# bubble attachment and dettachment in flotation

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Ž .Int. J. Miner. Process. 56 1999 133–164

Bubble–particle attachment and detachment inflotation

John Ralston ), Daniel Fornasiero, Robert HayesIan Wark Research Institute, UniÕersity of South Australia, The LeÕels Campus, Mawson Lakes, Adelaide,

SA 5095, Australia

Received 3 June 1998; received in revised form 12 October 1998; accepted 27 November 1998

Abstract

The mechanism by which particles and bubbles interact captures many of the central conceptsof colloid science and hydrodynamics and is an example of heterocoagulation. Hydrodynamics,

Ž .interfacial including capillary forces, particle and bubble behaviour and solution chemistry areall interwoven. The processes of attachment and detachment are focused upon here. We deal withthe identification of a flotation ‘domain’, the deformation of a bubble surface upon interactionwith a solid surface, the kinetics of three phase contact line expansion and the determination ofattachment efficiencies through to the direct measurement of bubble–particle interaction forces.The results, concepts and implications of this work are discussed. q 1999 Elsevier Science B.V.All rights reserved.

Keywords: bubble–particle; attachment; detachment

1. Introduction

The capture of particles by rising bubbles is the central process in froth flotation. Forefficient capture to occur between a bubble and a hydrophobic particle, they must firstundergo a sufficiently close encounter, a process which is controlled by the hydrody-namics governing their approach in the aqueous environment in which they are normallyimmersed. Should they approach quite closely within the range of attractive surfaceforces, the intervening liquid film between the bubble and the particle will drain, leading

) Corresponding author. Fax: q61-8-302-3683; E-mail: [email protected]

0301-7516r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-7516 98 00046-5

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Table 1Key papers in understanding fundamental flotation substeps

Ž .1948 A fundamental paper on the kinetics of the flotation process appeared by Sutherland, 1948 in Australia. This invoked induction time,described particle size effects in flotation and catalysed other similar approaches. While it was preceded by other efforts, it was the firstcomprehensive effort to describe recovery-size-time data in a fundamental manner.

Ž .1960–1961 In Moscow, Derjaguin and Dukhin 1960 produced a key paper on the theory of flotation of small and medium-size particles.Hydrodynamics, surface forces and diffusiophoresis were all used in this theory. This seminal work resulted in an acceleration offundamental flotation research worldwide.

Ž .1972 Blake and Kitchener 1972 , working together in London, published some very careful measurements of the thickness of aqueousfilms on hydrophobic quartz surfaces. Film thicknesses, measured as a function of salt concentration, were shown to depend on theelectrical double layer force. Film instability occurred on hydrophobic surfaces at film thicknesses less than about 60 nm. This value,which was smaller than the range of the electrical double layer force, represented the combined effects of hydrophobic force, surfaceheterogeneities and external disturbances. It hinted at the length dependence of hydrophobic forces, information which was subsequentlydetermined by surface force experiments after 1982.

Ž . Ž .1976 Scheludko 1967 and Scheludko et al. 1976 in Bulgaria considered how particles might become attached to a liquid surface anddeveloped the capillary theory of flotation.

Ž .1977 Anfruns and Kitchener 1986 published the first measurements of the absolute rate of capture of small particles in flotation. This wasthe first critical test of collision theory under conditions where the bubble and particle surface chemistry was characterised and controlled.

Ž .1983 Schulze 1984 published a key textbook on the physico-chemical substeps which are important in flotation, drawing on a wide rangeof hydrodynamic, surface chemical and engineering information.

Žcirca 1980–present There has been a strong interest in developing reliable collision models Dobby and Finch, 1986; Dobby and Finch, 1987; Yoon and.Luttrell, 1989; Dai et al., 1998 . The surface force apparatus and, recently, the atomic force microscope colloid probe technique, have

Žprovided very useful insight into electrical double layer, van der Waals and hydrophobic forces Christenson and Claesson, 1988; Butt,.1994; Ducker et al., 1994; Wood and Sharma, 1995; Fielden et al., 1996 . Thin film drainage has been investigated between a rigid and

Ž .a deformable interface Lin and Slattery, 1982; Chen, 1984; Hewitt et al., 1993 . Attachment efficiencies have been measured. ReliableŽ .methods for measuring contact angles on particles have been developed Crawford et al., 1987; Diggins et al., 1990 . Major theoretical

Žand experimental advances in describing dynamic contact angles on well defined surfaces have been made e.g., Blake, 1993; Hayes and.Ralston, 1993 .

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to a critical thickness at which rupture occurs. This is then followed by movement of theŽthree phase contact line the boundary between the solid particle surface, receding liquid

.phase and advancing gas phase until a stable wetting perimeter is established. Thissequence of drainage, rupture and contact line movement constitutes the second processof attachment. A stable particle–bubble union is thus formed. The particle may only bedislodged from this state if it is supplied with sufficient kinetic energy to equal orexceed the detachment energy, i.e., a third process of detachment can occur.

Ž .The capture or collection efficiency E of a bubble and a particle may be defined as

EsE PE PE 1Ž .c a s

where E is the collision efficiency, E is the attachment efficiency and E is thec a s

stability efficiency of the bubble–particle aggregate. This dissection of capture effi-Ž .ciency into three process efficiencies was used by Derjaguin and Dukhin 1960 and

focuses attention on the three zones of bubble–particle capture where, in order,hydrodynamic interactions, interfacial forces and bubble–particle aggregate stability aredominant, as illustrated in Fig. 1. We should note that these zones are not completelydiscrete, rather they grade into one another. Since surface forces are relatively shortrange, they do not have any significant influence on the collision step. This is whycollision and attachment may be treated as essentially independent processes.

Our knowledge of the various efficiencies has been enhanced by a continuous, strongŽ .research thrust, catalysed by Sutherland 1948 in 1948. Six of the most important

publications, described in Table 1, covering the period from 1948 to 1983, signalledmajor advances in the field and have directly led to the present high level of researchactivity. Two of the most important substeps in the froth flotation process, attachmentand detachment, are described below.

Ž . Ž . Ž .Fig. 1. Hydrodynamic 1 , diffusiophoretic 2 and surface force 3 zones of interaction between a bubble andŽ .a particle. From Derjaguin and Dukhin, 1960, with permission.

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2. Attachment

2.1. Zones of influence

Ž .Derjaguin and Dukhin 1960 identified zone 2 in Fig. 1 as that region wherediffusion effects are important. A strong electric field exists in this zone, for the liquidflow around the moving bubble gives rise to a tangential stream at its surface whichdestroys the equilibrium distribution of adsorbed ions there. Where surfactant is present,it is continually swept from the upper to the lower surface of the bubble. Transport ofionic surfactant to the moving bubble surface therefore takes place, leading to theestablishment of a concentration gradient. A strong electric field of order 3000 V cmy1

is established when the cation and anion diffusion coefficients differ, as they generallydo. Hence, charged particles entering zone 2 will experience an electrophoretic force inprecisely the same way as in an electrophoresis cell and would be either attractedtowards, or repelled from the bubble surface. Derjaguin and Dukhin coined the term‘diffusiophoresis’ for this phenomenon, i.e., the ‘diffusiophoretic force’ therefore acts onthe particle as an additional force. To date, however, evidence confirming the presenceor absence of diffusiophoresis in flotation is equivocal and sparse. Apart from noting itspossible contribution to capture efficiency it is not pursued further here.

