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The role of hydrodynamic and surface forces in bubble–particle interaction
Int. J. Miner. Process. 58 Ž2000. 129–143 www.elsevier.nlrlocaterijminpro
The role of hydrodynamic and surface forces in bubble–particle interaction R.-H. Yoon
)
Department of Mining and Minerals Engineering, Virginia Polytechnic Institute and State UniÕersity, Blacksburg, VA 24060, USA Received 2 August 1999; accepted 6 October 1999
Abstract In modeling flotation, the process of bubble–particle interaction is usually divided into three subprocesses, including collision, adhesion and detachment. Of these, the hydrodynamics of bubble–particle collision has been studied most extensively by many investigators, and the results are useful for the design and scale-up of flotation cells. The process of adhesion, on the other hand, is least understood because it is essentially controlled by the chemistry of the system, which is complex and difficult to model mathematically. However, it is possible to determine the probability of the bubble–particle adhesion from the induction times that can be measured experimentally under different chemical environments. Furthermore, the new information reported in the literature on the hydrophobic forces of both particles and bubbles allow prediction of adhesion probabilities using various surface chemistry parameters. Consideration of both the hydrodynamic and surface force parameters is essential in predicting flotation rates from first principles. q 2000 Elsevier Science B.V. All rights reserved. Keywords: hydrodynamic flotation; hydrophobic force; contact angle; flotation rate constant; effect of bubble size on flotation rate; critical film thickness for rupture
1. Introduction Froth flotation is widely used for separating different minerals from each other. However, its effectiveness is limited to a relatively narrow particle size range of 10–100 mm ŽTrahar and Warren, 1976; Wills, 1998.. The difficulty in floating fine particles is attributed to the low probability of bubble–particle collision, while the problem with coarse particle flotation is due to detachment. For flotation occurring under quiescent )
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conditions, one can calculate the probability of collision using stream functions. The stream functions used by earlier workers are applicable for bubbles that are either too large or too small ŽSutherland, 1949; Gaudin, 1957., while those developed in recent years are useful for flotation size bubbles ŽWeber and Paddock, 1983; Yoon and Luttrell, 1989.. However, most of the flotation machines are operated under intensely agitated conditions, which makes it difficult to use the interceptional collision models based on stream functions. Under such conditions, models based on microturbulence may be more useful ŽSchubert and Bischofberger, 1979.. Not all the particles colliding with air bubbles result in flotation. Only the hydrophobic particles adhere to the surface of air bubbles. Thus, the probability of adhesion determines the selectivity of a flotation process, while its recovery depends critically on the collision probability. Also, some of the particles are detached from the surface due to the inertia force and turbulence in a flotation cell. Both the adhesion and detachment processes are determined largely by the surface chemistry of the bubbles and particles present in a flotation cell. It is difficult, however, to predict the probabilities of these subprocesses using the surface chemistry parameters such as contact angle, z-potential, Hamaker constant, etc. Derjaguin and Dukhin Ž1961. were the first to describe the bubble–particle interactions in flotation by considering surface forces. They considered that before a particle can adhere on the surface of an air bubble, it must pass through three distinct zones, i.e., hydrodynamic, diffusiophoretic and wetting zones. In the wetting zone, three surface forces, namely, van der Waals, electrostatic and structural forces were considered. ŽHere, the term structural force is used to represent the surface forces that cannot be explained by the continuum theory.. Derjaguin and Duhkin considered that the structural force is hydrophilic, but they considered it to be zero in their model calculation. In a later publication, Derjaguin and Dukhin Ž1979. suggested that small particles can adhere on the surface of air bubbles without penetrating the wetting film. They thought that the fine particles can be held to the bubble surface with relatively weak forces such as van der Waals and electrostatic forces, because they are not subject to significant tearing-off forces due to their small inertia. Again, the authors did not consider the structural forces playing an important role in the bubble–particle interaction. Laskowski and Kitchener Ž1969. studied the interaction between air bubble and silica plate hydrophobized by trimethylchlorosilane ŽTMCS.. Under the conditions employed in their experiments, both the bubbles and particles were negatively charged. They showed also that the van der Waals dispersion forces in wetting films are repulsive. Yet, bubbles readily adhered on the hydrophobic silica surface, forming contact angles as large as 70–808. They speculated, therefore, that there must be a third force, other than the electrostatic and van der Waals dispersion force, that is responsible for the bubble–particle adhesion. This work of Laskowski and Kitchener is regarded in the literature as the first recognition of the existence of hydrophobic force. The first direct measurement of hydrophobic force was reported by Israelachvili and Pashley Ž1988.. They brought two curved mica surfaces immersed in dodecylamine hydrochloride solutions to within 10 nm and observed attractive forces that were much larger than the van der Waals force. They called the additional attractive force hydrophobic force. Since then, numerous other investigators reported measurements of hydrophobic forces for
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various surfactants on different substrates. More recently, Yoon and Aksoy Ž1999. suggested that the hydrophobic forces are also present in soap films. In this communication, the role of hydrodynamic and surface forces in flotation is discussed. Flotation tests have been conducted under different hydrodynamic conditions, and a method of determining the probability of bubble–particle adhesion developed. The effects of various surface forces on the rupture of wetting films are discussed with a sample calculation. The results of the present investigation suggest that both the hydrodynamic and surface forces must be considered to develop a comprehensive flotation model.
2. Hydrodynamics The probability Ž P . of a particle being collected by an air bubble in the pulp phase of a flotation cell can be given by P s Pc Pa Ž 1 y Pd . ,
Ž 1.
where Pc is the probability of bubble particle collision, Pa is the probability of adhesion and Pd is the probability of detachment. For fine particles, Pd can be negligibly small because of the low inertia, in which case Eq. Ž1. becomes: P s Pc Pa .
Ž 2.
Pc is determined by the hydrodynamics of the system, which is strongly affected by the particle size, bubble size and the turbulence of the system. Pa is also affected by the hydrodynamics, but is largely a function of the surface chemistry involved. 2.1. Probability of collision Sutherland Ž1949. was the first to derive an expression for Pc from a stream function, Pc s 3 DprD b ,
Ž 3.
where Dp is the diameter of the particle and D b is the diameter of the bubble. Because of the assumption of potential flow, this equation is valid only for bubbles that are much larger than those used in flotation practice. Gaudin Ž1957. also derived an expression for Pc , 2
Pc s 1.5 Ž DprD b . ,
Ž 4.
using the stream function for the Stokes flow condition. However, Eq. Ž4. is applicable only for very small bubbles. Recognizing the limited applicability of Eqs. Ž3. and Ž4., Flint and Howarth Ž1971. numerically solved the Navier–Stokes equations to determine Pc . This approach was adapted by Reay and Ratcliff Ž1973. to derive a relationship, 2
Pc A Ž DprD b . , which has been verified experimentally for the flotation of fine particles.
Ž 5.
