- Email: [email protected]
The relationship between flotation and adhesion of galena particles to the air-solution interface
International Journal of Mineral Processing, 25 (1989) 275-288
275
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
The R e l a t i o n s h i p b e t w e e n F l o t a t i o n and A d h e s i o n of Galena Particles to the A i r - S o l u t i o n Interface I.N I S H K O V
and R.J. P U G H
Central Laboratory of Mineral Processing, Bulgarian Academy of Sciences, Box 32, 1126 Sofia (Bulgaria) Institute for Surface Chemistry, Box 5607, S- 114 86 Stockholm (Sweden) (Received January 28, 1988; accepted after revision July 19, 1988)
ABSTRACT Nishkov, I. and Pugh, R.J., 1989. The relationship between flotation and adhesion of galena particles to the air-solution interface. Int. J. Miner. Process., 25: 275-288. The adhesion of aggregates of galena particles to an air-solution interface was investigated experimentally by measuring the criticalcentrifugalforce necessary to detach and sink the aggregates. Theoretical detachment-force calculations based on the Nutt (1960) and Scheludko (1968/69) theory for a single spherical particle could be correlated with the experiment results but the quantitative theoretical predictions appeared to give values one order of magnitude less than the experimental data. In addition, a series of flotationexperiments were carried out with the same particle-sizefractions using a small-scale flotationcell.It was found that the flotationefficiencycould be directly related to the experimentally determined detachment forces for a seriesof particle-sizefractions in the absence and presence of xanthate collector.It is suggested that the adhesion forces may play a more important role than the capture forces in determining the flotation performance of rough and angular particles.Finally the centrifuge technique could prove to be a useful method for characterizing the floatabilityof minerals.
INTRODUCTION
The flotation efficiency of mineral particles (generally in the size range of 10 to 150/~m) is controlled by two "critical force barriers". The initial force barrier is essentially involved in the prevention of contact between particle and bubble. This usually occurs in the preliminary stages of the capture process and if this repulsive force is sufficiently high, capture may be prevented. However, at a later stage in the process, a second force barrier may increase the flotation efficiency. In this case the forces may prevent the detachment of particles from the bubble during the subsequent transportation of the particle by the bubble to the froth. This "detachment-force barrier" becomes particularly important under turbulent flow conditions in the flotation cells where the bub0301-7516/89/$03.50
© 1989 Elsevier Science Publishers B.V.
27~
hie-particle aggregates are subjected to destructive tensile and shear forces. In such cases only particles with strong adhesion to the bubble will reach the froth. Consequently, most fundamental research in flotation may be classified as concerned with: (a) bubble-particle interaction before or leading to capture; or (b} bubble-particle interaction after contact or capture. The phenomena involved are quite complex and it is presently not clear which of the force barriers make the greatest contribution towards the overall flotation efficiency. In the case of the force barrier controlling the kinetics before contact., a series of elementary steps in the capture process have been identified and studied in the laboratory. This has lead to the quantification of such parameters as the induction time (i.e. the time required for a stable particle-bubble contact to form after initial collision), the kinetics of the expansion of the three-phase (air-liquid-solid) contact, contact angle hysteresis, etc. (Schulze, 1984). The existence of a repulsive energy' barrier which opposes thinning of a thin liquid film between bubble and particle and retards the capture process has been clearly confirmed (Scheludko et al., 1970; Blake and Kitchener, 1972) although "induction times" as usually measured, are unrealistically long compared with "collision times". However, it should be stressed that all of these studies have been based on idealized sphere-plate or sphere-sphere models. It must be emphasized that practical flotation conditions may be drastically different from this idealized situation. For example, detailed studies by Anfruns and Kitchener ( 1977 ) have shown that coarse, angular particles with sharp edges and corners can be captured by bubbles with 100% efficiency whereas the capture efficiency of strongly hydrophobized glass ballotini is usually much lower. This was attributed to the jagged projection of the asperities of the mineral particles causing local thinning and rupture of' the liquid film between air and solid and almost eliminating the influence of the "capture force barrier". Thus, the study suggests that for rough particles the detachment-force barrier may play the most important role in controlling flotation efficiency. The purpose of the present investigation is to test this hypothesis. The main objectives may be summarized as follows. (1) To attempt to correlate flotation efficiency of large aggregates of well defined coarse mineral particles which cling to the bubbles in froth flotation, to the detachment forces from the air-solution interface. In the present study it was found convenient to quantify the critical detachment force using a centrifugal technique. (2) To compare the magnitude of the force of detachment of the aggregates of particles with theoretical values for idealized smooth monodispersed particles. (3) To study the effects of particle size, collector concentration and activator on the magnitude of the detachment force and the flotation efficiency.
277
Theory of the balance o/forces on a smooth spherical particle suspended at the fluid interface Theoretical studies by Nutt (1960) and Scheludko et al. (1968/89) have expressed the mechanics of the attachment of hydrophobic spherical particle to a gas-liquid (G-L) interface or to a gas bubble in terms of an equilibrium between the gravitational force on the one hand, and the sum of the surface tension and buoyancy forces on the other. Fig. 1 shows schematically the attachment of a spherical particle to the G-L interface and the quantities involved in the calculation. To detach the particle, a centrifugal force must be applied which must exceed a certain critical value. This value is a function of the size and density of the particle, as well as of the surface tension of the liquid and its angle of contact with the particle surface. The floatability of the particle clearly will be related to the strength of its adhesion to the bubble and it has been suggested that measurements of the critical force (i.e. the maximum attachment force) could be used to assess the effectiveness of the flotation process. The maximum attachment force of a single spherical particle on the G-L interface (FAt) can be expressed as an approximation by the equation (Nutt, 1960; Scheludko et al., 1968/89): FAt :
(1)
7[(TRp( 1 - c o s 0 )
where Rp is the particle radius, a is the surface tension at the G-L interface and 0 is the equilibrium wetting angle. From experimental studies the maximum attachment force of siliconed glass beads (radius 0.02-0.1 cm) was found to be in good agreement with this equa-
A/r
J Fcen Fig. 1. The balance of force acting to a spherical particle attached to the solution/air interface. F,,~. represents the centrifugal force acting on the sphere, F~ is the surface tension vector force acting around the perimeter and F~, is the buoyancy force. Rp is the radius of the particle and 0 is the contact angle. For large centrifugal forces F,. >> F~+ Fh the particle is detached from the surface.