In zone 3, surface forces predominate once the thin film between the bubble and theparticle is reduced much below a few hundred nanometer. These forces can accelerate,retard or even prevent the thinning of the liquid film between the particle and thebubble. From a thermodynamic point of view, the free energy of a liquid film differsfrom the bulk phase from which it is formed. This excess free energy was originallycalled the ‘wedging apart’ or ‘disjoining’ pressure by Derjaguin and represents thedifference between the pressure within the film, p f, and that in the bulk liquid adjacentto the solid surface, p l. Note that for a bubble pushed against a flat solid surface,immersed in water, pb, the pressure within the bubble, is equal to p f. The Derjaguin and

ŽScheludko schools in Moscow and Sofia Scheludko, 1967; Derjaguin and Churaev,.1976 performed experimental measurements of disjoining pressures, providing both the

first real tests of the DLVO theory of surface forces, as well as the first accurateexperimental estimates of the Hamaker constant, respectively. The disjoining pressureŽ .p depends on the film thickness, h, and:

p h sp f yp l . 2Ž . Ž .Ž .For mechanical equilibrium in a stable film, p h )0 and dprdh-0.

If the liquid film is stable at all thicknesses, the liquid is said to completely wet thesolid. This occurs, for example, when an air bubble approaches a clean silica surfaceimmersed in water—in this instance, the Hamaker constant is negative and the corre-sponding van der Waals force is repulsive for the silicarwaterrair triple layer. For anunstable film, the thin film must drain then rupture and the resulting three phase line of

Ž .contact TPLC, vapour–water–solid must expand to form a wetting perimeter beforethe particle can adhere to the bubble.

Each of these events will have a characteristic time associated with them, the sum ofŽ .which must be less than the contact time t between the bubble and the particle ifc

flotation is to occur. The contact time is generally of the order of 10y2 s or less.

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The induction time, t is normally taken as the time required for bubble–particleind

attachment to occur, once the two are brought into contact, i.e., it is the sum of the thinŽ .film drainage and TPLC spreading times t and t and is synonymous with thefilm TPLC

attachment time. Rupture is a very fast process and is not a significant contributor tot .ind

2.2. Film drainage

When a bubble is pressed against a solid surface, through water, the intervening filmis generally not plane parallel. Rather the edge of the film thins quickly and a small,thicker dimple is trapped in the centre, for the bubble is deformable. This is essentially akinetic phenomenon, caused by the flow being greatest at the very edge of the film inthe initial stages of drainage. The existence of this dimple has been detected bytechniques such as laser interferometry. Hydrodynamic theories attempting to describethe profile and evolution of the dimple have been proposed but with very limited successin describing experimental data.

Most approaches have not considered the rigorous differential equation describing theŽ .drainage rate Buevich and Lipkina, 1978 , but instead have used various approxima-

tions to derive simpler differential equations which are amenable to analytical solution.Ž .Approximate models and the equations derived from them are as follows:

Ž . Ž .a Frankel and Mysels 1962 assumed that the rate of flow through the barrier ringper unit length of periphery was independent of the radius. Using this assumption, theyderived equations for the film thickness, h , at the centre of the draining film, and h , ato b

the barrier ring, as a function of time, t:1

60.0096hr 4ch s 3Ž .o ž /g r tb

120.0090hr r 2c b

h s 4Ž .b ž /g t

where r is the radius of the air bubble, r is the radius of the barrier ring, andb c

measured values are expressed in cgs units. g is the surface tension of water and h is itsviscosity.

Unfortunately, this model predicts that the shape of the dimple becomes morepronounced as thinning progresses, which is the reverse of the experimental observationŽ .Hewitt et al., 1993 .

Ž . Ž .b Hartland and Robinson 1977 approximated the shape of the dimple by threelinked parabolas, which led to the following equations for the film thickness as afunction of time:

162phr 4c

h s0.117 5Ž .o ž /g r tb

12r hr 2b c

h s0.370 . 6Ž .b ž /2pg t

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Ž . Ž .c Dimitrov and Ivanov 1978 assumed that all the energy dissipation occurred inthe barrier ring, and from this derived a model which predicted:

12r hr 2b c

h s . 7Ž .o ž /g t

Ž . Ž .d Jain and Ivanov 1980 developed a ‘step’ model for the draining film, whichpredicted:

12r hr 2b c

h s0.433 . 8Ž .b ž /g t

Ž . Ž . Ž . Ž .e Chen and Slattery 1982 , Lin and Slattery 1982 and Chen 1984 produced aseries of models in which they improved on Hartland’s approach by redefining theboundary conditions to produce a fourth-order differential equation which is identical in

Ž .form with that published earlier Buevich and Lipkina, 1978 . To solve the equation, LinŽ .and Slattery 1982 postulated that the shape of the dimple was independent of time at

some early stage in the drainage, and adjusted the arbitrary parameters in their solutionŽ .so that the predicted curve fitted an experimentally measured curve Platikanov, 1964 at

an early time.Ž .Hewitt et al. 1993 have examined the drainage behaviour between a hydrophilic

quartz surface and an approaching air bubble, using scanning interferometry. In Fig. 2,their experimental drainage rate curves in doubly distilled water in the centre of the

Fig. 2. Film thickness in the centre of the draining film in conductivity water against time after film formationŽ . y3 Ž . y3 Ž . y3 Ž .is shown for film radius values of B 0.22=10 m, I 0.24=10 m, ' 0.26=10 m and ^

y3 Ž .0.32=10 m. From Hewitt et al., 1993, with permission.

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Ž .Fig. 3. The inverse of the square of the film thickness at the centre of the dimple against time for film radiusŽ . y3 Ž . y3 Ž . y3 Ž . y3values of B 0.22=10 m, I 0.24=10 m, ' 0.26=10 m and ^ 0.32=10 m, i.e., for the

Ž .same data as shown in Fig. 2. From Hewitt et al., 1993, with permission.

draining film are shown for a series of experiments with increasing film radius at aŽ .hydrophilic quartz surface. For the early drainage rates for times -50 s , a plot of the

Ž .inverse of the square of the film thickness against time results in a straight line Fig. 3 ,the gradient of which increases with decreasing film radius. Absolute values of the

Ž .gradient in all cases exceed that predicted by either Reynolds theory Reynolds, 1886 orŽ .the theory of Dimitrov and Ivanov 1978 as shown in Table 2.

Observation of the drainage pattern which evolves when the bubble was pressedagainst the quartz plate did not reveal any detectable asymmetry in the bubble ‘shape’.Moreover, scanning the draining film on either side of the centre did not give detectably

Table 2Ž .y2The experimental gradient derived from the plot of film thickness vs. time is shown for various film radii

Ž . Žobserved; theoretical calculations of the gradient are shown for the theories of Reynolds 1886 theoretical. Ž . Ž .slope 1 and Dimitrov and Ivanov 1978 theoretical slope 2

Film radius Experimental slope Theoretical slope 1 Theoretical slope 2y3 10 y2 y1 13 y2 y1 12 y2 y110 m 10 m s 10 m s 10 m s

0.22 2.19 1.38 1.300.24 1.93 1.16 1.090.26 1.53 0.99 0.930.32 1.18 0.65 0.61

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different drainage rates at the same radii with all other conditions held constant. The twoŽ .surfaces i.e., airrwater and quartzrwater clearly ‘sense’ one another at large distances

of separation, exhibit linear inverse square behaviour of film thickness against time, but

Fig. 4. Film thickness of the drainage film shown against time after film formation is shown for drainage at theŽ . Ž .centre a and the boundary ring b at a hydrophilic surface in aqueous sodium chloride solutions of

Ž . y3 y3 Ž . y4 y3 Ž . y4 y3 Ž . y5concentrations of B 10 mol dm , I 5=10 mol dm , ' 2.5=10 mol dm and ^ 10y3 Ž .mol dm sodium chloride solutions. From Hewitt et al., 1993, with permission.