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More recently, Weber Ž1981. and Weber and Paddock Ž1983. derived an analytical expression for Pc for bubbles of low Reynolds numbers. For large Reynolds number bubbles, the Navier–Stokes equations were solved numerically using a curve-fitting technique. In this manner, Weber and Paddock were able to derive an expression for Pc , Pc s
3
Ž 3r16. Re
1q
2
Dp
ž /
Ž 6. Db 1 q 0.249Re 0.56 where Re is the Reynolds number of bubble. Eq. Ž6. is the first of its kind that can predict Pc for wide ranges of bubble and particle sizes. However, Weber and Paddock did not verify this relationship experimentally. The essence of predicting Pc accurately is to derive appropriate stream functions for different ranges of bubble sizes. The stream function for the Stokes and potential flow conditions had been known for a long time, but not for the bubbles of intermediate Reynolds numbers that are used for flotation. For this reason, Yoon and Luttrell Ž1989. derived a stream function for the intermediate Reynolds number range, 1 2 3 Re 0.72 1 1 2 2 c s u b R b sin u x y x q y qxy1 , Ž 7. 2 2 4 15 x x 2
ž
/
where c is the dimensionless stream function, u and x are the angular and dimensionless radial coordinates, respectively; R b is the bubble radius and u b is the bubble rise velocity. Using Eq. Ž7., Yoon and Luttrell derived an expression for Pc , Pc s
4 Re 0.72
3 q 2
15
Dp
2
ž /
,
Db
Ž 8.
which shows that Pc varies as Dy2 for small bubbles with Re s 0. However, as bubble b size becomes larger, Pc becomes less dependent on D b . It can be readily shown that for .46 very large bubbles, Pc varies Dy0 . b Thus, the power relationship between Pc and the D brDp ratio changes with bubble size. The relationship may be represented in a generalized from, Pc s A
Dp
ž / Db
n
,
Ž 9.
where A and n are the parameters that vary with Reynolds numbers. The values of A and n are given Table 1 for three different flow regimes, while Fig. 1 shows the values Table 1 Values of A and n of Eq. Ž9. for different flow conditions Flow conditions
A
Stokes Eq. Ž4.
2r3
Intermediate Eq. Ž7. Intermediate Eq. Ž6. Potential Eq. Ž3.
n 2
3r2qŽ4 Re 3 2 3
1q
0.72 .
r15
Ž 3r16 . Re
2 2
1q0.249 Re 0.56
1
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Fig. 1. Effects of bubble size Ž D b . on collision efficiency Ž Pc . under different flow conditions.
of Pc calculated for Dp s 11.4 mm using the four different expressions given in the table. Note that for the bubble sizes used in flotation practice, Sutherland’s equation ŽEq. Ž3.. overestimates Pc , while Gaudin’s ŽEq. Ž4.. underestimates it. However, Gaudin’s equation can still be useful for bubbles smaller than approximately 100 mm. Beyond this limit, the two equations derived by Weber and Paddock ŽEq. Ž6.. and Yoon and Luttrell ŽEq. Ž8.. may be useful. Although these two equations have been derived using completely different approaches and are in different functional forms, the predicted Pc values are in reasonable agreement with each other. All of the Pc expressions given in Table 1 are based on the interceptional collision model, which may be useful for flotation under relatively quiescent conditions. They may be applicable to flotation columns with large length-to-diameter ratios or for columns with sufficient baffles. Use of small bubbles may also help create quiescent conditions. For mechanically agitated cells, however, the turbulence model may be more appropriate. Based on the collision model derived by Abrahamson Ž1975., Schubert and Bischofberger Ž1979. used the following expression for predicting the number of bubble–particle collisions Ž Zpb . per unit volume of slurry and time: Dp q D b
2
/(
np2 q n b2 , Ž 10 . 2 where Np and Nb are the number of particles and bubbles per unit volume, respectively; Zpb s 5Np Nb
ž
(
(
and np2 and n b2 are the mean relative velocities of the particles and bubbles, respectively. Ž1976. and Schubert Ž1986. showed that the mean relative Liepe and Mockel ¨ velocities can be estimated by the following relationship:
(n
2 i
s 0.33
e 4r9 Di7r9 n 1r3
Dr
ž / r
2r3
Ž 11 .
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where the subscript i refers to bubble or particle, e the specific energy dissipation, r the density of the medium, and D r is the density difference between i and the medium and n is the kinematic viscosity of the medium. 2.2. Probability of adhesion The availability of stream functions also makes it possible to predict Pa . Using the stream function given by Eq. Ž7., Yoon and Luttrell Ž1989. calculated the sliding times of particles on bubble surfaces. If the sliding time is longer than the induction time, the particle will have long enough contact time to thin and rupture the disjoining film between the particle and bubble. Using this criterion, one can derive an expression for Pa in the following form: Pa s sin2 2tany1 exp
ž
y Ž 45 q 8 Re 0.72 u b t i 15D b Ž D brDp q 1 .
/
Ž 12 .
where t i is the induction time and u b is the bubble rise velocity. This expression is applicable for the intermediate Reynolds number range. The induction time is strongly a function of particle hydrophobicity and can be routinely determined in laboratory using a relatively simple device ŽYordan and Yoon, 1986.. Thus, Eq. Ž12. can be useful for determining Pa from the values of t i determined under various reagent conditions. Fig. 2 shows the Pa values predicted using Eq. Ž12. from the induction times measured on a coal sample oxidized in water over a long period of time. The calculated
Fig. 2. Values of Pa calculated using Eq. Ž12. from the induction times measured on a coal sample oxidized in water at ambient temperature.
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Pa values are in reasonable agreement with the microflotation results. Eq. Ž12. suggests that Pa is also a function of particle size and bubble size. It has been shown that as the particle size decreases Pa increases, and that Pa increases with decreasing bubble size until it decreases again as the bubble size becomes too small ŽYoon and Luttrell, 1989.. 2.3. Flotation rate Once P is known from the detailed knowledge of its subprocesses, the first order flotation rate constant Ž k . can be obtained using the following relationship ŽYoon et al., 1989; Yoon and Mao, 1996.: ks
3Vg 2 Db
Ps
1 4
Sb P
Ž 13 .
where Vg is the superficial gas rate which is the gas flow rate divided by the cross-sectional area of the flotation cell, and S b is the surface area of the bubbles rising out of a flotation cell per unit area and unit time. Since Pc varies as Dp2 at the bubble sizes commonly employed in flotation ŽTable 1., k will also vary as Dp2 at given Pa and Vg , providing an explanation for the difficulty in recovering fine particles. A common solution to this problem is to increase Vg as suggested by Eq. Ž13.. However, the improvement in k that can be brought about by this method is only linear, i.e., k A Vg . Also, there is a limitation in increasing Vg without flooding the cell. A better solution to the problem of fine particle flotation may be to use smaller bubbles. From Eq. Ž8., it can be seen that when bubble size is in the range of 100 mm or Ž . smaller, Pc varies as Dy2 b . Substituting this relationship into Eq. 13 , one can see that k varies as Dy3 b , which is a powerful message to use smaller bubbles. However, as the bubble size becomes larger, Pc will become progressively less dependent on bubble .46 size. If bubbles become very large, k varies as Dy1 . b For the purpose of illustrating the effect of bubble size on flotation rate constant, Eq. Ž13. has been used to calculate k as a function of D b . The results are given in Fig. 3 as a solid line. The calculation has been made for Dp s 8 mm and Pa s 0.6. Also shown are the ‘frothless’ flotation test results obtained using a methylated quartz sample with 8-mm median size ŽLuttrell, 1986.. When the agitation speed was as slow as 100 rpm, the experimental k values were in close agreement with those predicted, particularly with regard to the slope. As the agitation speed was increased, k became less dependent on D b . The results also show that strong agitation is detrimental to fine bubble flotation, while it helps the flotation with larger bubbles. This finding suggests that the benefits of suing small bubbles for flotation can be best realized under quiescent conditions. Even for the case of turbulent collision, decreasing bubble size should increase the collision rate and, hence, the rate constant. However, the flotation rate is not as strongly dependent on D b as the case of the interceptional collision under quiescent conditions. Eqs. Ž10. and Ž11. suggest that for the case of D b 4 Dp , the collision rate varies .61 approximately as Dy0 . Substituting this relationship into Eq. Ž13., one can see that b y1 .61 k ADb . This finding may suggest that the benefits of using small bubbles can be best
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Fig. 3. Effects of bubble size on the flotation rate constants Ž k . for a methylated quartz sample. The tests were conducted at different agitation speeds.
realized when the flotation cell can provide a relatively quiescent condition. As Ahmed and Jameson Ž1989. suggested, the remedy to low recovery rates associated with fine particles would be to either use small bubbles of the order of 100 mm under quiescent conditions, or to use moderate-to-high agitation with larger bubbles.
