W. Richard Bowen and Nidal Hilal 4

2.3 INTERACTION FORCES 49
A choice of boundary conditions is available at the particle surface. It
is important to choose physically meaningful conditions at the particle
surface, which may depend on the colloidal material being considered.
For metal sols in a solution, a constant surface potential boundary condition
is appropriate, given by:
� o r� �
α
constant
(2.32)
where � is the particle radius plus the distance to the OHP (� a � d)
(see Figure 2.4).
A constant surface charge boundary condition may be appropriate
when the surface charge is caused by crystal lattice defects, such as in
clay minerals.
� � ε ε �
o d r o
d
�� �� � constant
dr
r�α
(2.33)
A boundary condition where the zeta potential is held constant is also
possible [37, 48].
� d r=α
� ζ � constant
(2.34)
In the case of biomaterials and oxide surfaces, the charge can be generated
by surface dissociation reactions, which are influenced by the solution
conditions. This can be described by a boundary condition known as
charge regulation [49].
� � f ( � ) ≠ constant
o o
(2.35)
As an example of this, consider the protein bovine serum albumin
(BSA). The protein is made up of a number of different types of amino
acids. Only certain amino acids will take part in the ionisation reactions,
which will generate a charge on the protein surface. The development of
a charge regulation model for BSA requires the number of these chargegenerating
amino acids to be known. This data is available in the literature
from the amino acid sequence of the protein [50] or from titration
data [51]. The relevant equilibria reactions are illustrated by:
1
� �
for aspartic or glutamic acid � COOH �� �� �� ��
� COO � H (2.36)
2
for lysine � NH �� �� �� ��
� NH � H
K
� K
�
3 2
(2.37)