Page 188 - Introduction to Colloid and Surface Chemistry

Charged interfaces 177

The diffuse double layer

The electric double layer can be regarded as consisting of two
regions: 'an inner region which may include adsorbed ions, and a
diffuse region in which ions are distributed according to the influence
of electrical forces and random thermal motion. The diffuse part of
the double layer will be considered first.
Quantitative treatment of the electric double layer represents an
extremely difficult and in some respects unresolved problem. The
requirement of overall electroneutrality dictates that, for any dividing
surface, if the charge per unit area is -For on one side of the surface, it
must be —a- on the other side. It follows, therefore, that the
magnitude of a will depend on the location of the surface. Surface
location is not a straightforward matter owing to the geometric and
chemical heterogeneity which generally exists. It follows, further-
more, that electric double-layer parameters (potentials, surface
charge densities, distances) are not amenable to unequivocal
definition. Despite this, however, various simplifications and approx-
imations can be made which allow double-layer theory to be
developed to a high level of sophistication and usefulness.
The simplest quantitative treatment of the diffuse part of the
double layer is that due to Gouy (1910) and Chapman (1913), which
is based on the following model:

1. The surface is assumed to be flat, of infinite extent and uniformly
charged.
2. The ions in the diffuse part of the double layer are assumed to be
point charges distributed according to the Boltzmann distribution.
3. The solvent is assumed to influence the double layer only through
its dielectric constant, which is assumed to have the same value
throughout the diffuse part.
4. A single symmetrical electrolyte of charge number z will be
assumed. This assumption facilitates the derivation while losing
little owing to the relative unimportance of co-ion charge number.
Let the electric potential be »// 0 at a flat surface and ^ at a distance x
from the surface in the electrolyte solution. Taking the surface to be
positively charged (Figure 7.1) and applying the Boltzmann distribu-
tion,