3. Surface forces in flotation While improvements in the hydrodynamic conditions discussed in the foregoing paragraph can bring about higher recoveries, the selectivity of a flotation process depends on the control of surface forces. The electrostatic forces acting between particles and air bubbles can be calculated from their z-potentials, which are routinely measured in laboratories. When particles and bubbles are oppositely charged, they can be attracted with each other and form bubble–particle aggregates, which is a prerequisite for flotation. However, the electrostatic attraction is so weak that the particles can be readily detached from the surface of air bubbles, especially for large particles. Only when particles are sufficiently hydrophobic, the wetting films between the particles and bubbles rupture, and three-phase contacts are established. The free energy change associated with the three-phase contact provides the forces that are necessary to withstand the tearing-off forces created by the hydrodynamic drag or the turbulence in a flotation cell. Furthermore, both the particles and bubbles are negatively charged in many flotation systems. Therefore, it is unlikely that electrical forces are responsible for the rupture of wetting films. On the other hand, the Õan der Vaals dispersion forces are repulsive in
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bubble–particle interactions, as will be shown later. The classical DLVO theory ŽDerjaguin and Landau, 1941; Verwey and Overbeek, 1948. considers only the electrostatic and dispersion forces to describe the interaction between lyophobic particles in a medium. Thus, there must be a third force that is responsible for the rupture of wetting films during flotation. Recent work conducted at Virginia Tech suggested that the hydrophobic force is responsible for the destabilization of both wetting and soap films ŽYoon and Mao, 1996; Yoon and Aksoy, 1999.. Hydrophobic forces were first measured by Isrealashivili and Pashley in 1982 between mica surface coated with cetyltrimethylammonium bromide ŽCTAB.. The measurements were conducted using the surface force apparatus ŽSFA.. This apparatus can be used for transparent and molecularly smooth surfaces such as mica and sapphire. Atomic force microscope ŽAFM., which was originally designed to measure surface morphologies, was used to measure surface forces in 1991 by Ducker et al. Ž1991.. These investigators replaced the pyramidal tip with a glass sphere, so that the forces measured between the sphere and plate can be related to the DLVO theory and converted to surface free energies using the Derjaguin approximation ŽDerjaguin, 1934.. Rabinovich and Yoon Ž1994. used the AFM technique to measure the surface forces between hydrophobic surfaces, i.e., silaned glass sphere and silica plate. They found that the measured forces were strongly attractive and increased with increasing contact angle. Since then, numerous other investigators used AFM to measure the hydrophobic forces. It has been found that measured hydrophobic forces Ž FH . decay exponentially with the closest distance Ž H . separating the sphere and the flat surface. Thus, the experimental data are frequently fitted to a double-exponential function: FH rR s C1exp Ž yHrD1 . q C2 exp Ž yHrD 2 . ,
Ž 14 .
where R is the radius of the sphere, and C1 , C2 , D 1 , and D 2 are fitting parameters. D 1 and D 2 are decay lengths, and the pre-exponential parameter C1 is related to surface free energy at the solid–liquid interface. It has also been shown that measured hydrophobic forces can be described by the following relationship ŽRabinovich and Yoon, 1994., FH rR s yKr6 H 2 ,
Ž 15 .
in which K is the only fitting parameter. Eq. Ž15. is of the same form as the expression for the van der Waals dispersion forces. Therefore, the value of K can be directly compared with the Hamaker constant. Fig. 4 shows a relationship between K and contact angle Ž u ., which is the more conventional measure of hydrophobicity. Implications of the data shown in this figure are discussed in another communication ŽPazhianur and Yoon, 1999.. It may be useful to discuss the role of surface forces in bubble–particle interactions in a quantitative manner. A sample calculation to be made here will be the case of approaching an air bubble toward a flat hydrophobic surface. Yordan Ž1989. conducted an experiment, in which an air bubble of 2 mm in radius was pressed against a silica surface immersed in a 10y5 M DAH solution at pH 10. The z-potentials of the air bubble and silica were y45 and y20 mV, respectively, in 10y5 M DAH solutions.
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Fig. 4. Effects of advancing contact angle Ž u . on the hydrophobic force constant of Eq. Ž15.. The K values are from Yoon and Ravishankar Ž1993; 1996., Rabinovich and Yoon Ž1994., Vivek Ž1998. and Pazhianur Ž1999..
From these values, one can calculate the free energy of electrostatic interaction using the equation ŽHogg et al., 1966.: VE s p´o ´ R 2 Ž c 12 q c 22 .
2 c1c2
ln 2
c 12 q c 2
ž
1 q eyk H 1 y ey k H
/
q ln w 1 y ey2 k H x
Ž 16 .
where R 2 is the radius of bubble; c 1 and c 2 are the surface potentials of the silica plate and spherical air bubble, respectively; H is the shortest distance between the two surfaces; ´o Žs 8.854 = 10y1 2 C 2 Jy1 my1 . is the permittivity of free space; ´ is the dielectric constant of water; and k is the reciprocal Debye length. At 258C, ´ s 86.5 and ky1 s 96 nm. In this sample calculation, the z-potentials of the silica and air bubble were used to substitute c 1 and c 2 , respectively, as approximation. One can also calculate the free energy change due to the van der Waals dispersion interaction using the following equation: VD s y
R 2 A132 6H
,
Ž 17 .
where A132 is the Hamaker constant for the interaction of silica plate 1 interacting with air bubble 2 in water 3. This value can be obtained using the combining rule: A132 s
ž (A
11
( / ž (A
y A 33
22
( /
y A 33 ,
Ž 18 .
where A11 , A 22 and A 33 refer to the Hamaker constants of the silica, air bubble and water, respectively, in vacuum. In the bubble–particle adhesion experiment conducted by Yordan, A11 s 5.04 = 10y2 0 J, A 22 s 0, and A 33 s 4.38 = 10y2 0 J, which give A132 s y3.12 = 10y2 1 J. That A132 is a negative number indicates that V D becomes positive Žor repulsive. according to Eq. Ž17..
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Fig. 5. Effects of DAH concentration on the K 232 values of Eq. Ž19. Žfrom Yoon and Aksoy, 1999..
For the free energy change due to hydrophobic interaction Ž VH ., one can use the following expression, VH s y
R 2 K 132 6H
,
Ž 19 .
which is obtained from Eq. Ž15. from the force–potential energy relationship. Here, K 132 is the hydrophobicity constant for the interaction of silica plate and air bubble in water. It has been shown that K 132 can be obtained from the following relationship ŽYoon and Mao, 1996; Yoon et al., 1997.:
(
K 132 s K 131 K 232 ,
Ž 20 .
in which K 131 and K 232 are the hydrophobicity constants for particle–particle and bubble–bubble interactions in water. The silica plate used in the bubble–particle interaction experiment exhibited a contact angle of 808. From the K 131 vs. u plot given in Fig. 4, one can obtain a value of K 131 s 8 = 10y2 0 J at u s 808. One can also obtain the value of K 232 from Fig. 5, in which K 232 is plotted vs. DAH concentration. The data given in this figure were obtained from the equilibrium film thickness measurements conducted on the soap films stabilized using various concentrations of DAH ŽYoon and Aksoy, 1999.. As shown, K 232 decreases with increasing surfactant concentration. At 10y5 M DAH, K 232 s 5 = 10y1 8 J. Substituting the values of K 131 and K 232 into Eq. Ž20., one obtains the value of K 132 s 6.3 = 10y1 9. One can now calculate the free energy of bubble–particle interaction Ž V . by using the extended DLVO theory, which includes the contributions from the hydrophobic interaction Ž VH . between air bubble and silica in a 10y5 M DAH solution: VT s VE q VD q VH .