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yield gradients which differ from extant theoretical predictions, even in the case of filmdrainage for high-purity water.

Ž . ŽDrainage rates at both the film centre rs0 and at the boundary ring rs0.20=y3 .10 m at the quartz surface, as a function of salt concentrations, are shown in Fig. 4.

The drainage rate at the centre of the film decreases with increasing salt concentrationwhereas at the boundary ring the reverse is true.

Quantitatively, this effect can be explained in terms of the decreased electrostaticrepulsion with increasing salt concentration, allowing the film to drain more rapidly atthe boundary ring. The drainage rate at the centre is then controlled by the rate at whichthe aqueous solution can pass through the boundary ring. Film profiles for 2.5=10y4

mol dmy3 and 10y3 mol dmy3 salt solutions are shown in Fig. 5. These profilesprovide a graphic illustration of the point at which the ‘dimple’ becomes morepronounced as the salt concentration is increased.

In Fig. 4b, particularly, it is clear that the influence of salt on drainage rate is sensedŽ .at large distances of separation i.e., a micrometer or more . This is also evident in Fig.

4a, albeit not as graphically. Plots of inverse square of film thickness against time aregiven in Fig. 6a and b for drainage at the centre and the boundary ring, respectively. Thefit at the centre becomes more linear as the salt concentration increases, but still divergessignificantly from theory. These models predict 1rt n dependence with ns0.25 at thecentre and ns0.5 at the boundary ring, in disagreement with experiment. At equilib-

Žrium, the bubble surface flattens and, as noted by others Read and Kitchener, 1969;.Blake and Kitchener, 1972 , the system may be described as an asymmetric interface of

Ž .two parallel plates air and quartz , interacting through a thin liquid film.

Fig. 5. Film profiles are shown for drainage at a hydrophilic surface as a function of film radius shown fory4 y3 Ž . y3 y3 Ž .2.5=10 mol dm open symbols, solid lines and 10 mol dm closed symbols, dashed lines sodium

Ž . Ž . Ž . Ž . Ž . Žchloride solutions for times of I 20, ` 30, ^ 50, ` 100 and I 200 s after film formation. From.Hewitt et al., 1993, with permission.

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Reasonable agreement is obtained between experimental equilibrium film thicknessesŽ .i.e., when drainage has ceased and those calculated using the DLVO theory. Compres-sion of the double layer qualitatively explains the trends in the observed changes in thedrainage rates with variations in the ionic strength of the aqueous solution, however, the

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distances at which the differences in drainage rates become apparent are an order ofmagnitude greater than could be predicted or explained using the static or equilibriumDLVO model. Dissolved gas bubbles trapped on the solid surface and roughness may beimplicated.

Surface deformation of bubble surfaces can also occur under the influence ofŽ .electrostatic interactions and possibly other surface forces as well , aside from any

Ž .kinetic effects, as shown by Miklavcic et al. 1995 .Experimental evidence relating to film drainage in systems where soluble surfactants

are present is rather equivocal. Adsorption and desorption processes coupled withpossible molecular reorientation make any theoretical interpretation difficult. Unfortu-nately, these are the very systems which are of primary interest to mineral processing.Furthermore, additional complications ensue when one considers a particle approachinga bubble in flotation.

Ž .The nature of the bubble surface i.e., whether it is mobile or immobile willinfluence the evolution of the thin film between bubble and particle. This makes anysolution of the equation for film drainage difficult, particularly in the case of the angularparticles that are normally present in flotation. It is worth recalling at this point theobservations that smooth spheres float more slowly than angular particles under other-wise identical conditions, presumably because the asperities on the angular particles lead

Žto increased film drainage rates andror rupture Blake and Ralston, 1985; Anfruns and.Kitchener, 1986 .

An unstable film arises when there is a net attractive force between the particle andthe bubble. This normally occurs when there is an attractive hydrophobic force involved,

Ž .for the van der Waals and electrostatic forces the ‘DLVO’ components are repulsive,except in rare circumstances. The measurement of this hydrophobic force and itstheoretical origins are subjects of intense research effort. In recent times, it has becomepossible to measure the hydrophobic force, in a configuration relevant to the flotationprocess, by attaching a small particle to the cantilever in an atomic force microscopeŽ . Ž .AFM Fig. 7 . The particle is then pressed against a captive bubble and the force–sep-aration distance profile determined, for example, as shown by Butt, 1994, Ducker et al.,

Ž .1994 and Fielden et al. 1996 . In this fashion, the various surface forces may beexplored.

2.3. Forces between bubble and particle

The interaction between a silica particle and an air bubble is shown in Figs. 8 and 9Ž .Fielden et al., 1996 . It is monotonically repulsive on approach, in agreement with

Fig. 6. The inverse of the square of the film thickness against time after film formation is shown for drainageŽ . Ž .at the centre a and at the boundary ring b at a hydrophilic surface in aqueous sodium chloride solutions of

Ž . y3 y3 Ž . y4 y3 Ž . y4 y3 Ž . y5concentrations of B 10 mol dm . I 5=10 mol dm , ' 2.5=10 mol dm and ^ 10y3 Ž . Ž .mol dm sodium chloride solutions. Also shown are the models of ____ Frankel and Mysels 1962 and

Ž . Ž . Ž .- - - - Hartland and Robinson 1977 for drainage at the centre a . In the case of drainage at the boundary ringŽ . Ž . Ž . Ž . Ž .b , models are shown for the ____ Frankel and Mysels 1962 model and the - - - - Jain and Ivanov 1980

Ž . Ž .model see text . From Hewitt et al., 1993, with permission.

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Fig. 7. Schematic diagram of experimental arrangement for the measurement of forces between a particle and abubble with an atomic force microscope. The bubble and particle diameters were approximately 600 and 3

Žmm, respectively, and the cantilever typically had a spring constant of 0.050 Nrm. From Fielden et al., 1996,.with permission.

Žearlier bubble against plate observations Read and Kitchener, 1969; Blake and Kitch-.ener, 1972 . The repulsion is electrical in origin, revealed through the influence of

Fig. 8. Raw data for the air bubble–silica interaction in water. The interaction is monotonically repulsive onŽ .approach while on separation there is clear evidence of adhesion with F s3 mNrm normalised force . Thedet

loading force, F , prior to separation was 6 mNrm. B denotes the region of particle—‘constant compliance’l

region where the loading force is being monotonically increased. A is the region where repulsive forces areevident. C relates to the bubble–particle adhesion which becomes evident when the bubble is moved away

Ž .from the particle. From Fielden et al., 1996, with permission.