278 tion (Nutt, 1960). These experiments were also extended to determine the effect of aggregation of the particles on the force of a t t a c h m e n t at the air-liquid interface. From these studies it was found that the attachment force of the aggregates decreased as the number of particles increased, reaching a limiting value of about 50% of the force of a t t a c h m e n t of an individual particle. However, it is important to note that these experiments with aggregates were carried out on relatively large particles ( radius 0.04 cm). Varbanov et al. (1982) have also studied the behaviour of an aggregate of monodispersed glass spheres attached to the air-water interface. It was experimentally established that aggregates of particles with a radius smaller than 100 #m do not obey eqn. 1 and detach at a greater force then predicted by the theoretical model. In the case of large aggregates of irregularly shaped particles, initial studies showed a considerable difference in the detachment-force values for galena particles respectively in the absence and presence of xanthate collector (Alexandrova et al., 1984). However, generally these results were found to deviate from that of a monodispersed smooth particle. This may be explained by the differences in degrees of wetting of the irregularly shaped edges and corners of the aggregates which for example, might be especially favorable for adsorption of a hydrophobizing agent. In fact, the particles may have a number of orientations at the interface, each having a particular potential energy barrier and adhesive strength. In addition, it is important to consider particle interactions and collective effects due to aggregation of particles at the interface. With a practical system the problem is therefore unfortunately too complex for theoretical treatment. EXPERIMENTAL
Characteristics of mineral systems A sulphide mineral sample of hand-picked Swedish natural galena was supplied by Boliden Mineral AB. The sample was crushed in an agate mortar and TABLE I
Particle size fraction (#m)
Averageparticle size (/zm)
5- 12 40- 63 63- 90 90-125
9 48 72 102
279 separated by sieving under water into narrow particle-size fractions for experimental studies. Semi-quantitative spectroscopic analysis of the mineral showed the presence of the following impurities: Si 0.17%; Ca 0.01%; Fe 0.11%; Mg <0.01%; Mn <0.01%; Cu 0.01%; Zn 0.08%. The particle-size distribution and the average particle size were determined (Table I ) using a Particle Size Counter manufactured by Elzon Particle Data, USA. Microscopic examination showed the particles from all size fractions to be irregular in shape with rough and angular surfaces. The solid content of the suspensions for both attachment-forces measurements and flotation experiments was maintained a 6 g/1.
Reagents Potassium ethylxanthate (KEtX) (Hoechst, analytical grade) was used as collector. Fresh solutions were prepared daily for the flotation and attachment experiments. Copper sulphate (analytical grade) was used as activator. Sodium hydroxide (reagent grade) was used for pH adjustment. The experimental studies were carried out in a slightly alkaline solution (pH 8.5).
Attachment force measurements The centrifugal equipment consisted of a Beckman ultracentrifuge L5-50B (rotation up to 50.000 rpm) with a rotor SW50-1, equipped with buckets that swing out to a horizontal position as the rotor is accelerated. Thin wall ultra clear plastic non-wettable tubes (dimensions 13 × 51 mm, volume 2.5 cm 3) were found to be the most convenient for the present study. The radial distance (Rrot) of the meniscus was constant (8 cm) for all experiments. The suspension containing 0.6 g of galena in 100 ml of collector solution at a pre-adjusted pH, was stirred for 10 min in a 150-ml beaker using a magnetic stirrer in an attempt to simulate the shear field experienced in the micro flotation cell. In the studies of the influence of activator, before conditioning with collector a predetermined amount of copper sulphate was introduced and the suspension was conditioned for 5 min by stirring. After conditioning, the galena particles were allowed to sediment and then brought to the air-solution interface as a coagulated clump by tilting repeatedly the glass beaker. In order to establish an equilibrium contact angle the galena particles were then left for 30 min attached to the interface before transferring them with a spatula onto the surface of the initial solution in the centrifuge tubes. The aggregates were then subjected to the detaching action of centrifugal forces at a range of centrifuge speeds. For each experiment the centrifuge speed was increased gradually to a pre-set value and maintained at this speed for 5 min. The critical speed where the particles were thrown from the interface into the body of the solution was noted.
:L~!)
During the experiments it was observed that under gravity the particles tended to aggregate together into a loosely packed raft at. the interface. ()n the initial application of the centrifugal field, the raft became compressed in the middle of the meniscus and small particles appeared to be mechanically entrapped between the large particles, thus reducing the voids. At high centrifugal speeds, the coagulated aggregates became compressed and finally the aggregate appeared to be detached as a whole at the critical centrifuge speed. For a single particle of radius R,, the fbrce required to overcome the tbrces that bind the particle to the meniscus so that the particle becomes submerged in the liquid can be expressed by: 4 F ..... =
(2)
a = \ 30 ] Rrot
(3)
where: a is centrifugal acceleration; N is number of revolutions at which detaching takes place; Rrot is radial distance ( radius of rotation); Ap is the effective density of particles.
Flotation experiments In these experiments, a cylindrical glass cell (volume 100 cm ~) with porous glass frit bottom and an external gas source was used. Bubbles were generated by flowing nitrogen through the glass frit at a controlled flow rate of 50 cm3/ min. The flotation procedure was as follows. A standardized amount of the particle-size fractioned galena (0.6 g) was placed in the microflotation cell and the collector solution added. The pulp was then conditioned by magnetic stirring for 10 min. In the studies of the influence of activator concentration before conditioning with collector a predetermined amount of copper sulphate was introduced and the pulp was conditioned for 5 min. After conditioning, nitrogen gas was bubbled through the cell for 5 min in all flotation tests. Finally, the froth product was removed and collected and the quantity of galena determined. For studying the flotation kinetics of different particle-size fractions, the froth product was removed after 0.5, 1, 2, 3, 4, and 5 min, giving six separate concentrations for each experiment. No frother was used in these systems. The percentage of mineral floated was determined by gravimetric analysis and expressed as a wt% of dry mineral to the total weight of the original sample.
