W. Richard Bowen and Nidal Hilal 4

98 3. QUANTIFICATION OF PARTICLE–BUBBLE INTERACTIONs
3.7 EFFECt oF HyDroDynAmICS on
PArtICLE–BuBBLE IntErACtIonS
As all of the particle–bubble interaction measurements carried out
using an AFM-based system are undertaken in a liquid environment, it
is necessary to consider the implications of hydrodynamic forces on the
interactions being measured. During the process of froth flotation, the
particles and bubbles are circulated around the flotation chamber at some
speed, and to relate what happens in this industrial process to what happens
in AFM experiments, some measure of the effects of the speed of
interaction between particles and bubbles necessarily has to be made.
Nguyen and Evans [66] considered the hydrodynamic force acting on a
sphere approaching a bubble in fluid. The hydrodynamic drag force F h
on the particle follows the following relationship, dependent upon the
separation distance d from the bubble surface:
Fh RVf
� �6π� 1
(3.16)
where � is the dynamic viscosity of the surrounding fluid, V the velocity
with which the particle approaches the bubble surface (not to be confused
with the AFM piezo-drive speed), and f 1 a correction factor which accounts
for the deviation of the drag force from Stokes law. It is within this correction
factor that the term for the separation distance appears and was derived by
Nguyen and Evans to be approximated by:
f
1
R
� 1 �
4d
⎛ ⎡
⎢ ⎞
⎜
⎢ ⎝⎜
⎠⎟
⎣⎢
1
0. 719⎤
0. 719
⎥
⎦⎥
(3.17)
Deformation of the bubble as the approaching particle encounters the
thin wetting film around the bubble will lead to a reduction of V at small
separation distances [20] relative to the driving speed.
Nguyen et al. [12] studied the effect of the approach speed upon the
forces obtained when a spherical glass particle approached an air bubble
surface. In Figure 3.7 the effect on the measured force of different piezotranslation
speeds is demonstrated. As the approach speed was increased,
the repulsive particle–bubble forces at short separation distances were
also increased. This was possibly due to limitations on the ability of
the thin water film between particle and bubble to drain in the shorter
time frames imposed by the higher approach velocities. This demonstrates
that hydrodynamic forces were acting as an additional repulsive
force. At low piezo-approach speeds of 0.6 �m s �1 the hydrodynamic forces
were negligible and instead surface forces were dominant. This effect has