Nanoelectrochemistry: Fundamentals and Applications in Biology and Medicine - Applications of Electrochemistry and Nanotechnology in Biology and Medicine - page 24

layer that extends into the bulk liquid away from the surface. This
is illustrated in Fig. 4 , which sketches the electrostatic potential
( Fig. 4a ) and the ion concentrations ( Fig. 4b ) as a function of dis-
tance from a charged surface. The Poisson-Boltzmann formalism
provides a description of the diffuse part of the double layer that is
relatively simple and consistent with most observations without
the need for additional complications. Specifically, it predicts that
the electrostatic potential, M ( r ), falls off exponentially with dis-
tance from the surface. 119 The decay length, known as the Debye
length O d , therefore represents the characteristic thickness of the
double layer. Its value is given by

1/ 2

§

f 22

·

nZe
kT

¦

O ¨

i i

¸

(12)

d

HH

© ¹

i

0 B

where e is the charge of the electron, H is the relative permittivity
of the solvent, H 0 is the permittivity of free space, k B is
Boltzmann's constant, i represents the various ionic species pre-
sent in the electrolyte, and n i 0 and Z i are the number density and
valence of each species, respectively. To give some numerical ex-
amples, pure water at pH 7 has O d | 1 Pm, while a 0.1 M aqueous
solution of a monovalent salt has O d | 1 nm. The ionic charge in
the diffuse layer consists of two components, as illustrated in Fig.
4b : excess counterions that are attracted to the surface, and a re-
duction of the coion concentration below its bulk value due to re-
pulsion from the surface.
In contrast to the relative simplicity of the diffuse layer, the
structure of the compact layer is extremely difficult to describe
quantitatively. First, the nonlinearity of the Poisson-Boltzmann
equation manifests itself strongly in this region except at weakly
charged surfaces. The potential M ( r ) then exhibits a more compli-
cated functional form than a simple exponential decay, while the
ionic charge in this part of the double layer consists predominantly
of counterions. Second, and more importantly, the inherent molec-
ular scale of the compact layer means that a complete description
calls for a treatment that includes details at the molecular level.
Issues such as the finite size of the ions, crowding, the hydration
state of the ions, the structure of the solvent near the surface, spa-