281 RESULTS AND DISCUSSION
Studies of the attachment force in the absence and presence of collector The critical centrifugal force required to detach and submerge the aggregates of galena particles from the air-solution interface was determined for the various particle-size fractions. The number of revolutions (N) at which detachment takes place as a function of average particle size of the fraction constituting the aggregates is shown in Fig. 2. Theoretical numbers of revolution for the detachment of smooth spherical monodispersed particles were calculated (from N u t t theory) by equating Fat and Fc~, using eqns. 1, 2 and 3 to give: N
15
/
1-cos0
- nRp ~3azIpR~ot
(4)
These values are also shown in Fig. 2. In the application of the N u t t theory it is necessary to assume an equilibrium value of the contact angle during the detachment process. However, in the practical situation, as the three-phase contact expands along the surface of the particles a number of intermediate states displaying quasi-equilibrium contact +5
+4 Z
÷3"
+2-
7'2 average particle size
(prrO
Fig. 2. The log (number of revolutions) at which detachment occurs versus the averageparticle size of the various size fractionsof galenaparticles constituting the aggregates: 1 --- data obtained without collector;2 = data obtained with 10- '~M potassiumethyl xanthate; 3 = theoreticalcurve (Nutt model) assuming 0 = 8 ° (no collector); 4 = theoretical curve (Nutt model) assuming0 = 60° (10 -~ Mpotassium ethyl xanthate).
angles may occur. Since the contact angle cannot be quantified, for the purpose of the present investigation values of 0 = 8 for water and 0 = 60 for 10 :: M K E t X solution on a galena surface were selected according to the data of Wark and Cox (1932). The experimentally determined a t t a c h m e n t |brces of galena particles at the air-solution interface as a function of the average particle size respectively in the absence and presence of xanthate collector are presented in Table II and Fig. 3. In addition, data and theoretical plots for ideal T A B L E II
Particle
Co K E t X = 1 0 - :~M / I
C,~ K E t X = 0
size (pm) a t t a c h m e n t force ( d y n )
5 - 12 4 0 - 63 6 3 - 90 90-125
theory
experiment
9.9"10 ,4 5.3"10 -:~ 7.9"10 -:~ 1.1"10 -2
1.6.10-2 5.2"10 -~ 6.2"10 -~ 7.9"10 -2
F~p/Fth
16.1 9,8 7.8 7.2
a t t a c h m e n t force ( d y n ) theory
experiment
4.9"10 -2 2.6"10 -~ 4.0"10 ~ 5.6"10 - I
4.3"10 2.12 3.4 5.1
l
8.8 8.2 8.5 9.1
+1-
-1 ¢g tl.
/" J" 1./° --3' T
i
lo2
average particle size
(pm)
Fig. 3. The log (criticalcentrifugal force of attachment) versus the average particle size of the various size fractionsof galena particlesconstituting the aggregates (see Fig. 2 for legend).
283
100"
80
>o 60 o o
+ 4O 2O
4'8
7'2 (pro)
1(~2
average particle size
Fig. 4. T h e flotation recovery of galena particles after 5 rain flotation versus the average particle size of the various size fractions: A = no collector; • = 10 -3 M p o t a s s i u m ethyl xanthate.
monodispersed particle calculated from the Nutt theory are also shown for comparison purposes. From the data presented in Table II and Fig. 3 it can be seen that the experimental detachment forces for the aggregated particles appear to be greater than the theoretical values predicted for the idealized monodispersed model (but never more than by a factor of 16). This result is remarkable in view of the complexity of the galena system. Indeed with an aggregate of particles, a collective effect arising from capillary forces binding the particles together would also be expected to play a role within the balance of attachment forces. The interparticle interactions in a horizontal plane arising from the attachment of floating bubbles to an interface has been studied by Nicholson (1949). In addition Gifford and Scriven (1971) calculated the theoretical capillary attraction between small idealized floating particles consisting of parallel stationary cylinders of infinite length. These workers found that the capillary forces that hold the particles together were surprisingly high. Later Chan et al. ( 1981 ) extended Nicolson's theoretical approach and demonstrated its applicability to calculate the capillary forces between bodies of arbitrary shape and physical properties. However, the influence of these forces on their stability
r
1°°i
/
?
¢
# v
>,
80
I
I
> 0 r~
y ~-
60
0
40
/ 4 /
20j/
"tl ,
-2
log
Fat .
Fig. 5. The flotation recovery of galena particles after 5 min flotation versus the log (force of attachment ) with and without potassium ethyl xanthate collector. The points represent the average size of the various particle fractions constituting the aggregates. A = no collector; • = 10- :~ M potassium ethyl xanthate.
with respect to sinking is not known. Unfortunately, in the present experiments with agglomerates consisting of non-spherical coagulated particles, the situation is far too complex for theoretical analysis. From Fig. 3 it can also be observed that the detachment force increases with the average particle size of the various size fractions of particles constituting the aggregates for all the experimental studies. This trend tends to correlate with the theoretical data based on the monodispersed spherical model. Finally, the experimental detachment forces are greater for systems containing KEtX collector. These results generally agree with the theoretical curves. In Fig. 4 the flotation recoveries of different-size galena fractions in water and 10 -:~ M KEtX solution are shown as a function of the average particle size after 5 min flotation. For both water and collector solution the flotation recoveries appear to increase with the average particle size of the fractions constituting the aggregates. Fig. 5 illustrates the relationship between recovery after 5 min flotation and attachment force of aggregates of galena particles at the air-solution interface for the various particle fractions. This plot demonstrates the increase of flo-
285
ration recovery as the attachment force is increased. The experimental results suggest that the strength of adhesion of the aggregates to the interface can be related to the flotation efficiency.
Study of the effect of collector and activator on adhesion force The variation of the attachment force of an aggregate of galena particles at the air-solution interface, treated in advance with KEtX and copper sulphate as a function of initial concentration and average particle size is shown in Fig. 6. From these results it may be seen that the attachment force depends on particle size, initial collector and activator concentration. The major increase in attachment force evidently takes place over the range 10-s to 10 -4 M KEtX. The maximum force was obtained with a solution containing both flotation reagent and activator (10 -3 M KEtX and 10 -2 M CuS04). The increase of the attachment force of fine particles increases very slightly with increasing collector concentration. For the largest particles the increase
8"
¥
1
•
2
6. 3 •
5
7
!
6
N
5
~
4
i
e-
3-
'~
2-
O
1
162 average particle size (~m) Fig. 6. The a t t a c h m e n t force versus the average particle size of the various size fractions of galena particles: I = 10 -~ M p o t a s s i u m ethyl xanthate; 2 = 10 -4 M p o t a s s i u m ethyl xanthate; 3 = 10 -:~ M potassium ethyl xanthate; 4 = 2-10 -:~ M potassium ethyl xanthate; g = 10 -:~ M potassium ethyl xanthate + 10 -~ M copper sulphate.
'ooiA
,oo. •
?'
~o,
~
~
o
I
1
2
3
time
(min)
2
3
time
[min)
4
4
5
5
. 2o.
