W. Richard Bowen and Nidal Hilal 4

Bubble
3.2 PARTICLE–BUBBLE INTERACTIONs 83
Hydration
layers
between the particle and the bubble will add a hydrodynamic component
to the attachment kinetics. In flotation, if the time required for this film to
rupture (the induction time) is less than the actual contact time, then the
particle will be unable to attach to the bubble [4]. Further approach will
cause the hydration layers to become thinner and less stable. As this layer
becomes destabilised, the particle and bubble will be allowed to attach
directly, leading to the formation of a three phase contact (TPC) line. The
more hydrophobic the particle surface, the thinner and less stable any
hydration layer will be. Consequently, the more hydrophobic in character
a particle is, the more readily it will attach to air bubbles in solution. This
is the mechanism by which the flotation of fine particles is achieved [1, 2].
Derjaguin and Dukhin [5] explain that the main thermodynamic parameter
involved in the particle–bubble interaction is the disjoining pressure,
which is defined by them as the derivative of the free energy with respect
to the thickness h of this wetting layer per unit area. This disjoining pressure
is formed by a combination of pressures:
P � N � A � S
( h) ( h) ( h) ( h)
Particle
Bulk fluid
layer
FIGurE 3.1 Simple illustration of a basic particle–bubble interaction. On approach
there are three fluid layers present between the particle and the bubble. These are hydration
layers on both the particle and the bubble (represented by the dashed line) and a layer
of bulk fluid in the middle. The confinement and drainage of this bulk fluid layer leads to a
hydrodynamic component to the interactions.
(3.1)
where P is the disjoining pressure, N the ionic double layer repulsion
per unit area, A the contribution due to van der Waals interactions per
unit area and S (h) the contribution of particle surface hydrophobicity to
the disjoining pressure. S (h) will tend towards zero for very hydrophobic
surfaces.
As described by Sutherland [6, 7], the probability of a particle being
collected during flotation (P) is a composite of the probability of particle–
bubble collision (P c), the probability of such a collision leading to adhesion