Reduction of Contaminants in Soil and Water By Direct Electric Current - Electrokinetics for Petroleum and Environmental Engineers - page 37

charge density. The mathematical implication is then that the electrical
potential satisfies the Laplace equation [

=0] hence uniform concen-
tration distribution and uniform conductivity within the electrochemical
cell. Yet in real electrochemical systems with concentration gradients, the
current density of the system can be affected to cause current flow in the
opposite direction of the electric field (Newman, 1991). Using the PNP
equations and the electroneutrality condition, it can be shown mathemat-
ically that the concentration gradients give rise to a spatial variation of
conductivity, where a diffusion potential arises to ensure ion movement at
the same speed to overcome the charge separation and violation of electri-
cal neutrality (Newman, 1991). The charge density accumulation can't be
neglected for inter-phase layers, such as the electrical double layer. Using
a non-dimensionalized form of equation 2.2, it was shown that the electro-
neutrality condition is a direct result of the Debye screening length, where
the ratio of the Debye length, l D to the field length, L is described as follows
[Eq 2.3] (Chu, 2005):

∇

2

Φ

−∇ ej

2

=∑ zc

(2.3) ii

i

⎛
⎜

⎞
⎟

C
C

RT
FC

e

l Φ D i s

where, e

= , j = , c = , l =

i D 2

L RT / F

eq eq

where, R = universal gas constant, 8.3144 J/K mol ; T = absolute tempera-
ture, [K]; F = Faraday constant, 96,485 C/mol electrons; C eq = equilibrium
concentration in bulk.
As shown in equation 2.3, unless the Laplace compliance holds true -
which is not the case for most electrochemical systems owing to the pres-
ence of concentration gradients - the electroneutrality of the system can
be satisfied when the Debye length l D is so small compared to L (s.t., e <<)
that the left hand side of the equation becomes zero. Asymptotic analysis
of PNP equations numerically solved at several values of e showed that for
small values of e - as would be in macroscopic systems - the charge den-
sity r is zero for majority of an electrochemical cell of length L , except at
the boundaries where Faradaic reactions and Stern-layer capacitance are
considered (Chu, 2005). As the Debye length approaches to the width of
the electric double layer, non-zero charge density distribution appears in
the entire electrochemical cell (Chu, 2005). Hence, the electroneutrality
in an electrochemical system will hold when the charge density is small
compared to total ion concentration, C eq of the bulk fluid. As the charge
density approaches to total concentration within the electrochemical cell