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Ž . Ž .Fig. 9. Normalised force FrR vs. separation D for air bubble–silica interaction in aqueous electrolyte.y4 Ž . y2 Ž .Symbols represent experimental data measured in 10 M ', ^ and 10 M ` sodium chloride. The

Ž . Ž .lines correspond to constant charge and constant potential - - - fits to DLVO theory. The fitting wasŽ .performed with the asymmetric DLVO program of Grabbe and Horn 1993 . c was obtained from forcep

measurements between silica particles under identical conditions and was y100 and y27 mV at sodiumchloride concentrations of 10y4 and 10y2 M, respectively, with c maintained at y34 mV. The analyticalb

concentration of electrolyte was used in the fitting procedure. The Hamaker constant for the interaction wasy1.0=10y20 J. In the inset, the detachment force–loading force data, normalised by particle radius, R ,p

Žmeasured on separation of the air bubble and particle are displayed. From Fielden et al., 1996, with.permission.

electrolyte concentration. For an air bubble and silica, upon removal, a small adhesionŽ .-4 mNrm normalised force was typically observed which was found to be dependenton the loading force, F , prior to removal, and on the electrolyte concentration.l

The contrast between the approach and removal curves with respect to the sign of theinteraction is significant. An attractive ‘jump’ may occur on approach but may not bedetected in the experiment. A monotonic repulsion implies the presence of a stable thinfilm, and an attractive force would be required for the removal of this aqueous film. Anadhesion is generally observed upon removal and it is possible that the reversal of thepiezoscanner, and attached bubble, perturbs the film and initiates drainage. The magni-

Ž .tude of the force Fig. 9 is not well described by theory. While there is some

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uncertainty in the values of bubble potential used to generate the theoretical fits, thegreatest uncertainty relates to the difficulties inherent in deconvoluting the AFM data.

ŽAlthough the ‘constant compliance’ slope is typically found to be acceptably linear Fig..8 , given the deformability of the bubble interface, its significance is not as clear as for

the interaction of nondeformable surfaces. The point of deviation of the approach curvefrom the constant compliance slope cannot simply be equated to zero separation. For amonotonically repulsive force prior to constant compliance, the onset of constantcompliance will correspond to some finite but indeterminate separation. Note that thescale of this separation is significantly less than the range of the operative surfaceforces; that is, the film is thin rather than thick. This underestimation of the separationmeans that the measured force is less than predicted by theory. Conversely, the

Ž .deformation of the bubble surface due to an electrical repulsion Miklavcic et al., 1995will lead to flattening, an increase in the area of interaction, and consequently, anincrease in the measured force. In the classic large bubble against plate measurements,the system may be described as an asymmetric interface of two parallel plates interact-

Ž .ing through a liquid film Blake and Kitchener, 1972; Hewitt et al., 1993 . Bubbles ofŽ .the smallest convenient diameter were used by Fielden et al. 1996 because their higher

Laplace pressure minimised the degree of deformation. Since the position of theliquidrvapour interface was not directly measured, it was also likely that the deformabil-ity of the bubble was not correctly accounted for in the deconvolution procedure. In this

Ž .context, the reasonable agreement between experiment and theory Fig. 9 is probablydue to the mutually compensating nature of these unknown factors.

Ž .When comparing experiment and theory Fig. 9 , it is tempting to infer that onapproach both the bubble and silica particles are interacting under constant chargeconditions. However, if this were the case, then one might have expected the experimen-tal data to increase sharply at small separations, where the flattening of the bubble would

Ž .be greatest Miklavcic et al., 1995 . This is clearly not the case, with the experimentaldata maintaining exponentiality over the entire range of separation. Intuitively, onewould expect the bubble and particle to interact at constant potential and constantcharge, respectively.

There are two possible explanations for the small adhesion observed between an airbubble and silica. If one, or both, of the surfaces were interacting at constant potential,then DLVO theory predicts that a charge reversal, for surfaces with potentials ofidentical sign but different magnitude, should occur as the separation is decreased, onthe surface of lower potential. Given that the van der Waals force is monotonicallyrepulsive the resulting interaction would have a primary minimum, and a stable air

Ž .bubble–particle aggregate would be formed with a thin 3.5 nm intervening liquid film.Ž .This is the so-called ‘contactless flotation’ of Derjaguin and Dukhin 1960 and

Ž . Ž .Derjaguin et al. 1984 Fig. 10 . The resulting adhesion would be expected to decreasewith ionic strength and to be independent of loading force. While the measured

Ž .detachment force Fig. 9 was found to broadly correlate in this way with ionic strength,a clear dependence on loading force was observed. If the silica surface is not hy-

Ž .drophilic, a possibility discussed by Ducker et al. 1994 , and the receding water contactangle is greater than zero, then the role of capillary forces must also be considered ifsurface force, kinetic, and hydrodynamic factors enable establishment of a three phase

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Fig. 10. DLVO calculation for the interaction of a silica particle and an air bubble in 10y4 M univalentelectrolyte. The potential of the silica particle was y100 mV, and that of the air bubble, y34 mV at infiniteseparation. The charge on silica and the potential of the air bubble were held constant as the separation was

y2 0 Ždecreased. The Hamaker constant for the interaction was y1.0=10 J. From Fielden et al., 1996, with.permission.

Ž . Ž .line TPL . Once a TPL is formed, capillary forces will dominate the adhesion Fig. 11as the contact angle increases, particularly because of the effect of contact anglehysteresis. If a TPL has receded across a surface, either due to spontaneous dewetting orforced movement due to an externally applied load, on separation, the contact line will

Ž .remain pinned Fig. 12 until the TPL subtends the advancing contact angle. TheŽ .expected adhesion, which may be readily calculated see below under detachment will

Ždepend on loading force as well as on electrolyte concentration since the contact angles.are affected, Fokkink and Ralston, 1989 . For the salt concentration range examined

Ž .here, the changes in contact angle are expected to be small Fokkink and Ralston, 1989Ž .a few degrees . For the silica–bubble interaction, the detachment force–loading forcedata may indeed be reasonably fitted by the capillary force given an advancing contactangle of 108 and a finite receding angle close to 08. The possibility of contaminationcannot be ruled out. One of the weaknesses of the colloid probe technique is thatbecause of the small surface area of the particle, and additionally, in the present case thebubble, it is almost impossible for surfaces to be kept entirely contaminant free. Giventhat silica surfaces are well known to contaminate very readily, as observed from contactangle studies, it is entirely possible that in this work, slight contamination would giverise to an advancing contact angle of 108, providing support for a capillary force

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Ž .Fig. 11. Variation of capillary force, F , with v8 particle location and with water contact angle, u in aircapŽ . Ž . Žbubble–particle interaction. F rR s2pg siny 180y v sin 180y v qu . From Fielden et al., 1996,cap p lv

.with permission.

dominated adhesion. It would be convenient if the adhesion could be attributed to eithersurface or capillary forces simply on the basis of the magnitude of the measuredadhesion. However, the magnitudes of the expected forces are very similar for the twocases.

