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The electrode electrolyte interface behaves as a capacitor with charged electrode surface and accumulated counter electrode ions acting as two plates of capacitor and the water/solvent acting as a dielectric. In fact, the contact between a charged surface and an electrolyte solution results in ionic rearrangement near the surface and the formation of an electrical double layer. The double layer exists in all heterogeneous systems [19]. The actual electrode-electrolyte interfaces does not behave ideally and different models were proposed in different times to explain the structure of electrical double layer formed at the electrode-electrolyte interface.

1.5.1.1 Helmholtz model

The first model proposed for the double layer was by Helmholtz. It was the simplest model consisting of two planes; one charged electrode surface and the other a layer of compact adsorbed counter ions on the charged surface. The theory defines a linear drop of potential with distance from the surface. This model is equivalent to a parallel plate capacitor having charge density, ,

σ = εε଴. ψୌ/d Equation 1.9

H is the potential in volts, d is the inter plate spacing, ε is the dielectric constant

and 0is the permittivity of free space. The potential drop across the double layer is,

ψ_ୌ= σ. d/εε_଴ Equation 1.10

and C = εε଴/d = σ/ψୌ Equation 1.11

d and are constants for a specific capacitor and therefore, parallel plate model predicts a constant capacitance that does not change with potential which is not realised in the case of a solution double layer. The electrical double layer formed at Hg electrode in the 0.1 mol dm-3potassium chloride (KCl) solution as an electrolyte yields a potential capacitance plot (Figure 1.8) where capacitance is not constant over different potentials.

Figure 1.8: Capciatnce-potential plot of HMDE in 0.1 mol dm-3KCl.

1.5.1.2 Gouy-Chapman model

Louis George Gouy and David Chapman introduced a diffuse double layer model. According to this model, counter ions are attracted towards the charged surface and the co-ions are repelled by the charged surface described mathematically using Poisson-Boltzmann equation. The movement of counter ions towards the surface is attributed to the electrostatic forces only and the movement of co-ions and counter ions away from surface is due to the both electrostatic forces and diffusive forces. This model defines the exponential decay of the potential away from the surface to the bulk of solution [61].

ψ_଴ = ψ_ஔe(ି୏୶) _{Equation 1.12 a}

or ψஔ= ψ଴/e(ି୏୶) Equation 1.12 b

ψ0 is the total potential at the metal surface and ψ is the potential drop across the

diffuse double layer across the distance x from electrode surface. K is the Debye length and for dilute aqueous (ε = 78.49) solutions at 25 º

C. K = (2CN_୅zଶ_eଶ_/εε

଴kT)ଵ/ଶ Equation 1.13 a

K = (3.29 ∗ 10଻_)zCଵ/ଶ _{Equation 1.13 b}

According to Gouy Chapmann model capacitance of double layer is given by equation 1.13 [61]; 50 40 30 20 10 0

C

/µ

F

c

m

C = σ/ψஔ= (zଶeଶεε଴CN୅/kT)ଵ/ଶcosh(zeψஔ/kT) Equation 1.14

The Gouy-Chapman model considers only the electrostatic interactions and considers ions as point charges with no sizes in a diffuse cloud giving rise to an unlimited rise in differential capacitance with increased potential value (Figure 1.9) which is not observed experimentally. In addition, the properties of bulk water are assumed to be the same as those next to the surface giving no attention to the dielectric constant changes.

1.5.1.3 Stern model

Stern suggested the combination of the compact layer (Helmholtz) and the diffuse layer (Gouy and Chapman) models contributed to the experimental double layer model. He introduced an inner Stern layer and an outer diffuse layer resulting in a more realistic calculation of surface potential and surface charge.

ψ_଴ = ψ_ୱ+ ψ_ஔ Equation 1.15

Whereas, the Stern layer potential (

ψ

of double layer;

ψ_ୱ = ψ_ୌ = σ. d/ε_଴ε_୰ Equation 1.16 a And the diffuse layer potential (

ψ

ψஔ= (2kT/e) ∗ sinhିଵ[σ/(8cN୅ε଴ε୰kT)ଵ/ଶ] Equation 1.16 b

This model represents the interface as two capacitors connected in series; 1 C_ୢ = 1 C_ୌ+ 1 C_ஔ

The charges were again assumed as point like and the space within the layer as charge free. The dielectric permittivity was also assumed to be constant throughout the double layer by Stern but actually close to the charged surface, water molecules are depleted due to the accumulation of counter ions with significantly reduced relative permittivity and cannot move freely with their dipole moment oriented

towards the surface compared to the bulk of solution where all other orientations are equally probable [62].

Figure 1.9: Differential capacitance trends predicted from Helmholtz model (a) Gouy Chapmann model (b) and Stern model (c) of electrical double layer plotted against applied potential.

Later on other models were also proposed including Bockris and Grahame’s model. According to a more general picture of the double layer (Figure 1.10), the solution side of the double layer consists of many layers. The inner layer is closest to the electrode surface and contains mostly the solvent molecules and sometimes other ions as well. This layer is also called the compact layer, Helmholtz layer or Stern layer and a plane passing through this layer is called the IHP. Adjacent to the inner layer is a layer of non-specifically adsorbed solvated ions and the plane passing through this layer is termed the OHP. The interaction of the solvated ions with the charged surface is dependent on the nature of the ions as the only interactions possible are the long range electrostatic ones. Adjacent to the OHP is a diffuse layer

(a)

(b)

(c)

E

consisting of scattered solvate and non-solvated ions due to thermal agitation and this layer extends from the OHP into the bulk of solution. The thickness of the diffuse layer depends on the total ionic concentration of the solution [61].

Figure 1.10: Different models of electrical double layer.