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