2.3.1. Dehydroxylated silicaŽ .The interaction between dehydroxylated silica and an air bubble Fig. 13 was

repulsive at large separations but became attractive as the separation decreased, and asignificant adhesion was measured on removal. The force curves obtained for thisinteraction were found to be quite reproducible with respect to successive approach—re-moval cycles. A dependence of the interaction on electrolyte concentration was observedwhile the detachment force was an order of magnitude greater than that for theundehydroxylated silica–air bubble interaction. The detachment force was found todepend on loading force, but there was no clear dependence of adhesion on electrolyteconcentration evident. DLVO theory does not predict the dehydroxylation of silica tocause a significant change to the interaction, as only the potential of the particle surfaceŽ .obtained here from the nondeformable solid interaction has reduced slightly. However,there is clearly a marked change in the measured interaction on approach. If dehydrox-ylation of the silica surface caused it to interact under constant potential conditions, thencharge reversal of the bubble surface could occur at longer range and increase the depth

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Fig. 12. Loading–unloading cycle for contact angles of 0 and 328, as a function of v8 corresponding to thevalues of the receding and advancing contact angles, respectively, of water on dehydroxylated silica measured

Žin this work. F and F denote the loading force and detachment force, respectively. From Fielden et al.,l det.1996, with permission.

of the primary minimum. However, this is highly unlikely and, in the case of dehydrox-ylated silica, the attraction at decreasing separation is due to a hydrophobic force. Theuncertainties in the experimental data regarding deconvolution did not permit a quantita-tive analysis of this attractive component. An electrical force cannot be used to explainthe detachment force for dehydroxylated silica, simply because of the magnitude of themeasured adhesion, it is likely that capillary forces are responsible. If the hydrophobicattraction dominates the van der Waals repulsion at short range, then the aqueous filmseparating the particle and bubble would be expected to thin rapidly and a TPL would beestablished. Subsequently, capillary forces are expected to dominate the interaction. Ifinstead the van der Waals repulsion were to dominate the hydrophobic attraction, then aprimary minimum would occur and a stable thin film would exist between thebubble–particle aggregate. For a stable hydrophobic attraction, this latter possibilitywould give rise to a single-valued, electrolyte independent detachment force, in sharpcontrast to that observed. This explanation is also at odds with recent studies probing the

Žnature of hydrophobic surfaces Christenson and Claesson, 1988; Wood and Sharma,.1995 . However, because hydrophobic attractions are not yet completely understood and

are found to be variable in magnitude and range of operation even within a singleexperiment, this possibility cannot be entirely dismissed. In contrast, the alternativescenario supporting the formation of a TPL is given strong support because there isquantitative agreement between the measured and calculated values of the detachment

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Ž . Ž .Fig. 13. Normalised force FrR vs. separation D for interaction of dehydroxylated silica with an air bubbleŽ . y4 Ž . y2in aqueous electrolyte. Symbols represent experimental data measured in water = , 10 M ', D and 10

Ž . Ž . Ž .M `, v sodium chloride. The lines correspond to constant charge and constant potential - - - fits toDLVO theory. c was obtained from force measurements between dehydroxylated silica particles underp

identical conditions and was y85 and y32 mV at sodium chloride concentrations of 10y4 and 10y2 M,respectively, with c maintained at y34 mV. The analytical concentration of electrolyte was used in theb

fitting procedure. The Hamaker constant for the interaction was y1.0=10y20 J. In the case of 10y2 Melectrolyte, an outward shift of 15 nm for the fitted data allowed coincidence with the experimental data. Thisshift provides extra evidence of TPL formation. In the inset, the detachment force–loading force datameasured on separation of the air bubble and particle is displayed. In this case, the dashed line corresponds to

Ž . Ž .the capillary force calculated for contact angles of 08 receding and 298 advancing . The correspondingŽ .values of the contact angles, measured by the dynamic Wilhelmy method, were 08 receding and 328

Ž . Ž .advancing . From Fielden et al., 1996, with permission.

Ž .force-loading force relationship Fig. 13 inset . This was obtained assuming an advanc-Ž .ing contact angle of 298 compared with the measured value of 328 Fig. 12 .

2.4. TPLC

Ž .The kinetics of movement of the three phase line of contact TPLC are of centralimportance in many processes, apart from flotation. During the movement of the TPLC,a dynamic contact angle is established. Irrespective of whether the ‘surface chemical’,

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‘hydrodynamic’ or mixed ‘surface chemicalrhydrodynamic’ approaches are used, thereis as yet no general theory which adequately describes TPLC kinetics on all surfacesŽ .Blake, 1993; Hayes and Ralston, 1993 . One cannot generally calculate ab initio whatthe spreading velocity of the TPLC will be when an air bubble spreads over a mineralsurface immersed in water in the presence of a surfactant. Part of the problem at least isdue to the fact that poorly characterised experimental systems have been used, whereany generalisation has been obscured by the same time-dependent adsorptionrdesorp-tionrmolecular reorientation processes which complicate thin film drainage rate studies.Physical and chemical surface heterogeneities on the particle surface also stronglyinfluence the TPLC kinetics. Certainly, experimental evidence for t indicates that itTPLC

Ž .makes a significant contribution Schulze and Stechemesser, 1997 .At present, only the crudest estimates of film drainage time, t , and t can befilm TPLC

made. Hence, various experimental methods for determining t are frequently resortedind

to. Ye and Miller have developed a potentially valuable approach to the calculation ofcontact times, based on bubble deformation and restoration and have linked this to

Ž .induction time measurements and the flotation of coal Ye and Miller, 1989 .Experimental methods for determining induction times are generally based on either

pressing a bubble against a smooth mineral surface or against a bed of particles. TheŽ .disadvantages of all current methods for determining t include: 1 insufficientind

understanding of the process of bubble deformation and energy dissipation duringŽ .bubblerparticle collision; 2 insufficient information concerning the behaviour of the

Žattractive hydrophobic forces during the bubblerparticle interaction e.g., how the thinfilm of liquid evolves with time, during the time a particle slides or rolls around abubble; it may well be incorrect to assume that bubble–particle interaction ceases when

. Ž .the particle passes the bubble equator ; 3 data on t , e.g., influence of surfactant typefilmŽ .and concentration on thin-film drainage mechanisms and rate; and 4 data on t as aTPLC

function of hydrophobicity, physical and chemical surface heterogeneities and surfactantŽ .type. Recent work by Nguyen et al. 1997 indicates that a pseudo-line tension may play

a role in expansion of the three phase contact line.The most appropriate method for determining induction times is probably through

direct observation of bubblerparticle interactions in a flotation cell under well-definedŽ .conditions Schulze, 1984 . The necessary theory can then be developed. For the

present, the Sutherland and similar approaches serve as useful approximations indetermining t from experimental flotation data of the type normally generated.ind

Kinetic effects certainly have a strong influence on bubble–particle collision andattachment efficiencies. The latter provide the key to selective separations in flotation.We now focus on the experimental determination of attachment efficiencies and discusshow they may be described theoretically.

2.5. Attachment efficiency

2.5.1. Experimental measurementWhen flotation experiments are conducted within the flotation domain where E s1s

Ž .Hewitt et al., 1995; Dai et al., 1998 , as described below, then the attachment efficiencyŽ . Ž . Ž .E is the ratio of the collection efficiency E to the collision efficiency E , i.e.,a c

E sErE , where E can be experimentally measured. E was calculated using thea c c

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Ž .Generalised Sutherland Equation Dai et al., 1998 . This enables one to obtain experi-mental attachment efficiencies accurately and thus to test attachment efficiency modelswith confidence.

Ž Ž . . Ž .Fig. 14. Calculated Eq. 21 ; curves and experimental symbols attachment efficiency as a function ofparticle size for particles with various advancing water contact angles at different ionic strengths. d s1.52b

Ž .mm, pHs5.6. From Dai et al., 1998, with permission.

