanja Kallio tanja.kallio@aalto.fi
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  dx
2
Charge density of the solution is obtained summarizing over all species in the solution r
( x )
  i z i
Fc i
( x )
Ions are distributed in the solution obeying
Boltzmann distribution c i
( x )
 c i b exp


 z i
F

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  dx
2
 
F
 r

0
 i z i c i b exp


 z i
F

RT


The above eq is integrated using an auxiliary variable p d
 dx
 p
 d
2  dx
2
 dp dx
 dp d
 d
 dx
 p dp d

1
2
 pdp
 
F
 r

0
 i z i c i b exp


 z i
F

RT


 d
 dx
2

RT
 r

0
 i c i b exp


 z i
F

RT



B

Gouy-Chapman Theory (3/4)
Integration constant B is determined using boundary conditions: i) Symmetry requirement: electrostatic field must vanish at the midplane
 d

/ d x = 0 ii) electroneutrality: in the bulk charge density must summarize to zero


= 0
Thus
1
2 dx
2

RT
 r

0
 i c i b

 exp


 z i
F

RT



1


 x =0

Gouy-Chapman Theory (4/4)
Now it is useful to examine a model system containing only a symmetrical electrolyte d

 dx
2

2 RTc b
 r

0

 exp zF
RT d
 dx
 


8 RTc b
 r

0


1 / 2 sinh

 exp zF

2 RT zF

RT

2



8 RTc b
 r

0 sinh
2 zF

2 RT
(5.42)
The above eq is integrated giving tanh( zF

/ 4 RT ) tanh( zF

0
/ 4 RT )
 exp

  x
 where potential on the electrode surface, x = 0
  



2 c b z
2
F
 r

0
RT
2



 1 / 2

Gouy-Chapman Theory – potential profile
The previous eq becomes more pictorial after linearization of tanh

( x )
 
0 e
  x
140
120
100
80
60
40
20
0
0 1
 x
2 3
0
-20
-40
-60
-80
-100
-120
-140
0 1
 x
2 3
1
2 d dx
2

RT
 r

0
 i c i b

 exp


 z i
F

RT



1


1:1 electrolyte
2:1 electrolyte
1:2 electrolyte linearized

Gouy-Chapman Theory – surface charge
Electrical charge q inside a volume V is given by Gauss law q
  r

0
 
S
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 ( d f/ dx )
0 dS. Cosequently, double layer charge density is 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

8

RTc b  r

0
1

/ 2 sinh
 zF
2

RT
0

Gouy-Chapman Theory – double layer capacitance (1/2)
Capacitance of the diffusion layer is obtained by differentiating the surface charge eq
C d

  m
 
0

2 c b z
2
F
2  r

0
RT
1 / 2 cosh


 zF

2 RT
0


10
1:1 electrolyte
2:1 electrolyte
1:2 electrolyte
2:2 electrolyte
8
6
4
2
0
-100 -50

0
0
/ mV
50 100

Gouy-Chapman Theory – double layer capacitance (1/2)
C d

  m
 
0

2 c b z
2
F
2  r

0
RT
1 / 2 cosh

 zF
2

RT
0



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:
 
 x innerlayer


2
 x
2

0

(0) =

0

( x
2
) =

2 x = 0

0
OHL x = x
2

( x
2
) =

2

Inner layer effect on the capacity (2/2)
Surface charge density is obtained from the Gauss law
 m    r

0

2
 x
2

0


8 RTc b
 r b

0
1

/ 2 sinh
 zF

2
2 RT relative permeability in the inner layer relative permeability in the bulk solution

2 is solved from the left hand side eqs and inserted into the right hand side eq and C dl is obtained after differentiating
1
C dl



 
 
0 m



1

 x r

2
0

( 2
 r b

0 z
2
F
2 c b
/
1
RT ) cosh
 zF

2
/ 2 RT


1
C
2

1
C d

Surface charge density
C dl
 m 
E

C dl dE
E pzc
C dl
( E )
C dl,min
E pzc
 m ( E )
E

Effect of specific adsorption on the double layer capacitance (1/2)
From electrostatistics, continuation of electric field, for
| phase boundary
 r
 
0


  
 x


  r
 
0


  
 x


 q '
+
Specific adsorbed species are described as point charges located at point x
2
. Thus the inner layer is not charged and its potential profile is linear
 r

0

2
 x
2

0


8 RTc b
 r b

0
1

/ 2 sinh zF

2
2 RT
 q '
 q d
 q ´
  r

0

2
 
0 x
2
   m q d
+
+

Effect of specific adsorption on the double layer capacitance (2/2)
So the total capacitance is
C q ' dl
 


 q
  d
0





 q
 
0
'



C dl

C q '
H. A. Santos et al.,
ChemPhysChem8(2007)1540-
1547