Ab Initio Atomistic Thermodynamics for Fuel Cell Catalysis - Fuel Cell Catalysis: A Surface Science Approach - page 137

Within the interfaces, the explicit profile of the electrostatic potential f(x)asa
function of electrode distance is unknown ( possible shapes are indicated in Fig. 5.4
as dashed lines) and the only well-defined values are the asymptotic limits, which
are the potential values of the electrode f e and the bulk electrolyte f S , respectively,
and their difference (electrode potential)

Df ¼ f e f S (5 : 13)

A number of classical descriptions have been formulated to describe the shapes of the
potential drop across the electrochemical interface [Schmickler, 1996b; Bockris et al.,
2000], including the Helmholtz model, the Gouy - Chapman model, and the Gouy -
Chapman - Stern model. The earliest model, formulated by Helmholtz in 1879,
treats the interface mathematically as a simple capacitor formed by the electrode
and a single layer of nonspecifically adsorbed (solvated) ions, leading to the
so-called inner layer. Later, Gouy and Chapman (1910 - 1913) made significant
improvements by introducing the diffuse character of the electrolyte to the electric
double layer. In their model, the concentration of electrolyte ions, which are treated
as point charges, reach their bulk concentrations only at larger distances from the elec-
trode, leading to an exponential shape of the potential drop and a so-called diffuse
double layer. A classical and widely applied description is the Gouy - Chapman -
Stern model, which combines the Helmholtz inner layer with the Gouy - Chapman dif-
fuse layer. In this model, the finite size of the ions is taken into account, so that they
cannot approach the surface closer than their radius. The nonspecifically adsorbed ions
of the Gouy - Chapman diffuse double layer are not at the surface, but at some distance
(larger than their radius) away from the surface.
However, these classical models neglect various aspects of the interface, such as
image charges, surface polarization, and interactions between the excess charges
and the water dipoles. Therefore, the widths of the electrode/electrolyte interfaces
are usually underestimated. In addition, the ion distribution within the interfaces
is not fixed, which for short times might lead to much stronger electric fields near
the electrodes.

5.2.4 Extended Ab Initio Atomistic Thermodynamics

Now having specified the bulk electrode, the bulk electrolyte, and the interface
between them, our aim in this section is to quantify the atomistic structure of the
interface and derive an expression that allows us to evaluate its stability. Based
on (5.5), we will extend the ab initio atomistic thermodynamics approach to electro-
chemical systems.
For an electrode/electrolyte interface in equilibrium with the bulk electrode and the
electrolyte reservoir, and constrained by the potential difference Df, the most relevant
structures are those with low interfacial free energies

" #

Þ X
i

Þ¼ 1
A

g T, ð a fg , f fg GT, ð a fg , N fg N i m i (T, a i , f i ) (5 : 14)