XXXI International Mineral Processing Congress 2024 Proceedings/Washington, DC/Sep 29–Oct 3 2729
Substituting Eq. [9] into Eq. [15], one obtains an
expression for the free energy of bubble-particle interaction
as follows,
exp^2
exp^l
exp expc-
G
A h
C
D
h C
D
h
12rh 1
2 2r
1h
0
2
132
0
0 1
2
2
2
1
1
0 2
2
0
23
ff0l= lh
}
r
c i
D =+-
-+
++
=-^cos
c-
_}
m
h
h
m
i G
(16)
Assuming that the EDL force and the long-range compo-
nent of the hydrophobic force do not contribute signifi-
cantly to the free energy change, one obtains the following
relationship,
expc- G
A C
D
h
12rh 2r
1h
0
2
132 1
1
0
23 i
D =-+
=-c -^cos
m
(17)
The second term on the right of Eq. [17] gives the short-
range hydrophobic force, C1 representing the intensity (or
magnitude) of the hydrophobic force and D1 representing
the range over which the shortrange hydrophobic force is
discernable. Note also that the hydrophobic force becomes
attractive when C1 is negative, i.e.,
cos
expc-
C1
D1
h
A 2 6h
0 23
0
132 0
2 rc ih
=-
--^1
m
(18)
Eq. [18] shows that hydrophobic force becomes attractive
under the following condition,
2πγ23(1−cosθ) A132/6h02 (19)
a condition that is met because A132, representing the
Hamaker constant for bubble-particle interactions, is always
negative, while the contact angle term is always positive.
Eqs. [18] and [19] show that the hydrophobic force
becomes stronger under the following conditions. First, the
contact angle becomes larger. Second, the surface tension
of water (γ23) is kept high. Third, the magnitude of the
Hamaker constant is large. The third condition is usually
met for sulfide mineral flotation.
ACKNOWLEDGMENT
The authors greatly appreciate the financial support from
Mineral Refining Company (MRC), Richmond, Virginia.
REFERENCES CITED
Cassie, A. B. D., &Baxter, S. (1944). Wettability of
porous surfaces. Transactions of the Faraday society, 40,
546–551.
Chander, S., and D. W. Fuerstenau. “On the natural float-
ability of molybdenite.” Trans. AIME 252 (1972):
62–69.
Derjaguin, B. V. (1934). Friction and adhesion. IV. The
theory of adhesion of small particles. Kolloid Zeits, 69,
155–164.
Figure 6. Simulation of the first cell of a rougher flotation bank at an operating
porphyry copper flotation plant
Substituting Eq. [9] into Eq. [15], one obtains an
expression for the free energy of bubble-particle interaction
as follows,
exp^2
exp^l
exp expc-
G
A h
C
D
h C
D
h
12rh 1
2 2r
1h
0
2
132
0
0 1
2
2
2
1
1
0 2
2
0
23
ff0l= lh
}
r
c i
D =+-
-+
++
=-^cos
c-
_}
m
h
h
m
i G
(16)
Assuming that the EDL force and the long-range compo-
nent of the hydrophobic force do not contribute signifi-
cantly to the free energy change, one obtains the following
relationship,
expc- G
A C
D
h
12rh 2r
1h
0
2
132 1
1
0
23 i
D =-+
=-c -^cos
m
(17)
The second term on the right of Eq. [17] gives the short-
range hydrophobic force, C1 representing the intensity (or
magnitude) of the hydrophobic force and D1 representing
the range over which the shortrange hydrophobic force is
discernable. Note also that the hydrophobic force becomes
attractive when C1 is negative, i.e.,
cos
expc-
C1
D1
h
A 2 6h
0 23
0
132 0
2 rc ih
=-
--^1
m
(18)
Eq. [18] shows that hydrophobic force becomes attractive
under the following condition,
2πγ23(1−cosθ) A132/6h02 (19)
a condition that is met because A132, representing the
Hamaker constant for bubble-particle interactions, is always
negative, while the contact angle term is always positive.
Eqs. [18] and [19] show that the hydrophobic force
becomes stronger under the following conditions. First, the
contact angle becomes larger. Second, the surface tension
of water (γ23) is kept high. Third, the magnitude of the
Hamaker constant is large. The third condition is usually
met for sulfide mineral flotation.
ACKNOWLEDGMENT
The authors greatly appreciate the financial support from
Mineral Refining Company (MRC), Richmond, Virginia.
REFERENCES CITED
Cassie, A. B. D., &Baxter, S. (1944). Wettability of
porous surfaces. Transactions of the Faraday society, 40,
546–551.
Chander, S., and D. W. Fuerstenau. “On the natural float-
ability of molybdenite.” Trans. AIME 252 (1972):
62–69.
Derjaguin, B. V. (1934). Friction and adhesion. IV. The
theory of adhesion of small particles. Kolloid Zeits, 69,
155–164.
Figure 6. Simulation of the first cell of a rougher flotation bank at an operating
porphyry copper flotation plant