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Experimental attachment efficiency results determined as described above underŽ .various conditions are now presented Dai et al., 1998 . These results are then fitted to

Ž . Žthe modified Dobby–Finch Dobby and Finch, 1987 and the Yoon–Luttrell Yoon and.Luttrell, 1989 attachment efficiency models and the outcomes discussed.

A single bubble flotation experiment technique has been used for the determination ofthe collection efficiency. The samples used in the experiments were quartz particlesmethylated to different but known advancing water contact angles. The details concern-

Žing the experiments have been described elsewhere Hewitt et al., 1995; Dai et al.,.1998 .

Fig. 14 indicates that for a given particle size, the attachment efficiency monotoni-cally increases with contact angle, as well as with decreasing particle size, if the contactangle is kept constant. The influence of particle surface hydrophobicity on attachmentefficiency can be explained by its influence on the interaction energy between a particleand a bubble since the attractive hydrophobic interaction increases as the particle surfacehydrophobicity increases. This is accompanied by a decrease in induction time.

As is seen in Figs. 14 and 15, the attachment efficiency increases with electrolyteconcentration. An increase in electrolyte concentration reduces the interaction potentialenergy between the particle and bubble, as well as the critical film rupture thickness andincreases the film drainage rate.

The increase in attachment efficiency with electrolyte concentration is more pro-nounced for particles with greater hydrophobicity. Furthermore, as particle surfacehydrophobicity and electrolyte concentration are increased, the attachment efficiencybecomes less dependent upon particle size. In particular, for particles with contact angles

Ž Ž . . Ž .Fig. 15. Calculated Eq. 21 ; curves and experimental symbols attachment efficiency as a function ofŽ .particle size for particles with various advancing water contact angles u at different ionic strengths anda

w xbubble sizes. Curve 1 and symbol (: u s428, l s0, d s1.00 mm; curve 2 and symbol v: u s428,a b aw x w xl s0, d s0.77 mm; curve 3 and symbol I: u s468, l s0, d s0.77 mm; curve 4 and symbol B:b a b

w x w x y2u s748, l s0, d s0.77 mm; curve 5 and symbol ^: u s748, l s10 M, d s0.77 mm; pHs5.6.a b a bŽ .From Dai et al., 1998, with permission.

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w x y2larger than 708 and at KCl s10 M or higher, the attachment efficiency appears to beindependent of particle size.

The influence of bubble size on attachment efficiency is shown in Fig. 15. Fig. 15indicates that under otherwise similar conditions, smaller bubbles always give higherattachment efficiency than do larger bubbles. This is in good agreement with the

Ž .literature. For example, Yoon and Luttrell 1989 reported that for bubbles with diameterlarger than about 0.35 mm, the attachment efficiency decreases with increasing bubble

Ž .size. Hewitt et al. 1995 showed that bubbles of 0.75 mm in diameter always givehigher attachment efficiency than do 2.00 mm bubbles. In addition, Fig. 15 shows thatthe increase in attachment efficiencies caused by decreasing bubble size is morepronounced for smaller particles than for larger particles. Furthermore, Fig. 14 indicatesthat as the particle contact angle increases, the difference in the attachment efficiencybetween d s0.77 mm and d s1.52 mm becomes smaller. The attachment efficiencyb b

is less dependent upon bubble size for strongly hydrophobic particles, in agreement withŽ .the results of Hewitt et al. 1995 .

Figs. 14 and 15 show that for strongly hydrophobic particles, the influence of bubblesize on attachment efficiency becomes weaker when the electrolyte concentration isincreased. In fact, for the sample with a contact angle of 748 and at a KCl concentrationof 10y2 M, the difference in attachment efficiency between the two bubble sizes iswithin experimental error and the value of the attachment efficiency equals unity.

2.5.2. Attachment efficiency modelsŽ .The Dobby–Finch attachment efficiency model Dobby and Finch, 1987 is based on

the relative magnitude of the induction time and the sliding time. The latter is the timetaken for the particle to slide around the bubble surface, until it moves away from thebubble surface. Bubble surface deformation and thus particle rebound are ignored forfine particles. Assuming that particle–bubble collision occurs evenly over the section ofthe bubble surface between us0 and usu where u is measured from the frontc

stagnation point of the bubble, or the north pole in the case of a rising bubble, and u isc

the maximum possible collision angle, these authors proposed that the attachmentŽ .efficiency E is the ratio of the area corresponding to u to the area corresponding toa a

u . The basic equation of the model is expressed asc

sin2uaE s 9Ž .a 2sin uc

where u , termed the adhesion angle is a specific collision angle where if a particlea

collides at this angle, its sliding time is just equal to the induction time. So the angle ua

relates the sliding time and induction time to the attachment efficiency.Ž .According to Dobby and Finch, the maximum collision angle u is a function ofc

only the bubble Reynolds number. Expressions for u have been derived by fitting thecŽ .experimental results of Jowett 1980 where data show that the value of u decreasesc

linearly with the logarithm of the bubble Reynolds number. One obtains:u s78.1y7.37 log R for 20-R -400; 10Ž .c eb eb

u s85.5y12.49 log R for 1-R -20; 11Ž .c eb eb

u s85.0y2.50 log R for 0.1-R -1. 12Ž .c eb eb

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Ž .Nguyen 1994 improved this approach, accounting for the physical characteristics ofthe particles in addition to the Reynolds number.

ŽIn this study, to be consistent with the assumptions made in the collision model Dai.et al., 1998 , we used a maximum collision angle which is different to that used by

Ž .Dobby and Finch 1987 . This maximum collision angle, denoted by u , is a complextŽ .function of the bubble Reynolds number and satisfies the equation Dai et al., 1998 .

1r22 2sin u s2b 1qb yb 13Ž .Ž .t

Ž .in which the dimensionless number b , after modification Dai et al., 1998 is defined as

4d rp pbs 14Ž .

3Kd r yrŽ .b p f

where d and d are particle and bubble diameters, r and r are particle and fluidp b p f

densities, respectively, K is the Stokes number which characterises the ratio of theinertial forces of the particle to the viscous resistance of the fluid and is

r PÕ Pd2p b p

Ks 15Ž .9hdb

in which Õ is the bubble rising velocity and h is the dynamic viscosity of the fluid. Eq.bŽ .14 can be rewritten as

12 d r 1b fbs P P 16Ž .

d r yr Rp p f eb

Ž .where the bubble Reynolds number R iseb

R sÕ d r rh . 17Ž .eb b b f

As mentioned above, if a particle collides on the bubble surface at a certain angle andthen slides along the surface to the equator of the bubble, and if the sliding time is just

Ž .equal to the induction time, this particular collision angle is the adhesion angle u . Thea

sliding velocity, or sliding time, of a particle depends on the fluid flow regime at theŽ .bubble surface. It has been shown Dai et al., 1998 that the fluid flow at the bubble

surfaces under flotation conditions can be best described by potential flow. Therefore,the sliding time model for potential flow conditions should be used for the determination

Ž .of the adhesion angle u .aŽ . Ž .Dobby and Finch 1986 derived an equation for the sliding time t under potentialsl

flow conditions.

d qd up bt s P ln cot . 18Ž .sl 3 ž /2db

2 Õ qÕ q Õ qÕŽ . Ž .p b p b ž /d qdp b

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Ž .If the sliding time is substituted for the induction time t , the calculated angle isind

then the adhesion angle. Hence, u is determined by the equationa

3db

2 Õ qÕ q Õ qÕŽ . Ž .p b p b ž /d qdp bu s2 arccot exp t 19Ž .a ind d qdp b

3db

2 Õ qÕ q Õ qÕŽ . Ž .p b p b ž /d qdp bu s2 arctan exp yt . 20Ž .a ind d qdp b

Ž . Ž . Ž .Substitution of Eqs. 13 and 20 into Eq. 9 yields

3° ¶db2 Õ qÕ q Õ qÕŽ . Ž .p b p b ž /d qdp b2 B~ •sin 2 arctan exp yAPd Pp d qdp b

¢ ßE s 21Ž .a 1

222b 1qb ybŽ .

