W. Richard Bowen and Nidal Hilal 4

3.3 DETERMINATION OF PARTICLE–BUBBLE sEPARATION 87
the stiffness of the total system will behave as for any two linear springs
in series:
1 1 1
� �
ktot kc kbubble
(3.7)
where k b, k c and k tot are, respectively, the spring constants of the bubble,
the cantilever and the bubble and lever combined. This means that the
probe pressing against the bubble would lead to a gradient in the contact
region based on both the deflection of the lever and the ‘stiffness’ of the
bubble. The contact slope should thus be [21]:
∆x
k
�
∆z
k � k
bubble
c tot
(3.8)
where �x and �z are the changes in cantilever deflection and piezotranslation
distance, respectively. It is also assumed that the particle
approaches in a direction perpendicular to the interface. If this is not the
case, then the interaction becomes somewhat more complex due to the
potential slippage of the particle along the interface.
Attard and Miklavic [21, 22] in a thorough theoretical treatment of
bubble deformation concluded that for small deformations relative to
the bubble radius, air bubbles behave like linear Hookean springs, with
the deformation given by equation (1.1) in Chapter 1. Unlike with solids,
where the material properties determine stiffness, it is the interfacial
tension and the pressure drop across the interface which determine the
stiffness of the bubble. The same behaviour is also displayed by liquid
droplets. Interestingly, when a micro-manipulation rig was used to apply
large deformations (i.e. the deformation was �30% of the bubble diameter)
to air bubbles in aqueous solutions, they were found to have a pseudoelastic
behaviour [23]. However, deformation induced during AFM-based
measurements is orders of magnitude smaller than this. Currently it is
uncertain as to how large a deformation needs to occur to move from a
linear, Hookean, deformation to a pseudo-elastic deformation.
To calculate the separation distance, this linear deformation needs to
be taken into account. From the slope and intercept of the contact part of
the force curve, this can be estimated [12]:
x
d � �z � d
� �
c
�
c
c
(3.9)
where � c is the slope of the contact region on the force curve (� c � �x/�z)
and c the intercept; d c the particle–bubble distance when in ‘contact’,
i.e. the thickness of any wetting film, etc. or the depth to which the interface