W. Richard Bowen and Nidal Hilal 4

88 3. QUANTIFICATION OF PARTICLE–BUBBLE INTERACTIONs
has been penetrated. This still leaves the need to either obtain the thickness
of the wetting film from another source or assume that it is too thin to
be significant, which may not be the case, particularly for very hydrophilic
particles.
Perhaps the simplest method for finding the zero contact point is to use
the point at which the probe snaps-in during a jump to contact as suggested
by Butt [9]. In cases where there are only repulsive forces evident prior to
contact, there will be no snap-in and this method cannot be used. However,
in cases where estimates of particle contact angle are to be made from force
curves, this approach can be very useful. An interesting approach has been
described by Gillies et al. [24]. At large separations, interaction forces are
very weak, and thus the bubble or droplet will behave as though it is rigid
at such separations. The force versus distance data can be shifted along
the distance axis to coincide with the linear Poisson–Boltzmann theory for
rigid bodies. However, this technique requires determination of the surface
potentials from some other source, such as by electrophoresis, making its
implementation problematic in laboratories where the appropriate equipment
is not available.
3.4 DEtErmInAtIon oF ContACt AnGLE From
ForCE–DIStAnCE CurvES
As mentioned previously, the attachment of particles to air bubbles in
an aqueous environment is largely mediated by the degree of hydrophobicity
of the particle. A hydrophobic particle will prefer to be in contact
with the bubble, minimising its contact with surrounding water, whereas
a hydrophilic particle will retain a thin wetting film. One conventional
method of measuring the wettability, and hence the degree of hydrophobicity
of a surface is to measure the contact angle of a drop of water on
that surface, which is the angle formed by the TPC line [25–27]. When
the particle comes into contact with an air bubble, the TPC formed will
likewise also contain a contact angle.
Figure 3.3 shows a basic schematic representation for the particle–
bubble interaction. The particle has penetrated the bubble to a distance D.
The angle � represents the immersion angle of the particle, which gives
an indication of the position of the TPC line with regard to the particle.
The contact angle is indicated by � and corresponds to the angle of contact
internal to the droplet during conventional contact angle measurements.
The contact angle is related to the interfacial tensions of the three
interfaces present around the TPC by the Young equation:
� � �
cos� �
�
SV SL
LV
(3.10)