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