1
1
2
3
4
time
[min)
2
3
time
4
5
5
(mln)
Fig. 7. The flotation recovery versus the flotation time of galena particles: A. 5-12/~m size fraction; B. 40-63 p m size fraction; C. 63-90 z m size fraction; D. 90-125/~m size fraction. ! = no collector; 2 = 10- ~'M potassium ethyl xanthate; 3 = 10 4 M potassium ethyl xanthate; 4 = 10 :~ M potassium ethyl xanthate; 5 = 2.10 -:3 M potassium ethyl xanthate; 6 = 10 -:~ M potassium ethyl xanthate + 10 ~ M copper sulphate.
is considerably greater. This could be explained by the different surface areas and the adsorption of collector to the various particle-size fractions. The samples of fine particles would have a relatively large surface area and under the same treatment conditions (solid content in suspension, conditioning time and initial collector concentration ) the smaller size fractions will probably require a higher collector concentration to achieve an equivalent surface chemical potential. Finally, a series of flotation tests was carried out under the same conditions as the attachment-forces measurements. In Fig. 7 the results of a flotationkinetics study on the various particle fractions are presented. These plots show the flotation recovery of theparticle fractions at various collector and activator concentrations in terms of flotation time. From these plots it may be clearly seen that the fine particles have a lower rate of flotation than the large parti-
287 •
1
12
100-
¢~ O U ' I
ill c
6 I
..,
,~ 3
I
/,-
I
/ill
/E,~J
@-
i
i 40-~
I
'i 20-
1~
III I
-5
-3
log C O KEtX Fig. 8. the relationship between flotation recovery (after 1 min flotation), attachment force and potassium ethyl xanthate collectorconcentration.The symbolsrepresent the averageparticle size of the various fractions constituting the aggregates.A F and FR represent the attachment forces and flotation recovery, respectively. 1 = 5-12 pm; 2 = 40-63 ttm; 3 = 63-90 ttm; 4 = 90-125 pm.
cles. The relationship between the initial flotation efficiency (as evaluated by determining the recovery after 1 min of flotation) and a t t a c h m e n t force at various initial collector concentrations is represented by curves in Fig. 8. The small increase of a t t a c h m e n t force of the aggregates of fine particles corresponds to a slow increase in flotation kinetics of the same fine particles (5-12 pm). The steep increase of a t t a c h m e n t force of aggregates of large particles (90-125 pm) corresponds to a fast rate of flotation. The experimental results from a t t a c h m e n t force measurements and flotation tests indicate t h a t the magnitude of the adhesion force has an important influence over flotation response of galena particles, especially with respect to the flotation kinetics. CONCLUSIONS ( 1 ) The force of a t t a c h m e n t of coagulated aggregates of rough and angularshaped galena particles at the air-solution interface can be determined directly using an ultracentrifuge. (2) Theoretical detachment-force calculations based on the N u t t / S c h e ludko theory show a remarkable correlation with the experimental results but the quantitative theoretical predictions appear to be one order of magnitude less t h a n the experimental values.
(3~ F r o m this s t u d y it a p p e a r s that, the f l o t a t i o n efficiency for the g a l e n a x a n t h a t e s v s t e m can be directl3 c o r r e l a t e d with the s t r e n g t h of a t t a c h m e n t of the p a r t i c l e s to the a i r - s o l u t i o n interface. t 4 ) T h e s e a t t a c h m e n t - f o r c e results o b t a i n e d tbr rough a n d a n g u l a r - s h a p e d p a r t i c l e s w h e n c o n s i d e r e d in r e l a t i o n s h i p to t h e c a p t u r e e x p e r i m e n t s r e p o r t e d by A n f r u n s a n d K i t c h e n e r (1977) suggest t h a t the a d h e s i v e - f o r c e b a r r i e r m a y h a v e a m o r e i m p o r t a n t influence on t h e f l o t a t i o n efficiency t h a n the c a p t u r e force barrier. (5) Direct e x p e r i m e n t a l d e t e r m i n a t i o n of the critical a d h e s i o n fbrce by centrifugal t e c h n i q u e could be usefffl for s t u d y i n g selectivity of d i f f e r e n t t y p e s of m i n e r a l s in the f r o t h flotation process. ACKNOWLEDGEMENTS T h e a u t h o r s would like to t h a n k Prof. P. S t e n i u s a n d Dr. R. V a r b a n o v for discussions a n d critically r e a d i n g this m a n u s c r i p t . Finally, the Swedish B o a r d for T e c h n i c a l D e v e l o p m e n t is a c k n o w l e d g e d for f i n a n c i a l s u p p o r t .
REFERENCES Alexandrova, L., Pugh, R.J. and Varbanov, R., 1984. A method for the estimation of the effect of flotation reagent. Annu. Univ. Sofia, 78: 5-6. Anfrunds, J.F. and Kitchener, J.A., 1977. Rate of capture of small particles in flotation. Inst. Min. Metall. Trans., Sect. C, 86: 9-15. Blake, T.D. and Kitchener, J.A., 1972. Stability of aqueous films on hydrophobic methylated silica. J. Chem. Soc., Faraday Trans., 1 (68): 1435-1442. Chan, D.Y.C., Henry, I.D. and White, L.R., 1981. The interaction of colloidal particles collected at fluid interfaces. J. Colloid Interface Sci., 79: 410-418. Gifford, W.A. and Scriven, L.B., 1971. On the attraction of floating particles. Chem. Eng. Sci., 26: 287-297. Nicolson, M.M., 1949. Proc. Cambridge Philos. Soc., 46: 288. Nutt, C.W., 1960. The adhesion of solid particles to flat interfaces and bubbles. Chem. Eng. Sci., 12:133 144. Scheludko, A., Radoev, B. and Fabrikant, A. 1968/69. On the theory of flotation, II. Adhesion of particles to bubbles. Annu. Univ. Sofia, 63: 44-54. Scheludko, A., Tschaljowska, S. and Fabrikant, A., 1970. Contact between a gas bubble and a solid surface and froth flotation. In: Thin Liquid Films and Boundary Layers. Spec. Discuss. Faraday Soc., 1:112-117. Academic Press, London. Schulze, H.J., 1984. Analysis of the elementary stages of the flotation process. In H.J. Schulze (Editor) Physico-Chemical Elementary Processes in Flotation. Elsevier, Amsterdam, Ch. 5. Varbanov, R., Tschaljowska, S., Nishkov, I. and Somasundaran, P., 1982. On an experimentally established collective effect of microspheres floated on the liquid-gas interface. Annu. Univ. Sofia, 76: 44-47. Wark, I.W. and Cox, A.B., 1932. Principles of flotation, I. Trans. Am. Inst. Min. Metall. Eng., 112: 189-244.
