XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2725
pressure (Π) (surface force in mN per unit area in m2) in a
wetting film.
Flotation is based on controlling the surface forces (or
Π) present in the wetting films of water formed on min-
erals. According to the extended DLVO theory (Xu and
Yoon, 1989),
sinh^lhh coth^lhh
exp expc-
A
D1
C
D2
h C
D2
h
6rh 2
2 2rD2
d e h
3
132
2
1
2
2
2
1 2
1 2
ff0l }
}
r
P P P P =++
=--
+
-2}
++c
_}
m
icosech^lhh
m
H (9)
in which Πd, Πe, and Πh represent the disjoining pressures
due to the van der-Waals, electrical doublelayer (EDL), and
hydrophobic forces, respectively A132 is the Hamaker con-
stant for the interaction between mineral 1 and air bubble
2 in water 3. In wetting films, A132 0, indicating that
it takes energy to remove the water molecules adsorbed
on hydrophobic mineral surfaces by the van der Waals
force. Πe can be readily predicted using the Hogg-Healey-
Fuerstenau approximation (Hogg et al., 1966), in which ε0
is the permittivity of vacuum, ε the dielectric constant of
water, κ the reciprocal Debye length, ψ1 the double-layer
potential of the solid surface, and ψ2 is the same of the air
bubble. The values of ψ1 and ψ2 are usually substituted by
the ζ-potentials that can be readily measured in laboratory
experiments.
Force Measurement
Pan et al. (2011) developed a method of measuring Π in a
wetting film by monitoring the deformation of an air bubble
approaching a flat surface, which led to the development of
the force apparatus for deformable surfaces (FADS) shown
in Figure 1 (Pan and Yoon, 2016 Huang and Yoon, 2019).
The measurement begins by approaching an air bubble
toward the bottom surface of a cantilever spring by means
of piezo crystals while recording the optical fringes gener-
ated due to the changes in bubble curvature by means of a
high-speed camera. Recorded fringes were analyzed offline
using the microinterferometric technique to determine the
film thickness (h) as a function of time (t) with an accuracy
of 0.2 nm (Exerowa and Kruglyakov, 1997). In effect, the
FADS method is designed to measure the surface forces in
wetting films using the air/water interface as a force sensor,
with a spring constant of 72 Nm–1 at 25°C, which will give
an accuracy 3.6×10–11 N in force measurement. The FADS
is also capable of measuring the surface forces directly by
monitoring the deflection of the cantilever spring using
the fiber optic sensor. The force measurements using the
two different methods are in excellent agreement (Pan and
Yoon, 2016).
Figure 2 shows the results obtained with a silica sur-
face coated with methyl trichlorosilane (MTCS) to obtain a
receding contact angle (θr) of 40° at a 1,000 nm/s approach
speed. With the force curves and the spatiotemporal film
profiles shown in Figure 2-a and -b, respectively, one can
construct the disjoining pressure isotherm Π(h) (Figure 2c).
Disjoining pressure is defined as (Derjaguin, 1934),
,p T,n
i
2h
2G P =-b l (10)
which can be used to write the following relation,
G hhdh
1h
h
23
o
c i
D P^ =-
=-
3
3 ^cos
#
(11)
in which ∆G is the free energy change associated with con-
tact angle formation, γ23 is the interfacial tension between
Figure 1. Force apparatus For Deformable Surfaces (FADS)
pressure (Π) (surface force in mN per unit area in m2) in a
wetting film.
Flotation is based on controlling the surface forces (or
Π) present in the wetting films of water formed on min-
erals. According to the extended DLVO theory (Xu and
Yoon, 1989),
sinh^lhh coth^lhh
exp expc-
A
D1
C
D2
h C
D2
h
6rh 2
2 2rD2
d e h
3
132
2
1
2
2
2
1 2
1 2
ff0l }
}
r
P P P P =++
=--
+
-2}
++c
_}
m
icosech^lhh
m
H (9)
in which Πd, Πe, and Πh represent the disjoining pressures
due to the van der-Waals, electrical doublelayer (EDL), and
hydrophobic forces, respectively A132 is the Hamaker con-
stant for the interaction between mineral 1 and air bubble
2 in water 3. In wetting films, A132 0, indicating that
it takes energy to remove the water molecules adsorbed
on hydrophobic mineral surfaces by the van der Waals
force. Πe can be readily predicted using the Hogg-Healey-
Fuerstenau approximation (Hogg et al., 1966), in which ε0
is the permittivity of vacuum, ε the dielectric constant of
water, κ the reciprocal Debye length, ψ1 the double-layer
potential of the solid surface, and ψ2 is the same of the air
bubble. The values of ψ1 and ψ2 are usually substituted by
the ζ-potentials that can be readily measured in laboratory
experiments.
Force Measurement
Pan et al. (2011) developed a method of measuring Π in a
wetting film by monitoring the deformation of an air bubble
approaching a flat surface, which led to the development of
the force apparatus for deformable surfaces (FADS) shown
in Figure 1 (Pan and Yoon, 2016 Huang and Yoon, 2019).
The measurement begins by approaching an air bubble
toward the bottom surface of a cantilever spring by means
of piezo crystals while recording the optical fringes gener-
ated due to the changes in bubble curvature by means of a
high-speed camera. Recorded fringes were analyzed offline
using the microinterferometric technique to determine the
film thickness (h) as a function of time (t) with an accuracy
of 0.2 nm (Exerowa and Kruglyakov, 1997). In effect, the
FADS method is designed to measure the surface forces in
wetting films using the air/water interface as a force sensor,
with a spring constant of 72 Nm–1 at 25°C, which will give
an accuracy 3.6×10–11 N in force measurement. The FADS
is also capable of measuring the surface forces directly by
monitoring the deflection of the cantilever spring using
the fiber optic sensor. The force measurements using the
two different methods are in excellent agreement (Pan and
Yoon, 2016).
Figure 2 shows the results obtained with a silica sur-
face coated with methyl trichlorosilane (MTCS) to obtain a
receding contact angle (θr) of 40° at a 1,000 nm/s approach
speed. With the force curves and the spatiotemporal film
profiles shown in Figure 2-a and -b, respectively, one can
construct the disjoining pressure isotherm Π(h) (Figure 2c).
Disjoining pressure is defined as (Derjaguin, 1934),
,p T,n
i
2h
2G P =-b l (10)
which can be used to write the following relation,
G hhdh
1h
h
23
o
c i
D P^ =-
=-
3
3 ^cos
#
(11)
in which ∆G is the free energy change associated with con-
tact angle formation, γ23 is the interfacial tension between
Figure 1. Force apparatus For Deformable Surfaces (FADS)