XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2727
The free energy change (∆G) associated will be equal to the
energy gained when an unstable system with G =γ13 +γ23
transforms to a stable system with G =γ12, while the film
thickness decreases from h =∞ to h =h0, forming a new
solid/vapor interface with an interfacial tension (γ12) and a
receding contact angle θr. Thus, bubble-particle interaction
is a process in which a thick film (β film) becomes a thin
film (α-film), during the course of which an energy barrier
(E1) has to be overcome. In flotation, E1 is usually due to
the EDL force as shown in Figure 2b. It may be worthwhile
to note here that the α-film consists of 2–3 layers of water
molecules adsorbed on the mineral surface.
MODEL PREDICTIONS
Silica Flotation
The surface forces measured in the TLFs of water formed
on the MTSC-coated silica surface show that both the
EDL and vdW forces are repulsive, both of which are not
conducive to flotation. Only when the HP force is created
by coating the surface with a hydrophobizing agent, i.e.,
MTCS, disjoining pressure becomes negative, i.e., Π 0,
the wetting film ruptures to allow contact angle formation.
A physical meaning for this criterion for film rupture is that
positive pressure will stabilize the film. At Π =0, the free
energy of the film reaches a maximum and thus creates an
energy barrier (E1) as shown by Eq. [12]. The energy bar-
rier can be overcome by mechanical agitation afforded in a
flotation machine.
Figure 4 shows an example of the HP force playing a
decisive role in bubble-particle attachment. The experimen-
tal data presented in the figure were obtained by Fuerstenau
(1957) for the flotation of quartz. The author conducted
flotation tests on a mono-sized (–48+65 mesh) and fully
liberated quartz
particles using a modified Hallimond flotation tube.
Also shown in the figure are the contact angles and particle
ζ-potentials that can be used to estimate the HP and EDL
forces that are necessary to predict flotation recoveries using
the simplified model (Eqs. [13]–[14]). The surface force
parameters used for the simulation included the follow-
ing: A132 =–1.3×10–20 J, Ψ2 =–20 mV, κ–1 =9.6 nm and
K232= 4.07×10–17 J, which represented the Hamaker con-
stant for the quartz 1, air bubble 2 in water 3, the surface
potential of air bubble, and the Debye lengths, respectively.
The K132 values were determined as described by Huang
et al. (2022). As shown, the flotation recoveries predicted
using the simplified model and plotted in red are in good
agreement with the data obtained by Fuerstenau (1957). In
this simulation, the hydrodynamic resistance in film thin-
ning was not considered as particle sizes were large, and the
energy dissipation rare (f )in a Hallimond tube flotation
cell was assumed to be small 0.15 kWm–3.
Note in Figure 4 that the maximum flotation of quartz
was obtained at pH 10 where the HP force became strongly
attractive due to the high contact angle, θ =~80°, and the
weakly attractive EDL force due to the weakly positive
ζ-potential (~10 mV). Under these conditions, E1 must be
very small, which gave rise to the maximum recovery.
Molybdenite Flotation
Chander and Fuerstenau (1972) conducted a Hallimond
tube flotation experiment on a –270 mesh molybdenite
sample in the absence of any collector, which was possible
because the mineral is naturally hydrophobic. As shown in
Figure 5, the flotation recovery was maximum at pH in the
range of 5.5 and 6.5, which corresponded to the pH range
where the molybdenite sample exhibited the minimum
ζ‑potentials of about –40 mV. The authors attributed the
excellent correlation to the minimum double-layer repul-
sion between bubbles and particles.
It has been found that the simplified flotation model
represented by Eqs. [13]–[14] can be used to validate
Cander and Fuerstenau’s explanation quantitatively.
Simulations have been conducted using the following
model parameters: A132= –1.3×10–20 J, ψ2 =–50 mV, к–1 =
9.6 nm K232 =4.07×10–17 J θ =30° and ε =0.45 kW/m3.