275
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
The R e l a t i o n s h i p b e t w e e n F l o t a t i o n and A d h e s i o n of Galena Particles to the A i r - S o l u t i o n Interface I.N I S H K O V
and R.J. P U G H
Central Laboratory of Mineral Processing, Bulgarian Academy of Sciences, Box 32, 1126 Sofia (Bulgaria) Institute for Surface Chemistry, Box 5607, S- 114 86 Stockholm (Sweden) (Received January 28, 1988; accepted after revision July 19, 1988)
ABSTRACT Nishkov, I. and Pugh, R.J., 1989. The relationship between flotation and adhesion of galena particles to the air-solution interface. Int. J. Miner. Process., 25: 275-288. The adhesion of aggregates of galena particles to an air-solution interface was investigated experimentally by measuring the criticalcentrifugalforce necessary to detach and sink the aggregates. Theoretical detachment-force calculations based on the Nutt (1960) and Scheludko (1968/69) theory for a single spherical particle could be correlated with the experiment results but the quantitative theoretical predictions appeared to give values one order of magnitude less than the experimental data. In addition, a series of flotationexperiments were carried out with the same particle-sizefractions using a small-scale flotationcell.It was found that the flotationefficiencycould be directly related to the experimentally determined detachment forces for a seriesof particle-sizefractions in the absence and presence of xanthate collector.It is suggested that the adhesion forces may play a more important role than the capture forces in determining the flotation performance of rough and angular particles.Finally the centrifuge technique could prove to be a useful method for characterizing the floatabilityof minerals.
INTRODUCTION
The flotation efficiency of mineral particles (generally in the size range of 10 to 150/~m) is controlled by two "critical force barriers". The initial force barrier is essentially involved in the prevention of contact between particle and bubble. This usually occurs in the preliminary stages of the capture process and if this repulsive force is sufficiently high, capture may be prevented. However, at a later stage in the process, a second force barrier may increase the flotation efficiency. In this case the forces may prevent the detachment of particles from the bubble during the subsequent transportation of the particle by the bubble to the froth. This "detachment-force barrier" becomes particularly important under turbulent flow conditions in the flotation cells where the bub0301-7516/89/$03.50
© 1989 Elsevier Science Publishers B.V.
27~
hie-particle aggregates are subjected to destructive tensile and shear forces. In such cases only particles with strong adhesion to the bubble will reach the froth. Consequently, most fundamental research in flotation may be classified as concerned with: (a) bubble-particle interaction before or leading to capture; or (b} bubble-particle interaction after contact or capture. The phenomena involved are quite complex and it is presently not clear which of the force barriers make the greatest contribution towards the overall flotation efficiency. In the case of the force barrier controlling the kinetics before contact., a series of elementary steps in the capture process have been identified and studied in the laboratory. This has lead to the quantification of such parameters as the induction time (i.e. the time required for a stable particle-bubble contact to form after initial collision), the kinetics of the expansion of the three-phase (air-liquid-solid) contact, contact angle hysteresis, etc. (Schulze, 1984). The existence of a repulsive energy' barrier which opposes thinning of a thin liquid film between bubble and particle and retards the capture process has been clearly confirmed (Scheludko et al., 1970; Blake and Kitchener, 1972) although "induction times" as usually measured, are unrealistically long compared with "collision times". However, it should be stressed that all of these studies have been based on idealized sphere-plate or sphere-sphere models. It must be emphasized that practical flotation conditions may be drastically different from this idealized situation. For example, detailed studies by Anfruns and Kitchener ( 1977 ) have shown that coarse, angular particles with sharp edges and corners can be captured by bubbles with 100% efficiency whereas the capture efficiency of strongly hydrophobized glass ballotini is usually much lower. This was attributed to the jagged projection of the asperities of the mineral particles causing local thinning and rupture of' the liquid film between air and solid and almost eliminating the influence of the "capture force barrier". Thus, the study suggests that for rough particles the detachment-force barrier may play the most important role in controlling flotation efficiency. The purpose of the present investigation is to test this hypothesis. The main objectives may be summarized as follows. (1) To attempt to correlate flotation efficiency of large aggregates of well defined coarse mineral particles which cling to the bubbles in froth flotation, to the detachment forces from the air-solution interface. In the present study it was found convenient to quantify the critical detachment force using a centrifugal technique. (2) To compare the magnitude of the force of detachment of the aggregates of particles with theoretical values for idealized smooth monodispersed particles. (3) To study the effects of particle size, collector concentration and activator on the magnitude of the detachment force and the flotation efficiency.
277
Theory of the balance o/forces on a smooth spherical particle suspended at the fluid interface Theoretical studies by Nutt (1960) and Scheludko et al. (1968/89) have expressed the mechanics of the attachment of hydrophobic spherical particle to a gas-liquid (G-L) interface or to a gas bubble in terms of an equilibrium between the gravitational force on the one hand, and the sum of the surface tension and buoyancy forces on the other. Fig. 1 shows schematically the attachment of a spherical particle to the G-L interface and the quantities involved in the calculation. To detach the particle, a centrifugal force must be applied which must exceed a certain critical value. This value is a function of the size and density of the particle, as well as of the surface tension of the liquid and its angle of contact with the particle surface. The floatability of the particle clearly will be related to the strength of its adhesion to the bubble and it has been suggested that measurements of the critical force (i.e. the maximum attachment force) could be used to assess the effectiveness of the flotation process. The maximum attachment force of a single spherical particle on the G-L interface (FAt) can be expressed as an approximation by the equation (Nutt, 1960; Scheludko et al., 1968/89): FAt :
(1)
7[(TRp( 1 - c o s 0 )
where Rp is the particle radius, a is the surface tension at the G-L interface and 0 is the equilibrium wetting angle. From experimental studies the maximum attachment force of siliconed glass beads (radius 0.02-0.1 cm) was found to be in good agreement with this equa-
A/r
J Fcen Fig. 1. The balance of force acting to a spherical particle attached to the solution/air interface. F,,~. represents the centrifugal force acting on the sphere, F~ is the surface tension vector force acting around the perimeter and F~, is the buoyancy force. Rp is the radius of the particle and 0 is the contact angle. For large centrifugal forces F,. >> F~+ Fh the particle is detached from the surface.
