Gouy-Chapman Theory – double layer capacitance (1/2)
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Transcript 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 zF2RT0
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
zF2
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 zF2 / 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
zF2
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