W. Richard Bowen and Nidal Hilal 4

net molecular charge of the BSA molecule will also depend on whether
some surface groups on the molecule will also be involved with ionic
equilibria with other ions in the electrolyte solution. For BSA, chloride
binding will occur [53]. This chloride binding may be described by [54]:
Z
Cl
� �
( )
( )
� �zeψ
440 � [ Cl ] exp o
kT
� �zeψ
1� 44 � [ Cl ] exp o
kT
2.3 INTERACTION FORCES 51
�
�zeψo
( kT)
�zeψo
( kT
33 γ[
Cl ] exp
�
�
1�1. 1 γ [ Cl ] exp )
(2.44)
where � is the activity coefficient of the chloride ion at the particle surface
(determined from activity coefficient data for NaCl solutions).
Therefore, the overall surface charge number of a BSA molecule is:
Z T �Z AB �Z �
Cl
(2.45)
When solving the PBE using charge regulation as a boundary condition,
the compact part of the double layer around the BSA molecule needs
to be taken into account. Figure 2.4(b) illustrates the model assumed
for the compact part of the double layer. This model can be termed the
Zeroth-order Stern model [55], as we have a zone of thickness, d, that is
devoid of ions and represents a distance of closest approach to the particle
surface of charge density � o. From electroneutrality,
�o �� �d
From the solution of the PBE, � d can be determined as:
� ε ε � d
d � o r
dr
r�a� d
(2.46)
(2.47)
The surface potential of the particle, � o, may also be determined by
allowing for the capacitance of the fluid in the compact layer around the
sphere. For two concentric spheres, the capacitance, C, may be determined
using [56]:
aα
C � 4πεoεr α � a
(2.48)
where a is the radius of the inner sphere and � (� a � d) is the radius of
the outer sphere.
The capacitance can also be evaluated from the surface charge density as:
C Q
2
4πa
�o
� �
∆V
� � �
o d
(2.49)