278 tion (Nutt, 1960). These experiments were also extended to determine the effect of aggregation of the particles on the force of a t t a c h m e n t at the air-liquid interface. From these studies it was found that the attachment force of the aggregates decreased as the number of particles increased, reaching a limiting value of about 50% of the force of a t t a c h m e n t of an individual particle. However, it is important to note that these experiments with aggregates were carried out on relatively large particles ( radius 0.04 cm). Varbanov et al. (1982) have also studied the behaviour of an aggregate of monodispersed glass spheres attached to the air-water interface. It was experimentally established that aggregates of particles with a radius smaller than 100 #m do not obey eqn. 1 and detach at a greater force then predicted by the theoretical model. In the case of large aggregates of irregularly shaped particles, initial studies showed a considerable difference in the detachment-force values for galena particles respectively in the absence and presence of xanthate collector (Alexandrova et al., 1984). However, generally these results were found to deviate from that of a monodispersed smooth particle. This may be explained by the differences in degrees of wetting of the irregularly shaped edges and corners of the aggregates which for example, might be especially favorable for adsorption of a hydrophobizing agent. In fact, the particles may have a number of orientations at the interface, each having a particular potential energy barrier and adhesive strength. In addition, it is important to consider particle interactions and collective effects due to aggregation of particles at the interface. With a practical system the problem is therefore unfortunately too complex for theoretical treatment. EXPERIMENTAL
Characteristics of mineral systems A sulphide mineral sample of hand-picked Swedish natural galena was supplied by Boliden Mineral AB. The sample was crushed in an agate mortar and TABLE I
Particle size fraction (#m)
Averageparticle size (/zm)
5- 12 40- 63 63- 90 90-125
9 48 72 102
279 separated by sieving under water into narrow particle-size fractions for experimental studies. Semi-quantitative spectroscopic analysis of the mineral showed the presence of the following impurities: Si 0.17%; Ca 0.01%; Fe 0.11%; Mg <0.01%; Mn <0.01%; Cu 0.01%; Zn 0.08%. The particle-size distribution and the average particle size were determined (Table I ) using a Particle Size Counter manufactured by Elzon Particle Data, USA. Microscopic examination showed the particles from all size fractions to be irregular in shape with rough and angular surfaces. The solid content of the suspensions for both attachment-forces measurements and flotation experiments was maintained a 6 g/1.
Reagents Potassium ethylxanthate (KEtX) (Hoechst, analytical grade) was used as collector. Fresh solutions were prepared daily for the flotation and attachment experiments. Copper sulphate (analytical grade) was used as activator. Sodium hydroxide (reagent grade) was used for pH adjustment. The experimental studies were carried out in a slightly alkaline solution (pH 8.5).
Attachment force measurements The centrifugal equipment consisted of a Beckman ultracentrifuge L5-50B (rotation up to 50.000 rpm) with a rotor SW50-1, equipped with buckets that swing out to a horizontal position as the rotor is accelerated. Thin wall ultra clear plastic non-wettable tubes (dimensions 13 × 51 mm, volume 2.5 cm 3) were found to be the most convenient for the present study. The radial distance (Rrot) of the meniscus was constant (8 cm) for all experiments. The suspension containing 0.6 g of galena in 100 ml of collector solution at a pre-adjusted pH, was stirred for 10 min in a 150-ml beaker using a magnetic stirrer in an attempt to simulate the shear field experienced in the micro flotation cell. In the studies of the influence of activator, before conditioning with collector a predetermined amount of copper sulphate was introduced and the suspension was conditioned for 5 min by stirring. After conditioning, the galena particles were allowed to sediment and then brought to the air-solution interface as a coagulated clump by tilting repeatedly the glass beaker. In order to establish an equilibrium contact angle the galena particles were then left for 30 min attached to the interface before transferring them with a spatula onto the surface of the initial solution in the centrifuge tubes. The aggregates were then subjected to the detaching action of centrifugal forces at a range of centrifuge speeds. For each experiment the centrifuge speed was increased gradually to a pre-set value and maintained at this speed for 5 min. The critical speed where the particles were thrown from the interface into the body of the solution was noted.
:L~!)
During the experiments it was observed that under gravity the particles tended to aggregate together into a loosely packed raft at. the interface. ()n the initial application of the centrifugal field, the raft became compressed in the middle of the meniscus and small particles appeared to be mechanically entrapped between the large particles, thus reducing the voids. At high centrifugal speeds, the coagulated aggregates became compressed and finally the aggregate appeared to be detached as a whole at the critical centrifuge speed. For a single particle of radius R,, the fbrce required to overcome the tbrces that bind the particle to the meniscus so that the particle becomes submerged in the liquid can be expressed by: 4 F ..... =
(2)
a = \ 30 ] Rrot
(3)
where: a is centrifugal acceleration; N is number of revolutions at which detaching takes place; Rrot is radial distance ( radius of rotation); Ap is the effective density of particles.
Flotation experiments In these experiments, a cylindrical glass cell (volume 100 cm ~) with porous glass frit bottom and an external gas source was used. Bubbles were generated by flowing nitrogen through the glass frit at a controlled flow rate of 50 cm3/ min. The flotation procedure was as follows. A standardized amount of the particle-size fractioned galena (0.6 g) was placed in the microflotation cell and the collector solution added. The pulp was then conditioned by magnetic stirring for 10 min. In the studies of the influence of activator concentration before conditioning with collector a predetermined amount of copper sulphate was introduced and the pulp was conditioned for 5 min. After conditioning, nitrogen gas was bubbled through the cell for 5 min in all flotation tests. Finally, the froth product was removed and collected and the quantity of galena determined. For studying the flotation kinetics of different particle-size fractions, the froth product was removed after 0.5, 1, 2, 3, 4, and 5 min, giving six separate concentrations for each experiment. No frother was used in these systems. The percentage of mineral floated was determined by gravimetric analysis and expressed as a wt% of dry mineral to the total weight of the original sample.