Figure 3. Potential energy vs. distance diagram
The free energy change (∆G) associated will be equal to the
energy gained when an unstable system with G =γ13 +γ23
transforms to a stable system with G =γ12, while the film
thickness decreases from h =∞ to h =h0, forming a new
solid/vapor interface with an interfacial tension (γ12) and a
receding contact angle θr. Thus, bubble-particle interaction
is a process in which a thick film (β film) becomes a thin
film (α-film), during the course of which an energy barrier
(E1) has to be overcome. In flotation, E1 is usually due to
the EDL force as shown in Figure 2b. It may be worthwhile
to note here that the α-film consists of 2–3 layers of water
molecules adsorbed on the mineral surface.
MODEL PREDICTIONS
Silica Flotation
The surface forces measured in the TLFs of water formed
on the MTSC-coated silica surface show that both the
EDL and vdW forces are repulsive, both of which are not
conducive to flotation. Only when the HP force is created
by coating the surface with a hydrophobizing agent, i.e.,
MTCS, disjoining pressure becomes negative, i.e., Π 0,
the wetting film ruptures to allow contact angle formation.
A physical meaning for this criterion for film rupture is that
positive pressure will stabilize the film. At Π =0, the free
energy of the film reaches a maximum and thus creates an
energy barrier (E1) as shown by Eq. [12]. The energy bar-
rier can be overcome by mechanical agitation afforded in a
flotation machine.
Figure 4 shows an example of the HP force playing a
decisive role in bubble-particle attachment. The experimen-
tal data presented in the figure were obtained by Fuerstenau
(1957) for the flotation of quartz. The author conducted
flotation tests on a mono-sized (–48+65 mesh) and fully
liberated quartz
particles using a modified Hallimond flotation tube.
Also shown in the figure are the contact angles and particle
ζ-potentials that can be used to estimate the HP and EDL
forces that are necessary to predict flotation recoveries using
the simplified model (Eqs. [13]–[14]). The surface force
parameters used for the simulation included the follow-
ing: A132 =–1.3×10–20 J, Ψ2 =–20 mV, κ–1 =9.6 nm and
K232= 4.07×10–17 J, which represented the Hamaker con-
stant for the quartz 1, air bubble 2 in water 3, the surface
potential of air bubble, and the Debye lengths, respectively.
The K132 values were determined as described by Huang
et al. (2022). As shown, the flotation recoveries predicted
using the simplified model and plotted in red are in good
agreement with the data obtained by Fuerstenau (1957). In
this simulation, the hydrodynamic resistance in film thin-
ning was not considered as particle sizes were large, and the
energy dissipation rare (f )in a Hallimond tube flotation
cell was assumed to be small 0.15 kWm–3.
Note in Figure 4 that the maximum flotation of quartz
was obtained at pH 10 where the HP force became strongly
attractive due to the high contact angle, θ =~80°, and the
weakly attractive EDL force due to the weakly positive
ζ-potential (~10 mV). Under these conditions, E1 must be
very small, which gave rise to the maximum recovery.
Molybdenite Flotation
Chander and Fuerstenau (1972) conducted a Hallimond
tube flotation experiment on a –270 mesh molybdenite
sample in the absence of any collector, which was possible
because the mineral is naturally hydrophobic. As shown in
Figure 5, the flotation recovery was maximum at pH in the
range of 5.5 and 6.5, which corresponded to the pH range
where the molybdenite sample exhibited the minimum
ζ‑potentials of about –40 mV. The authors attributed the
excellent correlation to the minimum double-layer repul-
sion between bubbles and particles.
It has been found that the simplified flotation model
represented by Eqs. [13]–[14] can be used to validate
Cander and Fuerstenau’s explanation quantitatively.
Simulations have been conducted using the following
model parameters: A132= –1.3×10–20 J, ψ2 =–50 mV, к–1 =
9.6 nm K232 =4.07×10–17 J θ =30° and ε =0.45 kW/m3.
Figure 3. Potential energy vs. distance diagram