Ž 21 .
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Fig. 6. Potential energy of bubble–particle interaction for silica plate and air bubble: V E the electrostatic energy; V D the dispersion interaction energy; VH the hydrophobic interaction energy; and V the total interaction energy. The data used for the calculation were: c 1 sy45 mV, c 2 sy20 mV, ky1 s96 nm, ´ s86.5, A132 sy3.12=10y2 1 J, K 132 s6.3=10y1 9 J.
Substituting Eqs. Ž16., Ž17. and Ž19. into Eq. Ž21., one can calculate the values of V as a function of H, the results being plotted in Fig. 6. Also shown in this figure are the values of VE , VD and VH . One can see that VD contributes relatively little to V under the experimental conditions employed by Yordan Ž1989.. Note also that both the electrostatic and dispersion energies are repulsive, while the hydrophobic interaction is the only driving force for the rupture of the wetting film between air bubble and silica. The V vs. H plot shows a maximum of 2.5 = 10y1 5 J at H s 75 nm. At this distance, disjoining pressure becomes zero; therefore, the film ruptures spontaneously when the film thickness is reduced to this critical thickness Ž Hc . and below. However, the value of Hc measured experimentally by Yordan was 110 nm, which was substantially larger than the value shown in Fig. 6. This discrepancy suggests that there may be other factors affecting the stability of wetting films. A possible explanation may be that wetting films are destabilized by hydrodynamic effect, which is well recognized in the destabilization of soap films ŽScheludko, 1962; Vrij, 1966.. If the airrwater interface is perturbed due to the hydrodynamic fluctuation, the two surfaces of the wetting film may be subject to stronger hydrophobic force locally, causing a decrease in free energy. If the free energy change is net negative, the perturbation will grow spontaneously and the film will eventually rupture. The energy maximum in the V vs. H plot may represent an energy barrier Ž E . for bubble–particle adhesion. One can see that it should vary strongly with c 1 and c 2 and u . Judicious control of these variables should help improve flotation rate Žand, hence, recovery. and selectivity. It is also possible to increase the flotation rate by providing sufficient kinetic energies to overcome the energy barrier. The relationship between E and the kinetic energy Ž E k . requirement may be given by the following relationship: k s Aexp Ž yErE k . . Ž 22 .
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where k is the rate constant and A is a constant. The latter embodies the effects of hydrodynamic parameters such as particle size, bubble size, energy dissipation, etc. A more detailed discussion of the various hydrodynamic and surface force parameters have been discussed previously ŽYoon and Mao, 1996; Mao and Yoon, 1997..
4. Summary and conclusions Stream functions derived for the intermediate flow regime make it possible to predict the probability of bubble–particle collision over a range of bubble sizes employed in flotation practice. The availability of the stream functions also makes it possible to predict the probability of adhesion based on induction time measurements. Using these probability functions, one can predict the power relationship between flotation rate constant and bubble size. The flotation rate constants determined experimentally using a frotherless flotation cell are in reasonable agreements with those predicted based on the interceptional collision model. However, the agreement is found only when the cell was operated under a relatively quiescent condition. Under more turbulent conditions, the rate constants obtained using smaller bubbles tend to decrease, while those obtained using larger bubbles are increased. Many investigators showed the existence of attractive hydrophobic forces between surfactant-coated surfaces. Their magnitudes increase with increasing contact angle. This relationship has been used to estimate the hydrophobic force constant Ž K 131 . for a silica plate immersed in a 10y5 M DAH solution form its contact angle. It has also been shown that air bubbles are hydrophobic, with their hydrophobic force constants Ž K 232 . decreasing with increasing DAH concentration. From this relationship, it was possible to estimate the value of K 232 for the air bubbles formed at 10y5 M DAH. The hydrophobic force constants obtained in the present work have been used to model the process of approaching an air bubble toward a silica plate in a 10y5 M DAH solution. The model predicts a critical rupture thickness of 75 nm, which is lower than the value of 110 nm measured in experiment. The discrepancy may be attributed to the hydrodynamic fluctuation of wetting films.
References Abrahamson, J., 1975. Collision rates of small particles in vigorously turbulent fluid. Chem. Eng. Sci. 30, 1371–1379. Ahmed, N., Jameson, G.J., 1989. Flotation kinetics. Miner. Process. Extr. Metall. Rev. 5, 77–99. Derjaguin, B.V., 1934. Friction and adhesion: IV. The theory of adhesion of small particles. Kolloid-Z. 69, 155–164. Derjaguin, B.V., Dukhin, S.S., 1961. Theory of flotation of small and medium-size particles. Trans. Inst. Min. Metall. 70, 221–246. Derjaguin, B.V., Dukhin, S.S., 1979. Kinetic Theory of the Flotation of the Fine Particles, Proceedings 13th Int. Miner. Process. Cong., Warszawa, 2, pp. 1261–1287. Derjaguin, B.V., Landau, L., 1941. Acta Physicochim. URSS 14, 633–662.
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Ducker, W.A., Senden, T.J., Pashley, R.M., 1991. Nature 353, 239–241. Flint, L.R., Howarth, W.J., 1971. Collision efficiency of small particles with spherical air bubbles. Chem. Eng. Sci. 26, 1155–1168. Gaudin, A.M., 1957. Flotation, 2nd edn. McGraw-Hill, New York, 1957. Hogg, H., Healy, T.W., Fuerstenau, D.W., 1966. Trans. Faraday Soc. 62, 1638. Israelachvili, J.N., Pashley, R.M., 1988. The hydrophobic interaction is long range, decaying exponentially with distance. Nature ŽLondon. 300, 341–342. Laskowski, J., Kitchener, J.A., 1969. The hydrophilic–hydrophobic transition on silica. J. Colloid Interface Sci. 29, 670–679. Liepe, F., Mockel, O.H., 1976. Untersuchungen zum Stoffvereinigen in flussiger Phase: Teil 6. Der Einfluß ¨ ¨ der Turbulenz auf den Stoffverubergang suspendierter Teilchen. Chem. Tech. 28, 205–209. ¨ Luttrell, G.H., 1986. Hydrodynamic Studies and Mathematical Modeling of Fine Coal Flotation. PhD Thesis, Virginia Polytechnic Institute and State University. Mao, L., Yoon, R.-H., 1997. Predicting flotation rates using a rate equation derived from first principles. Int. J. Miner. Process. 50, 171–181. Pazhianur, R., 1999. Hydrophobic Force in Flotation. PhD Thesis, Virginia Polytechnic Institute and State University. Pazhianur, R., Yoon, R.-H., 1999. Hydrophobicity and the Long-Range Hydrophobic Force, in preparation. Rabinovich, Y.I., Yoon, R.-H., 1994. Use of atomic force microscopy for the measurements of hydrophobic forces. Colloid Surf. 93, 263–273. Reay, D., Ratcliff, G.A., 1973. Removal of fine particles from water by dispersed air flotation — effects of bubble size and particle size on collection efficiency. Can. J. Chem. Eng. 51, 179–185. Scheludko, A., 1962. Proc. Kon. Ned. Akad. Wetensch. B 65, 76. Schubert, H., 1986. Aufbereitung fester mineralischer Rohstoffe, Vol. 2, 3rd edn., Grundstoffverlag, Leipzig. Schubert, H., Bischofberger, C., 1979. On the Optimization of Hydrodynamics in Flotation Processes. Proceedings 13th Int. Miner. Process. Cong., Warszawa, 2, pp. 1261–1287. Sutherland, K.L., 1949. Physical chemistry of flotation: XI. Kinetics of the flotation process. J. Phys. Chem. 52, 394–425. Trahar, W.J., Warren, L.J., 1976. The floatability of very fine particles — a review. Int. J. Miner. Process. 3, 103–131. Verwey, E.J.W., Overbeek, J.Th.G., 1948. Theory of Stability of Lyophobic Colloids. Elsevier, Amsterdam. Vivek, S., 1998. Effects of Long-Chain Surfactants, Short-chain Alcohols and Hydrolyzable Cations on the Hydrophobic and Hydration Forces. PhD Thesis, Virginia Polytechnic Institute and State University. Vrij, A., 1966. Discuss. Faraday Soc. 42, 23. Weber, M.E., 1981. Collision efficiencies for small particles with a spherical collector at intermediate Reynolds numbers. Sep. Processes Technol. 2, 29–33. Weber, M.E., Paddock, D., 1983. Interception al and gravitational collision efficiencies for single collectors at intermediate Reynolds numbers. J. Colloid Interface Sci. 94, 328–335. Wills, B.A., 1998. Mineral Processing Technology, 4th edn. Pergamon, New York. Yoon, R.-H., Aksoy, B.S., 1999. Hydrophobic forces in thin water films stabilized by dodecylammonium chloride. J. Colloids Interface Sci. 211, 1–10. Yoon, R.-H., Luttrell, G.H., 1989. The effect of bubble size on fine particle flotation. Miner. Process. Extr. Metall. Rev. 5, 101–122. Yoon, R.-H., Mao, L., 1996. Application of extended DLVO theory: IV. Derivation of flotation rate equation from first principles. J. Colloids Interface Sci. 181, 613–626. Yoon, R.-H., Ravishankar, S.A., 1993. Application of extended DLVO theory: III. Effect of octanol on the long-range hydrophobic forces between deodecylamine-coated mica surfaces. J. Colloids Interface Sci. 165, 1–10. Yoon, R.-H., Ravishankar, S.A., 1996. Long-range hydrophobic forces between mica surfaces in dodeclammonium chloride solutions in the presence of dodecanol. J. Colloids Interface Sci. 179, 391–402. Yoon, R.-H., Luttrell, G.H., Adel, G.T., Mankosa, M.J., 1989. Recent advances in fine coal flotation. In: Chander, S. ŽEd.., Advances in Coal and Mineral Processing Using Flotation, Chap. 23. Society of Mining Engineers, Littleton, CO, pp. 211–218.
