W. Richard Bowen and Nidal Hilal 4

90 3. QUANTIFICATION OF PARTICLE–BUBBLE INTERACTIONs
angle when the particle is being extricated from the bubble. The jump-
in force is dominated by capillary forces, at least in the case of a particle with
a hydrophobic enough surface to be able to form a TPC line. As such, the
capillary force may be related directly to � r, as illustrated by equation (3.15)
[36–38]:
F � 2πR� sin� sin( � � �)
CAP r
(3.13)
where � represents the interfacial tension. When the net forces are zero
(i.e. F CAP � 0), � r will be equal to �; when this occurs, equation (1.14) may
be used to calculate � r [36, 39]:
cos
� r
( R Dr
)
�
R
�
(3.14)
Here the penetration depth D r can be extracted from force curves by
taking the distance from the initial jump-to-contact to the point at which
the force is zero, i.e. there is no net force acting on the cantilever. This is
illustrated in Figure 3.4. When the particle is withdrawn from the bubble,
the TPC line is advancing across the particle, hence contact angles determined
are called the advancing contact angle (� a). If adhesion occurs,
D a
Dr
FIGurE 3.4 A force curve, showing the relevant measurements to obtain D r and D a. D r
is found by measuring the distance from jump-in to the point at which a net force of zero,
equivalent in size to the free level force, is reached. From this distance on the approach
curve, the receding contact angle � r is obtained. From a similar measurement on the retract
curve (D a) the advancing contact angle � a may be obtained.