Collision and Attachment Interactions of Single Air Bubbles with Flat Surfaces in Aqueous Solutions - Drops and Bubbles in Contact with Solid Surfaces - page 222

Figure 7. Numerical results of Eq. (6) for the bubble surface profiles arrested underneath of a solid
surface in water by the electrical double layer and van der Waals repulsions given by Eqs (11) and
(7). The legend shows the shortest separation distance at the upper bubble apex, h( 0 ) .Seethetextfor
additional information about the model parameters.

D. Interfacial Interaction Forces

When a bubble approaches a solid surface, interfacial forces become significant at
small separation distances. These forces arise from molecular interactions between
charged and uncharged atoms and molecules of the interacting bodies and the sur-
rounding medium, which include (1) van der Waals (electrodynamic) interactions,
(2) electrostatic double-layer interactions, and (3) non-DLVO interactions. The first
two interactions form the key components of the celebrated DLVO (Derjagiun-
Landau-Verwey-Overbeek) theory of colloid stability. Summarised below are the
disjoining pressures of the interfacial interactions relevant to the bubble-surface
collision and attachment.

1. DLVO Disjoining Pressures

For bubble-solid surface interaction, the van der Waals interaction is repulsive. It
can be determined using the macroscopic (Hamaker) and the microscopic (Lifshitz)
theories. The combined Hamaker-Lifshitz approach is useful. For two planar par-
allel surface elements of a water film confined between the gas phase and the solid
phase, the combined theory gives:

A 1 d A

d H ,

vdw (H) =−

H 3 +

(7)

π π

H 2 6 12

where H is the thickness and A is the Hamaker-Lifshitz function, which is a weak
function of the thickness H . The function covers both the short- and long-range van
der Waals interactions and can be described as follows:
A = A 0 ( 1 2 κH) e − 2 κH + A ξ (H), + (8)

where κ is the Debye constant. The zero-frequency, A 0 , and non-zero frequency,