281 RESULTS AND DISCUSSION
Studies of the attachment force in the absence and presence of collector The critical centrifugal force required to detach and submerge the aggregates of galena particles from the air-solution interface was determined for the various particle-size fractions. The number of revolutions (N) at which detachment takes place as a function of average particle size of the fraction constituting the aggregates is shown in Fig. 2. Theoretical numbers of revolution for the detachment of smooth spherical monodispersed particles were calculated (from N u t t theory) by equating Fat and Fc~, using eqns. 1, 2 and 3 to give: N
15
/
1-cos0
- nRp ~3azIpR~ot
(4)
These values are also shown in Fig. 2. In the application of the N u t t theory it is necessary to assume an equilibrium value of the contact angle during the detachment process. However, in the practical situation, as the three-phase contact expands along the surface of the particles a number of intermediate states displaying quasi-equilibrium contact +5
+4 Z
÷3"
+2-
7'2 average particle size
(prrO
Fig. 2. The log (number of revolutions) at which detachment occurs versus the averageparticle size of the various size fractionsof galenaparticles constituting the aggregates: 1 --- data obtained without collector;2 = data obtained with 10- '~M potassiumethyl xanthate; 3 = theoreticalcurve (Nutt model) assuming 0 = 8 ° (no collector); 4 = theoretical curve (Nutt model) assuming0 = 60° (10 -~ Mpotassium ethyl xanthate).
angles may occur. Since the contact angle cannot be quantified, for the purpose of the present investigation values of 0 = 8 for water and 0 = 60 for 10 :: M K E t X solution on a galena surface were selected according to the data of Wark and Cox (1932). The experimentally determined a t t a c h m e n t |brces of galena particles at the air-solution interface as a function of the average particle size respectively in the absence and presence of xanthate collector are presented in Table II and Fig. 3. In addition, data and theoretical plots for ideal T A B L E II
Particle
Co K E t X = 1 0 - :~M / I
C,~ K E t X = 0
size (pm) a t t a c h m e n t force ( d y n )
5 - 12 4 0 - 63 6 3 - 90 90-125
theory
experiment
9.9"10 ,4 5.3"10 -:~ 7.9"10 -:~ 1.1"10 -2
1.6.10-2 5.2"10 -~ 6.2"10 -~ 7.9"10 -2
F~p/Fth
16.1 9,8 7.8 7.2
a t t a c h m e n t force ( d y n ) theory
experiment
4.9"10 -2 2.6"10 -~ 4.0"10 ~ 5.6"10 - I
4.3"10 2.12 3.4 5.1
l
8.8 8.2 8.5 9.1
+1-
-1 ¢g tl.
/" J" 1./° --3' T
i
lo2
average particle size
(pm)
Fig. 3. The log (criticalcentrifugal force of attachment) versus the average particle size of the various size fractionsof galena particlesconstituting the aggregates (see Fig. 2 for legend).
283
100"
80
>o 60 o o
+ 4O 2O
4'8
7'2 (pro)
1(~2
average particle size
Fig. 4. T h e flotation recovery of galena particles after 5 rain flotation versus the average particle size of the various size fractions: A = no collector; • = 10 -3 M p o t a s s i u m ethyl xanthate.
monodispersed particle calculated from the Nutt theory are also shown for comparison purposes. From the data presented in Table II and Fig. 3 it can be seen that the experimental detachment forces for the aggregated particles appear to be greater than the theoretical values predicted for the idealized monodispersed model (but never more than by a factor of 16). This result is remarkable in view of the complexity of the galena system. Indeed with an aggregate of particles, a collective effect arising from capillary forces binding the particles together would also be expected to play a role within the balance of attachment forces. The interparticle interactions in a horizontal plane arising from the attachment of floating bubbles to an interface has been studied by Nicholson (1949). In addition Gifford and Scriven (1971) calculated the theoretical capillary attraction between small idealized floating particles consisting of parallel stationary cylinders of infinite length. These workers found that the capillary forces that hold the particles together were surprisingly high. Later Chan et al. ( 1981 ) extended Nicolson's theoretical approach and demonstrated its applicability to calculate the capillary forces between bodies of arbitrary shape and physical properties. However, the influence of these forces on their stability
r
1°°i
/
?
¢
# v
>,
80
I
I
> 0 r~
y ~-
60
0
40
/ 4 /
20j/
"tl ,
-2
log
Fat .
Fig. 5. The flotation recovery of galena particles after 5 min flotation versus the log (force of attachment ) with and without potassium ethyl xanthate collector. The points represent the average size of the various particle fractions constituting the aggregates. A = no collector; • = 10- :~ M potassium ethyl xanthate.
with respect to sinking is not known. Unfortunately, in the present experiments with agglomerates consisting of non-spherical coagulated particles, the situation is far too complex for theoretical analysis. From Fig. 3 it can also be observed that the detachment force increases with the average particle size of the various size fractions of particles constituting the aggregates for all the experimental studies. This trend tends to correlate with the theoretical data based on the monodispersed spherical model. Finally, the experimental detachment forces are greater for systems containing KEtX collector. These results generally agree with the theoretical curves. In Fig. 4 the flotation recoveries of different-size galena fractions in water and 10 -:~ M KEtX solution are shown as a function of the average particle size after 5 min flotation. For both water and collector solution the flotation recoveries appear to increase with the average particle size of the fractions constituting the aggregates. Fig. 5 illustrates the relationship between recovery after 5 min flotation and attachment force of aggregates of galena particles at the air-solution interface for the various particle fractions. This plot demonstrates the increase of flo-
285
ration recovery as the attachment force is increased. The experimental results suggest that the strength of adhesion of the aggregates to the interface can be related to the flotation efficiency.
Study of the effect of collector and activator on adhesion force The variation of the attachment force of an aggregate of galena particles at the air-solution interface, treated in advance with KEtX and copper sulphate as a function of initial concentration and average particle size is shown in Fig. 6. From these results it may be seen that the attachment force depends on particle size, initial collector and activator concentration. The major increase in attachment force evidently takes place over the range 10-s to 10 -4 M KEtX. The maximum force was obtained with a solution containing both flotation reagent and activator (10 -3 M KEtX and 10 -2 M CuS04). The increase of the attachment force of fine particles increases very slightly with increasing collector concentration. For the largest particles the increase
8"
¥
1
•
2
6. 3 •
5
7
!
6
N
5
~
4
i
e-
3-
'~
2-
O
1
162 average particle size (~m) Fig. 6. The a t t a c h m e n t force versus the average particle size of the various size fractions of galena particles: I = 10 -~ M p o t a s s i u m ethyl xanthate; 2 = 10 -4 M p o t a s s i u m ethyl xanthate; 3 = 10 -:~ M potassium ethyl xanthate; 4 = 2-10 -:~ M potassium ethyl xanthate; g = 10 -:~ M potassium ethyl xanthate + 10 -~ M copper sulphate.
'ooiA
,oo. •
?'
~o,
~
~
o
I
1
2
3
time
(min)
2
3
time
[min)
4
4
5
5
. 2o.