Ž .in which b is determined by Eq. 16 and the induction time can be represented as

t sAPd B 22Ž .ind p

in accordance with the well-documented dependence of the induction time upon particleŽ .size e.g., Schulze, 1984 .

In the fitting process, the parameters A and B for the induction time are adjustedindependently so as to obtain best agreement between calculated and experimental Ea

values for each experimental condition. The values of A and B thus obtained are thenused to calculate the induction time. Finally, the calculated induction time and itsvariation with various factors are compared with those reported in literature. Whetheragreement with literature is obtained or not is a critical assessment of the validity of the

Ž Ž ..modified E model Eq. 21 .a

The values of A and B may be determined through an iterative procedure asŽ .described elsewhere Dai, 1998 . At the first stage, for a given experimental E vs.a

Ž .particle size curve, an initial set of values of A and B was tried in Eq. 21 to produce aŽ cal.calculated attachment efficiency E since the value of b can be easily determineda

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Ž . Ž . calfrom Eqs. 14 and 16 . Then E was compared with the experimental attachmentaŽ exp. cal expefficiency E . If the difference between E and E is larger than the preseta a a

tolerable error, the values of A and B are independently changed and Ecal is calculateda

again. This procedure is repeated until the difference between Ecal and Eexp is smallera a

than the tolerable error. The last set of A and B values is recorded and the sameprocedure starts over for another Eexp vs. d curve. The aim of the above process is toa p

Ž .find out how the values of A and B and thus the induction time change with processŽ .parameters such as particle contact angle, bubble size and ionic strength .

The calculation using the above procedure shows that the value of B is almostindependent of particle contact angle, bubble size and ionic strength. This is in

Ž .agreement with the theory of Jowett 1980 and the experimental results of Ye andŽ .Miller 1989 . Since these data are obtained by fitting to the experimental E which wasa

determined from experimental collection efficiency and a collision efficiency model, thedata are rather scattered due to experimental error and imperfection of the collisionmodel. However, as all values of B fall in the range between 0.45 and 0.75, they can beapproximately taken as a constant.

At the second stage, the computation program is such that a constant value of B is tobe sought for all Eexp vs. d curves and an individual value of A is to be determined fora p

each of these curves so as to get the best agreement between Ecal and Eexp as a whole.a a

It turns out that a value of Bs0.6 is obtained. This value is very close to the average ofthe values of B determined at the first stage and falls well in the range between 0 and

Ž .1.5, as theoretically proposed by Jowett 1980 but is smaller than the experimentalŽ .value of 1.8 to 2.2, as reported by Ye and Miller 1989 .

Ž .The calculated attachment efficiencies based on a constant value of B s0.6 andindividual values of A are displayed in Figs. 14 and 15 together with experimentalattachment efficiencies. In general, the agreement between calculated and experimentalvalues is satisfactory.

Ž .The Yoon–Luttrell Yoon and Luttrell, 1989 attachment efficiency model is alsobased on the relative magnitude of the induction time and the sliding time, and thus, isproposed for fine particles which do not rebound from the bubble surfaces. It has threeequations, each corresponding to a particular fluid flow regime. For the case of potentialflow conditions, the Yoon–Luttrell E model is described by the equationa

y3Õ tb ind2sin 2 arctan exp2 ½ 5d qdsin u p ba

E s s 23Ž .a 2 2sin u sin 90c

y3Õ tb indwhere u s2 arctan expa d qdp b

2 Ž .is the expression for the adhesion angle u . The ‘sin 90’ in the denominator of Eq. 23a

implies the assumption that u s908, or in other words, particle–bubble collision occursc

uniformly over the entire upper half of the bubble surface. This has been shown to be

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Ž . Ž .incorrect e.g., Dai et al., 1998; Dobby and Finch, 1987 . If Eq. 13 is substituted for ucŽ . Ž . Ž .or u and Eq. 22 for t , Eq. 23 becomest ind

By3Õ APdb p2sin 2 arctan exp½ 5d qdŽ .p bE s . 24Ž .a 1

222b 1qb ybŽ .

A similar procedure to that used for the modified Dobby–Finch E model wasaŽ .applied to Eq. 24 and the same calculation results as shown in the preceding section

were obtained. This means that the two models, after modification, are unified. As aŽ . Ž . Ž .matter of fact, from Eqs. 9 – 12 and 23 , it can be seen that the original concept of

the two models is very similar, the main difference being the value of the angle u . Thisc

difference has been eliminated by using the correct expression for the maximumŽ Ž .. Žcollision angle u Eq. 3 . Another difference is the expression for the angle of u cf.t a

Ž . Ž ..Eqs. 20 and 23 . This difference originates from, in the Yoon–Luttrell model, theneglect of the contribution of the particle settling velocity to its sliding velocity and theidentification of the latter to the tangential velocity of the fluid at the bubble surface. In

Ž . Ž .fact, if it is assumed that Õ qÕ fÕ and d qd fd , Eqs. 20 and 21 reduce tob p b b p bŽ .Eq. 23 . Since these approximations are valid only when d -d , such as in the presentp b

study, very close results have been obtained from the two independent approaches. It isobvious that the modified Dobby–Finch E model is more general than the modifieda

Yoon–Luttrell model. However, this does not mean that the modified Dobby–Finchmodel can be applied to cases where coarse particles are involved since under suchcircumstances, bubble surface deformation and thus particle rebound which are ignoredin the Dobby–Finch model may be important.

Neither of these models address the fundamental issues of thin film drainage, andthree phase contact line movement which are the major contributors to E , readily seena

Ž .through the link between t , t or t , t and tind c sl film TPLC

t t q tŽ .ind film TPLCE s1y s1y 25Ž .a t tc c

In arriving at a theoretical description of E , which correctly incorporates filma

drainage and contact line movement, the angular dependence of the various forces mustbe incorporated. This is yet to be attempted and is a major task for the future.

3. Stability and detachment

3.1. Flotation limits for coarse particles

The essential problem in understanding bubble–particle aggregate stability is todetermine whether or not the adhesive force, acting on the three phase contact line, is

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large enough to prevent the destruction of the aggregate under the dynamic conditionswhich exist in flotation.

It is important that the reader understands the physics of the problem before movingon to a mathematical description. Let us consider a smooth spherical particle located atthe fluid interface. Once the equilibrium wetting perimeter has been established follow-ing spreading of the three phase contact line, the static buoyancy of this volume of the

Ž .particle will act against the gravitational force Fig. 16 . The hydrostatic pressure of theliquid column of height Z acts against the capillary pressure. The ‘other detaching0

forces’ require further discussion—since they arise from the particle motion relative tothe bubble, velocity dependent drag forces will oppose the detachment of the particlefrom the bubble. An analysis of these forces is extremely complex and has not beenreported to date.