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Yoon, R.-H., Flinn, D.H., Rabinovich, Y.I., 1997. Hydrophobic interactions between dissimilar surfaces. J. Colloids Interface Sci. 185, 363–370. Yordan, J.L., 1989. Studies on the Stability of Thin films in Bubble–Particle Adhesion. PhD Thesis, Virginia Polytechnic Institute and State University. Yordan, J.L., Yoon, R.-H., 1986. Induction Time Measurements for the Quartz–Amine Flotation System. Preprint No. 86-105 for the 115th AIME Annual Meeting, New Orleans, LA, March 2–6.
The role of hydrodynamic and surface forces in bubble–particle interaction R.-H. Yoon
)
Department of Mining and Minerals Engineering, Virginia Polytechnic Institute and State UniÕersity, Blacksburg, VA 24060, USA Received 2 August 1999; accepted 6 October 1999
Abstract In modeling flotation, the process of bubble–particle interaction is usually divided into three subprocesses, including collision, adhesion and detachment. Of these, the hydrodynamics of bubble–particle collision has been studied most extensively by many investigators, and the results are useful for the design and scale-up of flotation cells. The process of adhesion, on the other hand, is least understood because it is essentially controlled by the chemistry of the system, which is complex and difficult to model mathematically. However, it is possible to determine the probability of the bubble–particle adhesion from the induction times that can be measured experimentally under different chemical environments. Furthermore, the new information reported in the literature on the hydrophobic forces of both particles and bubbles allow prediction of adhesion probabilities using various surface chemistry parameters. Consideration of both the hydrodynamic and surface force parameters is essential in predicting flotation rates from first principles. q 2000 Elsevier Science B.V. All rights reserved. Keywords: hydrodynamic flotation; hydrophobic force; contact angle; flotation rate constant; effect of bubble size on flotation rate; critical film thickness for rupture
1. Introduction Froth flotation is widely used for separating different minerals from each other. However, its effectiveness is limited to a relatively narrow particle size range of 10–100 mm ŽTrahar and Warren, 1976; Wills, 1998.. The difficulty in floating fine particles is attributed to the low probability of bubble–particle collision, while the problem with coarse particle flotation is due to detachment. For flotation occurring under quiescent )
Tel.: q1-540-231-7056; Fax: q1-540-231-3948; E-mail: [email protected]
0301-7516r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 1 - 7 5 1 6 Ž 9 9 . 0 0 0 7 1 - X
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conditions, one can calculate the probability of collision using stream functions. The stream functions used by earlier workers are applicable for bubbles that are either too large or too small ŽSutherland, 1949; Gaudin, 1957., while those developed in recent years are useful for flotation size bubbles ŽWeber and Paddock, 1983; Yoon and Luttrell, 1989.. However, most of the flotation machines are operated under intensely agitated conditions, which makes it difficult to use the interceptional collision models based on stream functions. Under such conditions, models based on microturbulence may be more useful ŽSchubert and Bischofberger, 1979.. Not all the particles colliding with air bubbles result in flotation. Only the hydrophobic particles adhere to the surface of air bubbles. Thus, the probability of adhesion determines the selectivity of a flotation process, while its recovery depends critically on the collision probability. Also, some of the particles are detached from the surface due to the inertia force and turbulence in a flotation cell. Both the adhesion and detachment processes are determined largely by the surface chemistry of the bubbles and particles present in a flotation cell. It is difficult, however, to predict the probabilities of these subprocesses using the surface chemistry parameters such as contact angle, z-potential, Hamaker constant, etc. Derjaguin and Dukhin Ž1961. were the first to describe the bubble–particle interactions in flotation by considering surface forces. They considered that before a particle can adhere on the surface of an air bubble, it must pass through three distinct zones, i.e., hydrodynamic, diffusiophoretic and wetting zones. In the wetting zone, three surface forces, namely, van der Waals, electrostatic and structural forces were considered. ŽHere, the term structural force is used to represent the surface forces that cannot be explained by the continuum theory.. Derjaguin and Duhkin considered that the structural force is hydrophilic, but they considered it to be zero in their model calculation. In a later publication, Derjaguin and Dukhin Ž1979. suggested that small particles can adhere on the surface of air bubbles without penetrating the wetting film. They thought that the fine particles can be held to the bubble surface with relatively weak forces such as van der Waals and electrostatic forces, because they are not subject to significant tearing-off forces due to their small inertia. Again, the authors did not consider the structural forces playing an important role in the bubble–particle interaction. Laskowski and Kitchener Ž1969. studied the interaction between air bubble and silica plate hydrophobized by trimethylchlorosilane ŽTMCS.. Under the conditions employed in their experiments, both the bubbles and particles were negatively charged. They showed also that the van der Waals dispersion forces in wetting films are repulsive. Yet, bubbles readily adhered on the hydrophobic silica surface, forming contact angles as large as 70–808. They speculated, therefore, that there must be a third force, other than the electrostatic and van der Waals dispersion force, that is responsible for the bubble–particle adhesion. This work of Laskowski and Kitchener is regarded in the literature as the first recognition of the existence of hydrophobic force. The first direct measurement of hydrophobic force was reported by Israelachvili and Pashley Ž1988.. They brought two curved mica surfaces immersed in dodecylamine hydrochloride solutions to within 10 nm and observed attractive forces that were much larger than the van der Waals force. They called the additional attractive force hydrophobic force. Since then, numerous other investigators reported measurements of hydrophobic forces for
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various surfactants on different substrates. More recently, Yoon and Aksoy Ž1999. suggested that the hydrophobic forces are also present in soap films. In this communication, the role of hydrodynamic and surface forces in flotation is discussed. Flotation tests have been conducted under different hydrodynamic conditions, and a method of determining the probability of bubble–particle adhesion developed. The effects of various surface forces on the rupture of wetting films are discussed with a sample calculation. The results of the present investigation suggest that both the hydrodynamic and surface forces must be considered to develop a comprehensive flotation model.
