Double layer capacitance

Sähkökemian peruseetKE-31.4100

Tanja Kalliotanja.kallio@aalto.fiC213

CH 5.3 – 5.4

Gouy-Chapman Theory (1/4)

D.C. Grahame, Chem. Rev. 41 (1947) 441

Charge density r(x) is given by Poisson equation

2

2

0)(dxdx r

r

Charge density of the solution is obtained summarizing over all species in the solution

ri

ii xFczx )()(

RTFz

cxc ibii exp)(

Ions are distributed in the solution obeying Boltzmann distribution

Gouy-Chapman Theory (2/4)

Previous eqs can be combined to yield Poisson-Boltzmann eq

i

ibii

r RTFz

czFdxd exp

02

2

ddpp

dxd

ddp

dxdp

dxdp

dxd

2

2

The above eq is integrated using an auxiliary variable p

d

RTFz

czFpdpi

ibii

rexp

0

BRTFz

cRTdxd

i

ibi

r

exp

21

0

2

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 = 0ii) electroneutrality: in the bulk charge density must summarize to zero = 0

Thus

x =0

i

ibi

r RTFz

cRTdxd 1exp

21

0

2

Gouy-Chapman Theory (4/4)

Now it is useful to examine a model system containing only a symmetrical electrolyte

RTzFRTc

dxd

RTzFRTc

RTzF

RTzFRTc

dxd

r

b

r

b

r

b

2sinh8

2sinh82expexp2

2/1

0

2

00

2

The above eq is integrated giving

xRTzFRTzF

exp

)4/tanh()4/tanh(

0

potential on the electrode surface, x = 0

2/1

0

222

RTFzc

r

b

where

(5.42)

Gouy-Chapman Theory – potential profileThe previous eq becomes more pictorial after linearization of tanh

xex 0)(

0 1 2 30

20

40

60

80

100

120

140

x

/ m

V

0 1 2 3-140

-120

-100

-80

-60

-40

-20

0

x

/ m

V1:1 electrolyte 2:1 electrolyte 1:2 electrolytelinearized

i

ibi

r RTFz

cRTdxd 1exp

21

0

2

Gouy-Chapman Theory – surface charge

Electrical charge q inside a volume V is given by Gauss law

SE dq r 0

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

00

dxdq r

RTzF

RTcq rbm

2sinh8 02/1

0

After inserting eq (5.42) for a symmetric electrolyte in the above eq surface charge density of an electrode is

Gouy-Chapman Theory – double layer capacitance (1/2)Capacitance of the diffusion layer is obtained by differentiating the surface charge eq

RTzF

RTFzc

C rbm

d 2cosh

2 02/1

022

0

-100 -50 0 50 1000

2

4

6

8

10

C/C0

0 / mV

1:1 electrolyte 2:1 electrolyte 1:2 electrolyte2:2 electrolyte

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

RTzF

RTFzc

C rbm

d 2cosh

2 02/1

022

0

Inner layer effect on the capacity (1/2)

+

+

+

-++

+

-

+

+x = 00

OHLx = x2

(x2) = 2

+ 2

02

innerlayer xx

If the charge density at the inner layer is zero potential profile in the inner layer is linear:

x = x2

(x2) = 2

(0) = 0

x = 0

Inner layer effect on the capacity (2/2)

RTzFRTc

xbr

br

m

2sinh8 22/1

02

020

Surface charge density is obtained from the Gauss law

relative permeability in the inner layer

relative permeability in the bulk solution

dbbrr

m

dl CCRTzFRTcFzx

C11

2/cosh)/2(11

2222

00

2

1

0

2 is solved from the left hand side eqs and inserted into the right hand side eq and Cdl is obtained after differentiating

Surface charge density

Cdl

E

Cdl,minm(E)

Cdl(E)

Epzc

E

Edl

m

pzc

dEC

Effect of specific adsorption on the double layer capacitance (1/2)

'00 qxx rr

+

+

+phase

phase

x = 0x = x

2

'2

sinh8 22/10

2

020 q

RTzFRTc

xbr

br

qdmrd x

qq

2

020´

From electrostatistics, continuation of electric field, for | phase boundary

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

Effect of specific adsorption on the double layer capacitance (2/2)

H. A. Santos et al., ChemPhysChem8(2007)1540-1547

'

00

' ' qdl

dqdl CCqqC

So the total capacitance is