W. Richard Bowen and Nidal Hilal 4

86 3. QUANTIFICATION OF PARTICLE–BUBBLE INTERACTIONs
3.3 DEtErmInAtIon oF PArtICLE–BuBBLE
SEPArAtIon
One of the main uncertainties involved when making single particle
bubble measurements is the determination of the separation distance
between the probe and the air–liquid interface. With a contact between
hard surfaces, there is normally a clear transition as contact is made, but
for deformable surfaces this is not necessarily obvious. First, deformation
of the bubble by both long-range interaction forces prior to contact and
compression by the probe after contact needs to be taken into account
(Figure 3.2). When a hydrophilic particle comes into contact with a bubble,
a thin wetting film may separate the particle from the bubble. As the
particle encounters this wetting film at small separations, the air–liquid
interface will become deformed due to hydrodynamic factors, producing
an apparent repulsion [20]. A hydrophobic particle may sit partially
inside the bubble to a distance dependent upon their contact angle, or
may be completely engulfed within the bubble. Together, these factors
make an exact determination of the separation distance and attachment
point more problematic than when considering interactions between
hard solid surfaces.
In the early paper by Ducker et al. [11], it was assumed that the bubble
was deformed in a linear manner, resembling a Hookean spring. As such
d 0 d n
FIGurE 3.2 On approach of a particle into the vicinity of the bubble, long-range interactions
will result in deformation of the bubble prior to contact. The nominal distance (d n) is the
distance that would exist between the particle and the bubble if deformation did not occur.
The actual distance (d 0) differs from d n by the size of the deformation. The size of the particle
here is important as the magnitude of interaction forces will scale with its radius (R p), if the
particle is a sphere. Note that in the illustration here the bubble is deforming as from a net
repulsive interaction. With a net attraction deformation will be towards the particle.
R p