W. Richard Bowen and Nidal Hilal 4

84 3. QUANTIFICATION OF PARTICLE–BUBBLE INTERACTIONs
(P a) and the probability that a particle–bubble aggregate will become
detached (P d):
P � P P ( 1 � P )
c a d
(3.2)
When conducting a bubble–particle interaction measurement using an
AFM or similar piece of equipment, the probability of collision is determined
by the user, and as such will not be considered further here. In
terms of kinetics the attachment of a particle to the bubble depends upon
the overcoming of an energy barrier by the kinetic energy imparted during
the collision [6, 8]:
⎛ E ⎞
1
Pa
� exp ⎜
⎜�
⎝⎜
E ⎠⎟
k
(3.3)
where E 1 is the energy barrier for adhesion and E k the kinetic energy of the
collision. The energy barrier is a result of the interaction of the repulsive
and attractive forces acting between the particle and the bubble. The sum of
forces between the particle and bubble can be summarised as follows:
F � Fd � Fe � Fh
(3.4)
where F d and F e are the London dispersion van der Waals force and the
electrical double layer force, respectively, representing traditional DLVO
theory forces; and F h the hydrophobic attractive force between the particle
and the bubble. When conducting AFM measurements between
colloidal particles and a bubble, these forces may be manifested in the
approach part of the force–distance curve. Models which describe longrange
forces may be fitted to force–distance data to investigate the nature
of the interaction under observation.
Finally, the probability of detachment can be described by the following
relationship [8]:
P
d
Wa E
� �
E
� ⎛ ⎞
1
exp ⎜ ’
⎝⎜
⎠⎟
k
(3.5)
Here E k ’ is the kinetic energy of detachment and is not to be confused
with E k the kinetic energy of collision; W a the work of adhesion. This latter
quantity W a is of interest when carrying out AFM experiments as it
can be directly related to adhesion measurements carried out. According
to Johnson–Kendall–Roberts (JKR) theory, this is related to adhesion by
the following association [9]:
F
ad
RW
� 3π
2
a
(3.6)