Collision and Attachment Interactions of Single Air Bubbles with Flat Surfaces in Aqueous Solutions - Drops and Bubbles in Contact with Solid Surfaces - page 235

Blake [3] studied the formulation of the combined theory based on the molecular-
kinetic approach by simple adding the viscous flows effect to the barriers created
by the solid-liquid attractions.
Another type of dissipation energy in precursor film was considered by de
Gennes [14]. The capillary force is balanced by the total energy dissipation which
comprises of the viscous dissipation, the dissipation at the advancing contact line
and in a precursor film.
Using a slightly different method, de Ruijter et al. [15], derived closed-form
equations for the evaluation of the droplet's base radius and specify regimes at
which different dissipation energies dominate. They derived the following equation
for the TPC velocity by combining these two models into one:

cos θ 0 − cos θ

Ca = , (28)

ζ 0 + 6 φ ln [ R(t)/a ]

where ζ 0 = k B T/Kλμ is the (dimensionless) friction coefficient of the liquid

molecules in the vicinity of the contact line, parameter a is the radius of the core
region where the dissipation is negligible and φ

sin 3 θ

cos 3 θ is the geometrical
factor. If the frictional forces are dominant, Eq. (28) reduces to molecular-kinetic
model and if the viscous forces control the spreading, it reduces to hydrodynamic
model. It was also shown that both energies of dissipation can have a dominant
effect on the wetting process at different time scale, namely, non-hydrodynamic dis-
sipation prevails at relatively short times and hydrodynamic dissipation dominates
at long time. The authors also found explicitly the crossover time, t 2 , separating
these two regimes. For spreading times longer than t 2 , most of the work is dissi-
pated due to viscous flow, and therefore a hydrodynamic regime is predicted and
for spreading time below t 2 the molecular-kinetic model is the governing channel
of dissipation. The value of t 2 depends naturally on both the friction coefficient and
the viscosity.
Recently, Roques-Carmes et al. [62] studied these different regimes experimen-
tally. They confirmed the existence of several spreading regimes which, depending
on the fluid particle volume and viscosity, the regimes sequence can be different
(Fig. 14). They also showed that the nature of the regimes is strongly influenced by
surfactant concentration. They estimated a critical contact angle for the crossover
between molecular-kinetic regime and hydrodynamic regime. A critical contact an-
gle of 35 ◦ was found. The critical value was also independent of the presence of
surfactant and the drop volume.

=

2 − 3cos θ +

G. Summary and Conclusions

In this chapter the interaction of air bubbles with solid surfaces has been reviewed.
It focuses on the rising bubbles interacting with a planar solid surface placed hor-
izontally relative to gravity in aqueous solutions. Bubbles motion in the solutions
is influenced by the bubble size and shape, and surfactants and the other impurities