W. Richard Bowen and Nidal Hilal 4

suspension, and the overall charge density within the cell is zero (electroneutral).
This kind of approach gives a mean field approximation that
accounts for multiparticle interactions to yield the configurational electrostatic
free energy per particle [78]. By equating the configurational free
energy with the pairwise summation of forces in hexagonal arrays, an
expression for the repulsive force between two particles can be obtained,
which implicitly takes into account the multiparticle effect [60].
1 ⎛ ⎛
o ze�β ( D)
⎞ ⎞
FR ( D) � Sβ ( D) n kT ⎜
⎜cosh
⎜ � 1
3 ⎝⎜
⎝⎜
kT ⎠⎟
⎠⎟
(2.52)
where S�(D) is the surface area of the spherical cell around the particle,
no is the ion number concentration, z is the valence of the ions, e is the
elementary electronic charge and ��(D) is the potential at the surface of
the spherical cell.
In order to evaluate the above equation, the size of the cell and the
potential at the cell surface need to be known. The radius of the fluid
shell can be determined with the volume fraction approach [62]. The
potential at the outer boundary of the cell may be determined by solving
the non-linear PBE in spherical coordinates numerically, using the electroneutrality
boundary condition at the cell surface (i.e. d�/ dr r�β
� 0)
and the appropriate boundary condition at the particle surface.
The interaction energy may now be determined using:
V ( D) � � F ( D) dD
R
∫
D
∞
R
(2.53)
This interaction energy implicitly takes multiparticle interactions into
account.
2.3.3 dlVo theory
2.3 INTERACTION FORCES 53
DLVO theory is named after Derjaguin and Landau [66] and Verwey
and Overbeek [24], who were responsible for its development during
the 1940s. This theory describes the forces present between charged surfaces
interacting through a liquid medium. It combines the effects of the
London dispersion van der Waals attraction and the electrostatic repulsion
due to the overlap of the double layer of counterions. The central
concept of the DLVO theory is that the total interaction energy of two
surfaces or particles is given by the summation of the attractive and
repulsive contributions. This can be written as:
VT � VA � VR
(2.54)