Biomedical Engineering Reference
In-Depth Information
no real physical basis—it only describes a difference between DLVO and experi-
mental data for surface forces. Indeed, the double exponential reflects the presence
of surface nanobubbles [2, 8, 21, 52, 68, 79]. The nanobubbles have been found
significant for coagulation of solid particles [26, 67], and drainage and rupture of
liquid films [54]. However, removing nanobubbles or degassing aqueous solution
of wetting films and foam films is difficult to experimentally carry out because the
gas bubbles are always present in the systems. Therefore, the empirical correlations
presented for the non-DLVO interfacial interaction in the above are useful.
E. Drainage and Rupture of Intervening Liquid Films
The wetting film formed between the rising bubble and the solid surface initially
thins under the influence of gravity and capillary suction at the film periphery. When
the film thickness reduces to 300-200 nm, the interfacial intermolecular forces start
affecting the film drainage. Then the films normally become unstable and rupture
at a critical thickness within the range of 200-10 nm, depending on concentration
of surfactant and surface impurities. Theories on the film drainage and rupture are
summarised below.
1. Theories on Drainage and Rupture of Wetting Films
The Navier-Stokes equations can be simplified and solved for the 2D fluid flow
within the film. The pressure field can be inferred from the flow solution and, hence,
the hydrodynamic equation for film drainage and rupture can be established. Such
a theoretical description for the film drainage rate was obtained long time ago and
is referred to as the Stefan-Reynolds equation as follows [47]:
2
h
3
3
μR
2
(P
σ
−
d
h
d
t
=−
),
(17)
where
R
is the (constant) film radius and
P
σ
is the capillary pressure at the film
periphery. The Stefan-Reynolds (lubrication) theory is restricted to planar parallel
and tangentially immobile film surfaces. For films with non-planar parallel film
surfaces, the film thickness is also a function of the film radial coordinate,
r
,and
the lubrication theory gives
h
3
r
∂P
∂r
.
∂h
∂t
=
1
3
rμ
∂
∂r
(18)
The dynamic pressure inside the film,
P
, in Eq. (18) depends on the lo-
cal deform
ation of
the gas-liquid interface and is described as
P(h)
=
P
g
−
∂r
(rh
/
√
1
γ
r
∂
h
2
)
(h)
, where the first term is the (constant) gas pressure
inside the bubble and the prime describes the derivative with respect to
r
. Equa-
tion (18) can be numerically solved to validate and predict the spatial and temporal
film profiles influenced by the van der Waals and double layer disjoining pressures
[66]. An example is shown in Fig. 8.
+
−
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