2. Hydrodynamics The probability Ž P . of a particle being collected by an air bubble in the pulp phase of a flotation cell can be given by P s Pc Pa Ž 1 y Pd . ,
Ž 1.
where Pc is the probability of bubble particle collision, Pa is the probability of adhesion and Pd is the probability of detachment. For fine particles, Pd can be negligibly small because of the low inertia, in which case Eq. Ž1. becomes: P s Pc Pa .
Ž 2.
Pc is determined by the hydrodynamics of the system, which is strongly affected by the particle size, bubble size and the turbulence of the system. Pa is also affected by the hydrodynamics, but is largely a function of the surface chemistry involved. 2.1. Probability of collision Sutherland Ž1949. was the first to derive an expression for Pc from a stream function, Pc s 3 DprD b ,
Ž 3.
where Dp is the diameter of the particle and D b is the diameter of the bubble. Because of the assumption of potential flow, this equation is valid only for bubbles that are much larger than those used in flotation practice. Gaudin Ž1957. also derived an expression for Pc , 2
Pc s 1.5 Ž DprD b . ,
Ž 4.
using the stream function for the Stokes flow condition. However, Eq. Ž4. is applicable only for very small bubbles. Recognizing the limited applicability of Eqs. Ž3. and Ž4., Flint and Howarth Ž1971. numerically solved the Navier–Stokes equations to determine Pc . This approach was adapted by Reay and Ratcliff Ž1973. to derive a relationship, 2
Pc A Ž DprD b . , which has been verified experimentally for the flotation of fine particles.
Ž 5.
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More recently, Weber Ž1981. and Weber and Paddock Ž1983. derived an analytical expression for Pc for bubbles of low Reynolds numbers. For large Reynolds number bubbles, the Navier–Stokes equations were solved numerically using a curve-fitting technique. In this manner, Weber and Paddock were able to derive an expression for Pc , Pc s
3
Ž 3r16. Re
1q
2
Dp
ž /
Ž 6. Db 1 q 0.249Re 0.56 where Re is the Reynolds number of bubble. Eq. Ž6. is the first of its kind that can predict Pc for wide ranges of bubble and particle sizes. However, Weber and Paddock did not verify this relationship experimentally. The essence of predicting Pc accurately is to derive appropriate stream functions for different ranges of bubble sizes. The stream function for the Stokes and potential flow conditions had been known for a long time, but not for the bubbles of intermediate Reynolds numbers that are used for flotation. For this reason, Yoon and Luttrell Ž1989. derived a stream function for the intermediate Reynolds number range, 1 2 3 Re 0.72 1 1 2 2 c s u b R b sin u x y x q y qxy1 , Ž 7. 2 2 4 15 x x 2
ž
/
where c is the dimensionless stream function, u and x are the angular and dimensionless radial coordinates, respectively; R b is the bubble radius and u b is the bubble rise velocity. Using Eq. Ž7., Yoon and Luttrell derived an expression for Pc , Pc s
4 Re 0.72
3 q 2
15
Dp
2
ž /
,
Db
Ž 8.
which shows that Pc varies as Dy2 for small bubbles with Re s 0. However, as bubble b size becomes larger, Pc becomes less dependent on D b . It can be readily shown that for .46 very large bubbles, Pc varies Dy0 . b Thus, the power relationship between Pc and the D brDp ratio changes with bubble size. The relationship may be represented in a generalized from, Pc s A
Dp
ž / Db
n
,
Ž 9.
where A and n are the parameters that vary with Reynolds numbers. The values of A and n are given Table 1 for three different flow regimes, while Fig. 1 shows the values Table 1 Values of A and n of Eq. Ž9. for different flow conditions Flow conditions
A
Stokes Eq. Ž4.
2r3
Intermediate Eq. Ž7. Intermediate Eq. Ž6. Potential Eq. Ž3.
n 2
3r2qŽ4 Re 3 2 3
1q
0.72 .
r15
Ž 3r16 . Re
2 2
1q0.249 Re 0.56
1
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Fig. 1. Effects of bubble size Ž D b . on collision efficiency Ž Pc . under different flow conditions.
of Pc calculated for Dp s 11.4 mm using the four different expressions given in the table. Note that for the bubble sizes used in flotation practice, Sutherland’s equation ŽEq. Ž3.. overestimates Pc , while Gaudin’s ŽEq. Ž4.. underestimates it. However, Gaudin’s equation can still be useful for bubbles smaller than approximately 100 mm. Beyond this limit, the two equations derived by Weber and Paddock ŽEq. Ž6.. and Yoon and Luttrell ŽEq. Ž8.. may be useful. Although these two equations have been derived using completely different approaches and are in different functional forms, the predicted Pc values are in reasonable agreement with each other. All of the Pc expressions given in Table 1 are based on the interceptional collision model, which may be useful for flotation under relatively quiescent conditions. They may be applicable to flotation columns with large length-to-diameter ratios or for columns with sufficient baffles. Use of small bubbles may also help create quiescent conditions. For mechanically agitated cells, however, the turbulence model may be more appropriate. Based on the collision model derived by Abrahamson Ž1975., Schubert and Bischofberger Ž1979. used the following expression for predicting the number of bubble–particle collisions Ž Zpb . per unit volume of slurry and time: Dp q D b
2
/(
np2 q n b2 , Ž 10 . 2 where Np and Nb are the number of particles and bubbles per unit volume, respectively; Zpb s 5Np Nb
ž
(
(
and np2 and n b2 are the mean relative velocities of the particles and bubbles, respectively. Ž1976. and Schubert Ž1986. showed that the mean relative Liepe and Mockel ¨ velocities can be estimated by the following relationship:
(n
2 i
s 0.33
e 4r9 Di7r9 n 1r3
Dr
ž / r
2r3
Ž 11 .
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where the subscript i refers to bubble or particle, e the specific energy dissipation, r the density of the medium, and D r is the density difference between i and the medium and n is the kinematic viscosity of the medium. 2.2. Probability of adhesion The availability of stream functions also makes it possible to predict Pa . Using the stream function given by Eq. Ž7., Yoon and Luttrell Ž1989. calculated the sliding times of particles on bubble surfaces. If the sliding time is longer than the induction time, the particle will have long enough contact time to thin and rupture the disjoining film between the particle and bubble. Using this criterion, one can derive an expression for Pa in the following form: Pa s sin2 2tany1 exp
ž
y Ž 45 q 8 Re 0.72 u b t i 15D b Ž D brDp q 1 .
/
Ž 12 .
where t i is the induction time and u b is the bubble rise velocity. This expression is applicable for the intermediate Reynolds number range. The induction time is strongly a function of particle hydrophobicity and can be routinely determined in laboratory using a relatively simple device ŽYordan and Yoon, 1986.. Thus, Eq. Ž12. can be useful for determining Pa from the values of t i determined under various reagent conditions. Fig. 2 shows the Pa values predicted using Eq. Ž12. from the induction times measured on a coal sample oxidized in water over a long period of time. The calculated
Fig. 2. Values of Pa calculated using Eq. Ž12. from the induction times measured on a coal sample oxidized in water at ambient temperature.
