1. Double layer capacitance Sähkökemian peruseet KE-31.4100 Tanja Kallio tanja.kallio@aalto.fi C213 CH 5.3 – 5.4

2. Gouy-Chapman Theory (1/4) D.C. Grahame, Chem. Rev. 41 (1947) 441 Charge density  (x) is given by Poisson equation Charge density of the solution is obtained summarizing over all species in the solution Ions are distributed in the solution obeying Boltzmann distribution

3. Gouy-Chapman Theory (2/4) Previous eqs can be combined to yield Poisson-Boltzmann eq The above eq is integrated using an auxiliary variable p

4. 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 x =0 

5. Gouy-Chapman Theory (4/4) Now it is useful to examine a model system containing only a symmetrical electrolyte The above eq is integrated giving potential on the electrode surface, x = 0 where (5.42)

6. Gouy-Chapman Theory – potential profile The previous eq becomes more pictorial after linearization of tanh 1:1 electrolyte 2:1 electrolyte 1:2 electrolyte linearized

7. Gouy-Chapman Theory – surface charge Electrical charge q inside a volume V is given by Gauss law 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) 0 dS. Cosequently, double layer charge density is After inserting eq (5.42) for a symmetric electrolyte in the above eq surface charge density of an electrode is

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

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

10. Inner layer effect on the capacity (1/2) + + + - + + + - + + x = 0  0 OHL x = x 2  (x 2 ) =  2 + If the charge density at the inner layer is zero potential profile in the inner layer is linear: x = x 2  (x 2 ) =  2  (0) =  0 x = 0

11. Inner layer effect on the capacity (2/2) Surface charge density is obtained from the Gauss law 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

12. Surface charge density C dl E C dl,min m(E)m(E) C dl (E) E pzc

13. Effect of specific adsorption on the double layer capacitance (1/2) + + + phase  phase  x = 0 x = x 2 qdqd From electrostatistics, continuation of electric field, for  phase boundary 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

14. Effect of specific adsorption on the double layer capacitance (2/2) H. A. Santos et al., ChemPhysChem8(2007)1540- 1547 So the total capacitance is