1
1
2
3
4
time
[min)
2
3
time
4
5
5
(mln)
Fig. 7. The flotation recovery versus the flotation time of galena particles: A. 5-12/~m size fraction; B. 40-63 p m size fraction; C. 63-90 z m size fraction; D. 90-125/~m size fraction. ! = no collector; 2 = 10- ~'M potassium ethyl xanthate; 3 = 10 4 M potassium ethyl xanthate; 4 = 10 :~ M potassium ethyl xanthate; 5 = 2.10 -:3 M potassium ethyl xanthate; 6 = 10 -:~ M potassium ethyl xanthate + 10 ~ M copper sulphate.
is considerably greater. This could be explained by the different surface areas and the adsorption of collector to the various particle-size fractions. The samples of fine particles would have a relatively large surface area and under the same treatment conditions (solid content in suspension, conditioning time and initial collector concentration ) the smaller size fractions will probably require a higher collector concentration to achieve an equivalent surface chemical potential. Finally, a series of flotation tests was carried out under the same conditions as the attachment-forces measurements. In Fig. 7 the results of a flotationkinetics study on the various particle fractions are presented. These plots show the flotation recovery of theparticle fractions at various collector and activator concentrations in terms of flotation time. From these plots it may be clearly seen that the fine particles have a lower rate of flotation than the large parti-
287 •
1
12
100-
¢~ O U ' I
ill c
6 I
..,
,~ 3
I
/,-
I
/ill
/E,~J
@-
i
i 40-~
I
'i 20-
1~
III I
-5
-3
log C O KEtX Fig. 8. the relationship between flotation recovery (after 1 min flotation), attachment force and potassium ethyl xanthate collectorconcentration.The symbolsrepresent the averageparticle size of the various fractions constituting the aggregates.A F and FR represent the attachment forces and flotation recovery, respectively. 1 = 5-12 pm; 2 = 40-63 ttm; 3 = 63-90 ttm; 4 = 90-125 pm.
cles. The relationship between the initial flotation efficiency (as evaluated by determining the recovery after 1 min of flotation) and a t t a c h m e n t force at various initial collector concentrations is represented by curves in Fig. 8. The small increase of a t t a c h m e n t force of the aggregates of fine particles corresponds to a slow increase in flotation kinetics of the same fine particles (5-12 pm). The steep increase of a t t a c h m e n t force of aggregates of large particles (90-125 pm) corresponds to a fast rate of flotation. The experimental results from a t t a c h m e n t force measurements and flotation tests indicate t h a t the magnitude of the adhesion force has an important influence over flotation response of galena particles, especially with respect to the flotation kinetics. CONCLUSIONS ( 1 ) The force of a t t a c h m e n t of coagulated aggregates of rough and angularshaped galena particles at the air-solution interface can be determined directly using an ultracentrifuge. (2) Theoretical detachment-force calculations based on the N u t t / S c h e ludko theory show a remarkable correlation with the experimental results but the quantitative theoretical predictions appear to be one order of magnitude less t h a n the experimental values.
(3~ F r o m this s t u d y it a p p e a r s that, the f l o t a t i o n efficiency for the g a l e n a x a n t h a t e s v s t e m can be directl3 c o r r e l a t e d with the s t r e n g t h of a t t a c h m e n t of the p a r t i c l e s to the a i r - s o l u t i o n interface. t 4 ) T h e s e a t t a c h m e n t - f o r c e results o b t a i n e d tbr rough a n d a n g u l a r - s h a p e d p a r t i c l e s w h e n c o n s i d e r e d in r e l a t i o n s h i p to t h e c a p t u r e e x p e r i m e n t s r e p o r t e d by A n f r u n s a n d K i t c h e n e r (1977) suggest t h a t the a d h e s i v e - f o r c e b a r r i e r m a y h a v e a m o r e i m p o r t a n t influence on t h e f l o t a t i o n efficiency t h a n the c a p t u r e force barrier. (5) Direct e x p e r i m e n t a l d e t e r m i n a t i o n of the critical a d h e s i o n fbrce by centrifugal t e c h n i q u e could be usefffl for s t u d y i n g selectivity of d i f f e r e n t t y p e s of m i n e r a l s in the f r o t h flotation process. ACKNOWLEDGEMENTS T h e a u t h o r s would like to t h a n k Prof. P. S t e n i u s a n d Dr. R. V a r b a n o v for discussions a n d critically r e a d i n g this m a n u s c r i p t . Finally, the Swedish B o a r d for T e c h n i c a l D e v e l o p m e n t is a c k n o w l e d g e d for f i n a n c i a l s u p p o r t .
REFERENCES Alexandrova, L., Pugh, R.J. and Varbanov, R., 1984. A method for the estimation of the effect of flotation reagent. Annu. Univ. Sofia, 78: 5-6. Anfrunds, J.F. and Kitchener, J.A., 1977. Rate of capture of small particles in flotation. Inst. Min. Metall. Trans., Sect. C, 86: 9-15. Blake, T.D. and Kitchener, J.A., 1972. Stability of aqueous films on hydrophobic methylated silica. J. Chem. Soc., Faraday Trans., 1 (68): 1435-1442. Chan, D.Y.C., Henry, I.D. and White, L.R., 1981. The interaction of colloidal particles collected at fluid interfaces. J. Colloid Interface Sci., 79: 410-418. Gifford, W.A. and Scriven, L.B., 1971. On the attraction of floating particles. Chem. Eng. Sci., 26: 287-297. Nicolson, M.M., 1949. Proc. Cambridge Philos. Soc., 46: 288. Nutt, C.W., 1960. The adhesion of solid particles to flat interfaces and bubbles. Chem. Eng. Sci., 12:133 144. Scheludko, A., Radoev, B. and Fabrikant, A. 1968/69. On the theory of flotation, II. Adhesion of particles to bubbles. Annu. Univ. Sofia, 63: 44-54. Scheludko, A., Tschaljowska, S. and Fabrikant, A., 1970. Contact between a gas bubble and a solid surface and froth flotation. In: Thin Liquid Films and Boundary Layers. Spec. Discuss. Faraday Soc., 1:112-117. Academic Press, London. Schulze, H.J., 1984. Analysis of the elementary stages of the flotation process. In H.J. Schulze (Editor) Physico-Chemical Elementary Processes in Flotation. Elsevier, Amsterdam, Ch. 5. Varbanov, R., Tschaljowska, S., Nishkov, I. and Somasundaran, P., 1982. On an experimentally established collective effect of microspheres floated on the liquid-gas interface. Annu. Univ. Sofia, 76: 44-47. Wark, I.W. and Cox, A.B., 1932. Principles of flotation, I. Trans. Am. Inst. Min. Metall. Eng., 112: 189-244.