2724 XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3
the measurement of strong attractive forces without the
artifact. It appears that nanobubbles are detected when
two surfaces approach each other at high approach speeds.
More recently, we have developed the force apparatus for
deformable surfaces (FADS), which is designed to mea-
sure the surface forces in the wetting films formed between
an air bubble and a hydrophobic surface (Pan and Yoon,
2016). Thus, the FADS method is ideally suited for the
measurement of the surface forces present in the thin liq-
uid films (TLFs) of water confined between air bubbles and
hydrophobic surfaces.
Based on the encouraging results obtained using the
FADS, we have developed a flotation model using the
hydrophobic force as a kinetic parameter for bubble-par-
ticle interaction (Huang et al., 2022). With this approach,
it was possible to derive a first principles model that can
predict for the first time the grade vs. recovery curves,
which is useful for daily plant operations. The model has
been validated in a full-scale plant operation (Gupta et al.,
2022). The input to the model is the liberation matrix of a
plant feed, from which the contact angles of the different
liberation classes of particles using the Cassie-Baxter equa-
tion (1944), which in turn were used to determine the cor-
responding hydrophobic force parameters.
In the present work, we have studied the effect of EDL
forces on flotation using the flotation model. The simula-
tion results obtained using the model showed that control
of particle ζ-potentials can significantly improve flotation
recoveries as suggested by DW Fuerstenau in 1957.
EXPERIMENTAL
Pressure Balance in a Thin Liquid Film
When an air bubble approaches a flat surface in water, a
thin liquid film (TLF), also known as wetting (or flotation)
film, is formed in between. The curvature changes associ-
ated with conforming to the flat geometry create an excess
(or capillary) pressure (pc) in the film, causing it to thin. As
the thinning continues to a film thickness (h) of ~250 nm,
surface forces begin to play a role. If they are repulsive, a
positive disjoining pressure (Π 0) is created and opposes
film thinning. The thinning stops at an equilibrium thick-
ness (he), at which pc =Π. If the disjoining pressure is lower
than the capillary pressure (Π pc), the film thins further
and ruptures at a critical thickness (hc), where the disjoin-
ing pressure becomes negative (Π 0), forming a finite
contact angle (θ).
Thus, pc serves as the driving force for film thinning,
while Π offers a resistance to film thinning.
In this regard, one may write a pressure balance in the
direction perpendicular to the film as follows,
p =pc −Π (2)
in which p represents the excess pressure that drives film
thinning. The rate of film thinning can be predicted using
the Reynolds equation,
rh 12nr
1 3 2t
2h
2r
2
2r
2p
=c m (3)
in which μ is the dynamic viscosity of water and r is the
radial distance from the film center. Under the condition of
flat films, Eq. [3] can be reduced to an approximate form
as follows,
dt
dh p
3nR
2h
f
2
3
=-(4)
in which Rf is the radius of the flat film. Integrating Eq.
[3] twice under the boundary conditions that pr=∞ =0 and
∂p/∂r|r=0 =0, one obtains p as follows,
p
rh
r dr 12 1
r
r
r
r
3
0 2t
2h
3 ==
b ldr ##(5)
Integrating Eq. [5] over the cross-sectional area of the film
gives the force pressing the film.
An objective of the present work is to measure the
surface forces in wetting films to find ways to improve the
bubble-particle attachment and contact angle formation.
To this end, an air bubble is slowly brought to a surface
within the distance range where surface forces can cause
the bubble to deform. By monitoring the bubble deforma-
tion, one can determine the capillary pressure (pc) using the
YoungLaplace equation,
p R r r 2c
c 2r
2
2r
2h c =-b l (6)
in which γ is the surface tension of water, and R is the bub-
ble radius. The first term to the right of the equal sign rep-
resents the macroscopic capillary pressure, while the second
term represents the effects of local curvature changes.
Integrating Eq. [6] across a TLF with a radius of rmax
formed at time t will give the total (or Laplace) force (F(t))
exerted on the bubble,
F th p 2
c
r
0
max ^^r,t hrdr #(7)
Substituting Eq. [2] into Eq. [7], one obtains (Manica et
al., 2008),
F th p 2r rmax
0
P^r,th@rdr =+^^r,th 6 #(8)
The first term on the right of Eq. [8] gives the hydrody-
namic force, while the second term gives the disjoining
the measurement of strong attractive forces without the
artifact. It appears that nanobubbles are detected when
two surfaces approach each other at high approach speeds.
