Ab Initio Atomistic Thermodynamics for Fuel Cell Catalysis - Fuel Cell Catalysis: A Surface Science Approach - page 135

from the electrode, as the difference between the work function F and the vacuum level
f vac :

m e ¼ ef vac F (5 : 10)

Charging the capacitor is equivalent to applying an external potential difference
between the two electrodes. In the system thus constrained (here by the applied poten-
tial), the vacuum levels of the electrodes are shifted to f e for the negative and f e for
the positive electrode, such that the difference between them corresponds exactly to
the externally applied potential difference. This finally leads to an accumulation of
charge (excess charge) at or near the electrode surfaces, which gives rise to a constant
electric field between the electrodes as long as they are separated by vacuum or a solid
dielectric. With respect to the excess charges, it should be noted that although the elec-
tric field tends to move all charge towards the surfaces of the electrodes, the Coulomb
interactions between the excess charges and with the nuclei of the metal atoms result in
a more diffuse charge layer. This might include several surface layers rather than pro-
ducing surface-aligned excess charges. Thus, the only reasonable quantity to consider
here is the surface charge density s e .
Since the externally applied potential difference influences the nuclei and electrons
of the electrodes in the same way, not only the vacuum levels but the entire electro-
static potentials of both electrodes, and therefore the Fermi levels, are shifted, too.
Consequently, the electrochemical potentials of the electrons become

m e ¼ ef e F (5 : 11)

In the case that the electrodes are of different materials, one has to distinguish between
the corresponding work functions in this equation.
The reverse process of discharging the capacitor occurs when both electrodes are
electrically connected. Through this connection, the system is allowed to reach its
equilibrium state, and charge flows from the negative to the positive electrode until
both Fermi levels are aligned again.
On the basis of the charged capacitor, we will now discuss the changes induced by
filling the space between both electrodes with a liquid electrolyte. Because of its
experimental relevance, we will consider a single electrode/electrolyte interface
only, where the electrostatic potential of the electrode will simply be designated as
f e (Fig. 5.4).
As described above, the electrolyte usually contains anions and cations, which are
partially or fully solvated, water molecules and various species being involved in
electrocatalytic reactions. The excess charge on the electrode surface is compensated
by an accumulation of corresponding electrolyte counter-ions, leading to overall
charge neutrality.
Since the capacitor without electrolyte shows a linear behavior of the electrostatic
potential between the electrodes, f(x), at the hypothetical moment when the electro-
lyte is added to the system (t ¼ 0 s), the electrochemical potentials of the ions,

m a ¼ m a þ q a f(x) and m c ¼ m c þ q c f(x) (5 : 12)