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135
Pa values are in reasonable agreement with the microflotation results. Eq. Ž12. suggests that Pa is also a function of particle size and bubble size. It has been shown that as the particle size decreases Pa increases, and that Pa increases with decreasing bubble size until it decreases again as the bubble size becomes too small ŽYoon and Luttrell, 1989.. 2.3. Flotation rate Once P is known from the detailed knowledge of its subprocesses, the first order flotation rate constant Ž k . can be obtained using the following relationship ŽYoon et al., 1989; Yoon and Mao, 1996.: ks
3Vg 2 Db
Ps
1 4
Sb P
Ž 13 .
where Vg is the superficial gas rate which is the gas flow rate divided by the cross-sectional area of the flotation cell, and S b is the surface area of the bubbles rising out of a flotation cell per unit area and unit time. Since Pc varies as Dp2 at the bubble sizes commonly employed in flotation ŽTable 1., k will also vary as Dp2 at given Pa and Vg , providing an explanation for the difficulty in recovering fine particles. A common solution to this problem is to increase Vg as suggested by Eq. Ž13.. However, the improvement in k that can be brought about by this method is only linear, i.e., k A Vg . Also, there is a limitation in increasing Vg without flooding the cell. A better solution to the problem of fine particle flotation may be to use smaller bubbles. From Eq. Ž8., it can be seen that when bubble size is in the range of 100 mm or Ž . smaller, Pc varies as Dy2 b . Substituting this relationship into Eq. 13 , one can see that k varies as Dy3 b , which is a powerful message to use smaller bubbles. However, as the bubble size becomes larger, Pc will become progressively less dependent on bubble .46 size. If bubbles become very large, k varies as Dy1 . b For the purpose of illustrating the effect of bubble size on flotation rate constant, Eq. Ž13. has been used to calculate k as a function of D b . The results are given in Fig. 3 as a solid line. The calculation has been made for Dp s 8 mm and Pa s 0.6. Also shown are the ‘frothless’ flotation test results obtained using a methylated quartz sample with 8-mm median size ŽLuttrell, 1986.. When the agitation speed was as slow as 100 rpm, the experimental k values were in close agreement with those predicted, particularly with regard to the slope. As the agitation speed was increased, k became less dependent on D b . The results also show that strong agitation is detrimental to fine bubble flotation, while it helps the flotation with larger bubbles. This finding suggests that the benefits of suing small bubbles for flotation can be best realized under quiescent conditions. Even for the case of turbulent collision, decreasing bubble size should increase the collision rate and, hence, the rate constant. However, the flotation rate is not as strongly dependent on D b as the case of the interceptional collision under quiescent conditions. Eqs. Ž10. and Ž11. suggest that for the case of D b 4 Dp , the collision rate varies .61 approximately as Dy0 . Substituting this relationship into Eq. Ž13., one can see that b y1 .61 k ADb . This finding may suggest that the benefits of using small bubbles can be best
136
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Fig. 3. Effects of bubble size on the flotation rate constants Ž k . for a methylated quartz sample. The tests were conducted at different agitation speeds.
realized when the flotation cell can provide a relatively quiescent condition. As Ahmed and Jameson Ž1989. suggested, the remedy to low recovery rates associated with fine particles would be to either use small bubbles of the order of 100 mm under quiescent conditions, or to use moderate-to-high agitation with larger bubbles.
3. Surface forces in flotation While improvements in the hydrodynamic conditions discussed in the foregoing paragraph can bring about higher recoveries, the selectivity of a flotation process depends on the control of surface forces. The electrostatic forces acting between particles and air bubbles can be calculated from their z-potentials, which are routinely measured in laboratories. When particles and bubbles are oppositely charged, they can be attracted with each other and form bubble–particle aggregates, which is a prerequisite for flotation. However, the electrostatic attraction is so weak that the particles can be readily detached from the surface of air bubbles, especially for large particles. Only when particles are sufficiently hydrophobic, the wetting films between the particles and bubbles rupture, and three-phase contacts are established. The free energy change associated with the three-phase contact provides the forces that are necessary to withstand the tearing-off forces created by the hydrodynamic drag or the turbulence in a flotation cell. Furthermore, both the particles and bubbles are negatively charged in many flotation systems. Therefore, it is unlikely that electrical forces are responsible for the rupture of wetting films. On the other hand, the Õan der Vaals dispersion forces are repulsive in
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bubble–particle interactions, as will be shown later. The classical DLVO theory ŽDerjaguin and Landau, 1941; Verwey and Overbeek, 1948. considers only the electrostatic and dispersion forces to describe the interaction between lyophobic particles in a medium. Thus, there must be a third force that is responsible for the rupture of wetting films during flotation. Recent work conducted at Virginia Tech suggested that the hydrophobic force is responsible for the destabilization of both wetting and soap films ŽYoon and Mao, 1996; Yoon and Aksoy, 1999.. Hydrophobic forces were first measured by Isrealashivili and Pashley in 1982 between mica surface coated with cetyltrimethylammonium bromide ŽCTAB.. The measurements were conducted using the surface force apparatus ŽSFA.. This apparatus can be used for transparent and molecularly smooth surfaces such as mica and sapphire. Atomic force microscope ŽAFM., which was originally designed to measure surface morphologies, was used to measure surface forces in 1991 by Ducker et al. Ž1991.. These investigators replaced the pyramidal tip with a glass sphere, so that the forces measured between the sphere and plate can be related to the DLVO theory and converted to surface free energies using the Derjaguin approximation ŽDerjaguin, 1934.. Rabinovich and Yoon Ž1994. used the AFM technique to measure the surface forces between hydrophobic surfaces, i.e., silaned glass sphere and silica plate. They found that the measured forces were strongly attractive and increased with increasing contact angle. Since then, numerous other investigators used AFM to measure the hydrophobic forces. It has been found that measured hydrophobic forces Ž FH . decay exponentially with the closest distance Ž H . separating the sphere and the flat surface. Thus, the experimental data are frequently fitted to a double-exponential function: FH rR s C1exp Ž yHrD1 . q C2 exp Ž yHrD 2 . ,
Ž 14 .
where R is the radius of the sphere, and C1 , C2 , D 1 , and D 2 are fitting parameters. D 1 and D 2 are decay lengths, and the pre-exponential parameter C1 is related to surface free energy at the solid–liquid interface. It has also been shown that measured hydrophobic forces can be described by the following relationship ŽRabinovich and Yoon, 1994., FH rR s yKr6 H 2 ,
Ž 15 .
in which K is the only fitting parameter. Eq. Ž15. is of the same form as the expression for the van der Waals dispersion forces. Therefore, the value of K can be directly compared with the Hamaker constant. Fig. 4 shows a relationship between K and contact angle Ž u ., which is the more conventional measure of hydrophobicity. Implications of the data shown in this figure are discussed in another communication ŽPazhianur and Yoon, 1999.. It may be useful to discuss the role of surface forces in bubble–particle interactions in a quantitative manner. A sample calculation to be made here will be the case of approaching an air bubble toward a flat hydrophobic surface. Yordan Ž1989. conducted an experiment, in which an air bubble of 2 mm in radius was pressed against a silica surface immersed in a 10y5 M DAH solution at pH 10. The z-potentials of the air bubble and silica were y45 and y20 mV, respectively, in 10y5 M DAH solutions.