More recently, we have developed the force apparatus for
deformable surfaces (FADS), which is designed to mea-
sure the surface forces in the wetting films formed between
an air bubble and a hydrophobic surface (Pan and Yoon,
2016). Thus, the FADS method is ideally suited for the
measurement of the surface forces present in the thin liq-
uid films (TLFs) of water confined between air bubbles and
hydrophobic surfaces.
Based on the encouraging results obtained using the
FADS, we have developed a flotation model using the
hydrophobic force as a kinetic parameter for bubble-par-
ticle interaction (Huang et al., 2022). With this approach,
it was possible to derive a first principles model that can
predict for the first time the grade vs. recovery curves,
which is useful for daily plant operations. The model has
been validated in a full-scale plant operation (Gupta et al.,
2022). The input to the model is the liberation matrix of a
plant feed, from which the contact angles of the different
liberation classes of particles using the Cassie-Baxter equa-
tion (1944), which in turn were used to determine the cor-
responding hydrophobic force parameters.
In the present work, we have studied the effect of EDL
forces on flotation using the flotation model. The simula-
tion results obtained using the model showed that control
of particle ζ-potentials can significantly improve flotation
recoveries as suggested by DW Fuerstenau in 1957.
EXPERIMENTAL
Pressure Balance in a Thin Liquid Film
When an air bubble approaches a flat surface in water, a
thin liquid film (TLF), also known as wetting (or flotation)
film, is formed in between. The curvature changes associ-
ated with conforming to the flat geometry create an excess
(or capillary) pressure (pc) in the film, causing it to thin. As
the thinning continues to a film thickness (h) of ~250 nm,
surface forces begin to play a role. If they are repulsive, a
positive disjoining pressure (Π 0) is created and opposes
film thinning. The thinning stops at an equilibrium thick-
ness (he), at which pc =Π. If the disjoining pressure is lower
than the capillary pressure (Π pc), the film thins further
and ruptures at a critical thickness (hc), where the disjoin-
ing pressure becomes negative (Π 0), forming a finite
contact angle (θ).
Thus, pc serves as the driving force for film thinning,
while Π offers a resistance to film thinning.
In this regard, one may write a pressure balance in the
direction perpendicular to the film as follows,
p =pc −Π (2)
in which p represents the excess pressure that drives film
thinning. The rate of film thinning can be predicted using
the Reynolds equation,
rh 12nr
1 3 2t
2h
2r
2
2r
2p
=c m (3)
in which μ is the dynamic viscosity of water and r is the
radial distance from the film center. Under the condition of
flat films, Eq. [3] can be reduced to an approximate form
as follows,
dt
dh p
3nR
2h
f
2
3
=-(4)
in which Rf is the radius of the flat film. Integrating Eq.
[3] twice under the boundary conditions that pr=∞ =0 and
∂p/∂r|r=0 =0, one obtains p as follows,
p
rh
r dr 12 1
r
r
r
r
3
0 2t
2h
3 ==
b ldr ##(5)
Integrating Eq. [5] over the cross-sectional area of the film
gives the force pressing the film.
An objective of the present work is to measure the
surface forces in wetting films to find ways to improve the
bubble-particle attachment and contact angle formation.
To this end, an air bubble is slowly brought to a surface
within the distance range where surface forces can cause
the bubble to deform. By monitoring the bubble deforma-
tion, one can determine the capillary pressure (pc) using the
YoungLaplace equation,
p R r r 2c
c 2r
2
2r
2h c =-b l (6)
in which γ is the surface tension of water, and R is the bub-
ble radius. The first term to the right of the equal sign rep-
resents the macroscopic capillary pressure, while the second
term represents the effects of local curvature changes.
Integrating Eq. [6] across a TLF with a radius of rmax
formed at time t will give the total (or Laplace) force (F(t))
exerted on the bubble,
F th p 2
c
r
0
max ^^r,t hrdr #(7)
Substituting Eq. [2] into Eq. [7], one obtains (Manica et
al., 2008),
F th p 2r rmax
0
P^r,th@rdr =+^^r,th 6 #(8)
The first term on the right of Eq. [8] gives the hydrody-
namic force, while the second term gives the disjoining