The net adhesive force, F , is equal to the sum of the attachment forces, F , minusad a

the detachment forces, F , i.e.:d

F sF yF . 26Ž .ad a d

An equilibrium position is achieved if F is zero. The particle will not remain attachedad

to the bubble if F is negative but will be present in the liquid phase.ad

The mathematical description of the various forces which dictate the equilibriumposition of particles at liquid–vapour or liquid–liquid interfaces has followed anevolutionary trail. Analogous processes of interest, for example, include pigment‘flushing’, where a solid particle is induced to transfer from one liquid phase to anotherby appropriate surface modification with surfactants and the stabilisation of emulsiondroplets by solid particles.

Ž .It was Princen 1969 who proposed the first extensive and generalised treatment ofthe forces acting on a particle at fluid interfaces. This theory was developed further by

Ž .Schulze 1977, 1984 .

Fig. 16. Location of a smooth spherical particle of radius, R at a fluid interface. The parameters shownpŽ . Ž .enable the force and energy balance to be calculated see text . From Schulze, 1984, with permission.

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It is now pertinent to consider the case of a spherical particle at liquid–air interface.The force balance will be used and then linked to the energy balance. We assume thatthe system is in a quasistatic state and that the contact angle corresponds to that obtainedfor a static system. The dynamic contact angle can depart significantly from the staticvalue, depending in part on the velocity of the TPLC. If the particle oscillates around itsequilibrium position, the TPLC would be expected to move to some extent.

Hence, a full analysis would need to account for the velocity dependent drag forcesmentioned above and link these to the contact angle dynamics. This is not a tractableproblem at present so that a simpler approach is necessary.

Let us suppose that a spherical particle of radius R is attached to a bubble of radiusp

R where R is much greater than R , as shown in Fig. 16. From understanding theb b pŽ .forces e.g., capillary, buoyancy, capillary pressure, etc. which operate on the particle,

it is possible to calculate the energy of detachment.The energy of detachment, E , corresponds to the work done in forcing a particle todet

Ž .move from its equilibrium position, h v at the liquid–vapour interface, as a functioneqŽ . Ž .of the central angle v Fig. 16 , to some critical point, h v , where detachmentcrit

occurs and the particle moves into the liquid phase. The sum of the various forces, ÝF,is related to E by:det

Ž .h vcritE s ÝFdh v . 27Ž . Ž .HdetŽ .h veq

The detachment process takes place when the kinetic energy of the particle equalsŽ . 3 2E . The kinetic energy of the particle is given by 2r3 p R r V , where V is thedet p p t t

Ž .relative turbulent velocity of the particle, acquired due to stresses on the bubble–par-ticle aggregate in the turbulent field of the flotation cell, as the aggregate collides withother bubbles or aggregates or due to other modes of excitation. V corresponds to thet

velocity of gas bubbles in the flotation cell. r and r refer to the densities of thep fl

particle and fluid, respectively.The maximum floatable particle diameter based on the kinetic theory, D , ismax, K

given as:

3 2 2 r 3hŽ .h v pcrit 3 3 2D s2 p R r g 1y ycos vq sin vHmax , K p fl2 ½ ž3 r 2 R2pr V Ž .h v fl pp t eq

1

3 2g 32y sinvsin vqu yp R sinv y2 R r g dhŽ . Ž .p b fl2 2 5ž // Ra R bp

28Ž .

where a is the capillary or Laplace constant,

p gfl.( g

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Ž .Eq. 28 may also be solved by numerical integration or by plotting each of thekinetic and detachment energies as a function of R at constant g and r and specifiedp p

V . This equation has been shown to adequately describe the detachment of a spheret

from a liquid–vapour interface by Schulze as well as the behaviour of hydrophobicangular quartz particles, between approximately 35 to 120 mm in diameter, under

Ž .flotation conditions by Crawford and Ralston 1988 .

3.2. Flotation limits for fine particles

The only theoretical study to date dealing with the limit of floatability of fineŽ .particles was proposed by Scheludko et al. 1976 . The limit is the critical work of

expansion required to initiate a primary hold or three phase contact line duringbubble–particle approach, a requirement which is met by the kinetic energy of theparticles. The matching of these two quantities enables a minimum particle diameter,D for flotation to be obtained:min, K

123k 3

D s2 29Ž .min , K 2 � 4V D rg 1ycosut

where k is the line tension, opposing expansion of the three phase contact, D r is thedifference between particle and fluid densities and u is the contact angle at the TPLC. Itwas Gibbs who first identified the importance of line tension, using as example the lineof intersection of the three surfaces of discontinuity which exist when two gas bubblesadhere together.

Molecules which are present in a line have a free energy which is different fromthose at a surface—in fact, there is an excess linear free energy and a linear tension inan analogous fashion to that of excess surface free energy and surface tension.

In fact,

EFks 30Ž .ž /EL T ,V ,W

where F is the Helmholtz free energy, L is the contact line, W is the thermodynamicwork, T is the temperature and V is the volume. The Young–Dupre equation becomes:

kg yg sg cos u" . 31Ž .Sr V Sr L L r V r

The line tension is important for small contact radii and can oppose or reinforceg cosu. It counteracts the formation of the three phase contact line in Scheludko’sLr V

theory, the latter neglecting thin film drainage and other hydrodynamic effects. TheŽ .experimental data of Crawford and Ralston 1988 for hydrophobic, angular quartz

particles between about 10 to 30 mm in diameter follow a general trend which isŽ . Ž .predicted by Eq. 29 . Drelich and Miller 1992 have shown that acceptable agreement

is achieved if a pseudo-line tension, embracing surface heterogeneities, replaces k inŽ . Ž .Eq. 29 . This in turn enables D in Eq. 29 to be reexpressed in terms of a criticalmin

bubble radius below which attachment does not occur. Reconciliation between theory

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and experiment is then achieved, although the concept of pseudo-line tension needs to beplaced on a firmer experimental footing.

Hysteresis, the difference between static and dynamic contact angles caused bysurface heterogeneities, is important in both attachment and detachment. Contact linepinning can occur during movement of the TPLC, inhibiting attachment or detachment,i.e., the bubble–particle union may be stabilised. A quantitative description of the effectof surface heterogeneities on TPLC movement is not yet available, although recent work

Ž .holds promise Petrov et al., in press .

4. The future

In terms of our fundamental understanding, thin film drainage is poorly understoodwhen one of the interfaces is both physically and chemically heterogeneous, the otherdeformable. The nature of the hydrophobic interaction between a particle and a bubblerequires both experimental and theoretical verification, particularly the influence ofdissolved gases. There is no reliable model at present to describe the movement of athree phase contact line over a rough surface, nor to describe the thin film drainageprocess between a bubble and a particle. The angle dependence of the interactionbetween a bubble and particle must be addressed during film drainage. Thus, majorresearch challenges exist which, if they are to be successfully overcome, must embracesystems where surfactants are both present and absent.

From a separation technology point of view, froth flotation will continue to be one ofthe principal means by which ores are successfully beneficiated for many years to come.Increasingly, the technique is also being used in the deinking of paper, soil remediation,plastics recycling and heavy metal ion contamination, to name but a few examples. Bothresearch and practice are expected to accelerate strongly over the next decades as newtechniques and theoretical approaches are used.

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