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138
Fig. 4. Effects of advancing contact angle Ž u . on the hydrophobic force constant of Eq. Ž15.. The K values are from Yoon and Ravishankar Ž1993; 1996., Rabinovich and Yoon Ž1994., Vivek Ž1998. and Pazhianur Ž1999..
From these values, one can calculate the free energy of electrostatic interaction using the equation ŽHogg et al., 1966.: VE s p´o ´ R 2 Ž c 12 q c 22 .
2 c1c2
ln 2
c 12 q c 2
ž
1 q eyk H 1 y ey k H
/
q ln w 1 y ey2 k H x
Ž 16 .
where R 2 is the radius of bubble; c 1 and c 2 are the surface potentials of the silica plate and spherical air bubble, respectively; H is the shortest distance between the two surfaces; ´o Žs 8.854 = 10y1 2 C 2 Jy1 my1 . is the permittivity of free space; ´ is the dielectric constant of water; and k is the reciprocal Debye length. At 258C, ´ s 86.5 and ky1 s 96 nm. In this sample calculation, the z-potentials of the silica and air bubble were used to substitute c 1 and c 2 , respectively, as approximation. One can also calculate the free energy change due to the van der Waals dispersion interaction using the following equation: VD s y
R 2 A132 6H
,
Ž 17 .
where A132 is the Hamaker constant for the interaction of silica plate 1 interacting with air bubble 2 in water 3. This value can be obtained using the combining rule: A132 s
ž (A
11
( / ž (A
y A 33
22
( /
y A 33 ,
Ž 18 .
where A11 , A 22 and A 33 refer to the Hamaker constants of the silica, air bubble and water, respectively, in vacuum. In the bubble–particle adhesion experiment conducted by Yordan, A11 s 5.04 = 10y2 0 J, A 22 s 0, and A 33 s 4.38 = 10y2 0 J, which give A132 s y3.12 = 10y2 1 J. That A132 is a negative number indicates that V D becomes positive Žor repulsive. according to Eq. Ž17..
R.-H. Yoonr Int. J. Miner. Process. 58 (2000) 129–143
139
Fig. 5. Effects of DAH concentration on the K 232 values of Eq. Ž19. Žfrom Yoon and Aksoy, 1999..
For the free energy change due to hydrophobic interaction Ž VH ., one can use the following expression, VH s y
R 2 K 132 6H
,
Ž 19 .
which is obtained from Eq. Ž15. from the force–potential energy relationship. Here, K 132 is the hydrophobicity constant for the interaction of silica plate and air bubble in water. It has been shown that K 132 can be obtained from the following relationship ŽYoon and Mao, 1996; Yoon et al., 1997.:
(
K 132 s K 131 K 232 ,
Ž 20 .
in which K 131 and K 232 are the hydrophobicity constants for particle–particle and bubble–bubble interactions in water. The silica plate used in the bubble–particle interaction experiment exhibited a contact angle of 808. From the K 131 vs. u plot given in Fig. 4, one can obtain a value of K 131 s 8 = 10y2 0 J at u s 808. One can also obtain the value of K 232 from Fig. 5, in which K 232 is plotted vs. DAH concentration. The data given in this figure were obtained from the equilibrium film thickness measurements conducted on the soap films stabilized using various concentrations of DAH ŽYoon and Aksoy, 1999.. As shown, K 232 decreases with increasing surfactant concentration. At 10y5 M DAH, K 232 s 5 = 10y1 8 J. Substituting the values of K 131 and K 232 into Eq. Ž20., one obtains the value of K 132 s 6.3 = 10y1 9. One can now calculate the free energy of bubble–particle interaction Ž V . by using the extended DLVO theory, which includes the contributions from the hydrophobic interaction Ž VH . between air bubble and silica in a 10y5 M DAH solution: VT s VE q VD q VH .
Ž 21 .
140
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Fig. 6. Potential energy of bubble–particle interaction for silica plate and air bubble: V E the electrostatic energy; V D the dispersion interaction energy; VH the hydrophobic interaction energy; and V the total interaction energy. The data used for the calculation were: c 1 sy45 mV, c 2 sy20 mV, ky1 s96 nm, ´ s86.5, A132 sy3.12=10y2 1 J, K 132 s6.3=10y1 9 J.
Substituting Eqs. Ž16., Ž17. and Ž19. into Eq. Ž21., one can calculate the values of V as a function of H, the results being plotted in Fig. 6. Also shown in this figure are the values of VE , VD and VH . One can see that VD contributes relatively little to V under the experimental conditions employed by Yordan Ž1989.. Note also that both the electrostatic and dispersion energies are repulsive, while the hydrophobic interaction is the only driving force for the rupture of the wetting film between air bubble and silica. The V vs. H plot shows a maximum of 2.5 = 10y1 5 J at H s 75 nm. At this distance, disjoining pressure becomes zero; therefore, the film ruptures spontaneously when the film thickness is reduced to this critical thickness Ž Hc . and below. However, the value of Hc measured experimentally by Yordan was 110 nm, which was substantially larger than the value shown in Fig. 6. This discrepancy suggests that there may be other factors affecting the stability of wetting films. A possible explanation may be that wetting films are destabilized by hydrodynamic effect, which is well recognized in the destabilization of soap films ŽScheludko, 1962; Vrij, 1966.. If the airrwater interface is perturbed due to the hydrodynamic fluctuation, the two surfaces of the wetting film may be subject to stronger hydrophobic force locally, causing a decrease in free energy. If the free energy change is net negative, the perturbation will grow spontaneously and the film will eventually rupture. The energy maximum in the V vs. H plot may represent an energy barrier Ž E . for bubble–particle adhesion. One can see that it should vary strongly with c 1 and c 2 and u . Judicious control of these variables should help improve flotation rate Žand, hence, recovery. and selectivity. It is also possible to increase the flotation rate by providing sufficient kinetic energies to overcome the energy barrier. The relationship between E and the kinetic energy Ž E k . requirement may be given by the following relationship: k s Aexp Ž yErE k . . Ž 22 .
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where k is the rate constant and A is a constant. The latter embodies the effects of hydrodynamic parameters such as particle size, bubble size, energy dissipation, etc. A more detailed discussion of the various hydrodynamic and surface force parameters have been discussed previously ŽYoon and Mao, 1996; Mao and Yoon, 1997..
4. Summary and conclusions Stream functions derived for the intermediate flow regime make it possible to predict the probability of bubble–particle collision over a range of bubble sizes employed in flotation practice. The availability of the stream functions also makes it possible to predict the probability of adhesion based on induction time measurements. Using these probability functions, one can predict the power relationship between flotation rate constant and bubble size. The flotation rate constants determined experimentally using a frotherless flotation cell are in reasonable agreements with those predicted based on the interceptional collision model. However, the agreement is found only when the cell was operated under a relatively quiescent condition. Under more turbulent conditions, the rate constants obtained using smaller bubbles tend to decrease, while those obtained using larger bubbles are increased. Many investigators showed the existence of attractive hydrophobic forces between surfactant-coated surfaces. Their magnitudes increase with increasing contact angle. This relationship has been used to estimate the hydrophobic force constant Ž K 131 . for a silica plate immersed in a 10y5 M DAH solution form its contact angle. It has also been shown that air bubbles are hydrophobic, with their hydrophobic force constants Ž K 232 . decreasing with increasing DAH concentration. From this relationship, it was possible to estimate the value of K 232 for the air bubbles formed at 10y5 M DAH. The hydrophobic force constants obtained in the present work have been used to model the process of approaching an air bubble toward a silica plate in a 10y5 M DAH solution. The model predicts a critical rupture thickness of 75 nm, which is lower than the value of 110 nm measured in experiment. The discrepancy may be attributed to the hydrodynamic fluctuation of wetting films.
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