2 Structure of electrified interface

1. The electrical double layer 2. The Gibbs adsorption isotherm 3. Electrocapillary equation 4. Electrosorption phenomena 5. Electrical model of the interface

2.1 The electrical double layer Historical milestones

-

The concept electrical double layer Quincke – 1862

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Concept of two parallel layers of opposite charges Helmholtz 1879 and Stern 1924

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Concept of diffuse layer Gouy 1910; Chapman 1913

-

Modern model Grahame 1947

Presently accepted model of the electrical double layer

2.2 Gibbs adsorption isotherm Definitions

a b s G – total Gibbs function of the system G a, G b, G s Gibbs functions of phases a,b,s Gibbs function of the surface phase s: G s = G – { G a + G b }

Gibbs Model of the interface

The amount of species j in the surface phase: n j s = n j – { n j a + n j b } Surface excess Distance Hypothetical surface Gibbs surface excess G j G j = n j s /A A – surface area

Gibbs adsorption isotherm Change in G brought about by changes in T,p, A and n j dG=-SdT + Vdp + g dA + Sm j dn j g = 

G



A



T

– surface energy – work needed to create a unit area by cleavage ,

p

,

n

m

j

= 

G



n j

 

T

,

p

,

n i



j

- chemical potential dG a =-S a dT + V a dp + + Sm j dn j a dG b =-S b dT + V b dp + + Sm j dn j b and dG s = dG – {dG a + dG b }= S s dT + g dA + + Sm j dn j s

Derivation of the Gibbs adsorption isotherm dG s = -S s dT + g dA + + Sm j dn j s Integrate this expression at costant T and p G s = A g + Sm j n j s Differentiate G s dG s = Ad g + g dA + S n j s d m j + Sm j dn j s The first and the last equations are valid if: Ad g + S n j s d m j = 0 or d g = G j d m j

Gibbs model of the interface - Summary

2.3 The electrocapillary equation Cu’ Ag AgCl KCl, H 2 O,L Hg Cu’’

s M = F( G H g G e )

Lippmann equation

Differential capacity of the interface

C

=

d

s

M dE

= 

d

2 g

dE

2

Capacity of the diffuse layer Thickness of the diffuse layer

2.4 Electrosorption phenomena

2.5 Electrical properties of the interface

In the most simple case – ideally polarizable electrode the electrochemical cell can be represented by a simple RC circuit

Implication – electrochemical cell has a time constant that imposes restriction on investigations of fast electrode process Time needed for the potential across the interface to reach The applied value : E c - potential across the interface E - potential applied from an external generator

Time constant of the cell t = R u C d

E c

=

E R u C d

1  exp   

t R u C d

  Typical values R u =50 W; C=2 m F gives t =100 m s

Current flowing in the absence of a redox reaction – nonfaradaic current In the presence of a redox reaction – faradaic impedance is connected in parallel to the double layer capacitance. The scheme of the cell is: The overall current flowing through the cell is : i = i f + i nf Only the faradaic current –i f contains analytical or kinetic information