Drag force on a spherical particle moving through a foam: The role of wettability

International Journal of Mineral Processing 102–103 (2012) 78–88

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Drag force on a spherical particle moving through a foam: The role of wettability Peter M. Ireland ⁎, Graeme J. Jameson Centre for Multiphase Processes/Centre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, Australia

a r t i c l e

i n f o

Article history: Received 7 May 2011 Received in revised form 8 September 2011 Accepted 25 September 2011 Available online 1 October 2011 Keywords: Foam Drag Flotation Contact angle Particle

a b s t r a c t Experimental measurements are presented of the drag force on a spherical probe particle moving through an aqueous foam at the low speed limit, for particle/bubble radius ratios of 1.5–3.0. In froth flotation, this force is critical for understanding selectivity of these particles. The effect of surface wettability and particle diameter is both explored. A ‘hybrid’ model of the force is developed and compared with the experimental data. This combines forces due to yield of the foam ‘continuum’, both in the bulk and at the foam–particle interface, with forces due to attachment/detachment of individual foam nodes from the particle. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The drag force experienced by a particle in motion through a foam depends on its size. Particles that are substantially smaller than the bubbles can move relatively easily through the Plateau borders. Movement of larger particles through the foam requires substantial deformation and structural rearrangement, and each particle may be in simultaneous contact with multiple bubbles. Historically, much of the work on particle mobility in foams has been motivated directly or indirectly by the need to understand the mechanisms of froth flotation, and that application is therefore worth discussing briefly at the outset. Froth flotation is widely used to separate mineral particles based on their surface properties. Hydrophobic particles tend to attach to bubbles, and rise to the top of the flotation cell, whereas hydrophilic particles stay in the liquid phase and drain back into the tails. Thus, previous work on particle mobility in foams has also focused strongly on the effect of the wetting contact angle at the particle surface, since this is the key to selectivity in flotation. In the past, flotation was used almost exclusively for particles that were much smaller than the bubbles (Jameson, 2010; Jowett, 1980), which explains the emphasis in the literature on the mobility of particles in that size regime (Honaker and Ozsever, 2003; Honaker et al., 2006; Meloy et al., 2007; Nguyen and Evans, 2002; Nguyen and Evans, 2004). More recently, a great deal of attention has been given to separating much larger particles. In many cases, this can yield substantial energy savings in downstream processes. However, the

⁎ Corresponding author. Tel.: + 61 2 4921 5653; fax: + 61 2 4960 1445. E-mail address: [email protected] (P.M. Ireland). 0301-7516/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2011.09.010

physical mechanisms of selectivity must be re-assessed in this regime. In this context, the critical determinant of a particle's floatability is the force needed to perturb its position in the foam irreversibly, and whether its weight exceeds this force. The role played by surface wettability in floatability of these larger particles in the froth layer has thus far been poorly understood. The purpose of the present study is to understand it better, particularly as differences in wettability traditionally form the very basis of flotation selectivity. We concentrate on spherical particles in this study, although it is recognized that particle shape, particularly sharp edges and asperities, can have a profound influence on interaction with foam films (Morris et al., 2011). For particles that are much larger than the bubbles, the foam could in principle be treated as a continuous deformable medium and its rheology described by an appropriate continuum model and boundary conditions. For instance, aqueous foams are often described in terms of yield stress, viscosity, and so on (e.g., Herzhaft, 1999; Khan et al., 1988; Princen and Kiss, 1989). A great deal of work has been done on the drag force on spherical bodies moving through various media, including Newtonian, Bingham and Herschel–Bulkley fluids (Chhabra, 2006). However, as the particle size decreases toward that of the bubbles, variations in the force due to the foam structure become increasingly important. A number of authors have investigated the drag force on a body moving through a foam, taking into account the foam's discrete character and the topological changes that must occur for motion to take place. Dollet and co-workers (Dollet et al., 2005a,b) measured the force on 2D objects of various shapes in a flow of two-dimensional foam. Davies and Cox (2009) and Cox et al. (2006) performed simulations of the force on disks (effectively, ‘2D spheres’) moving relative to a 2D foam. Raufaste et al. (2007) simulated and performed experiments on the force between disks and 2D foams, with special emphasis on the effect of liquid fraction. The effect of liquid fraction was not examined

P.M. Ireland, G.J. Jameson / International Journal of Mineral Processing 102–103 (2012) 78–88

in the present experimental study. Instead, all experiments were carried out in a foam with a constant liquid fraction of less than 0.05. The present work is based on an experimental study of the drag force on spherical bodies moving through a foam (i.e., in 3D). The measurement technique is similar to others developed independently to study drag on objects in foams (Cantat and Pitois, 2005; de Bruyn, 2004). However, we focus on a variable whose effect on the drag force has not been studied in detail — the wetting contact angle of the sphere surface by the interstitial liquid. Previous work (Ireland, 2008; Ireland and Jameson, 2009) has dealt with the interaction forces between foams, foam films, and solid flat surfaces of variable wettability. An interesting aspect of those studies was evidence of a minimum ‘yield tension’ required for a foam film contact line to move across a surface. This effect is similar to, but subtly distinct from, the more familiar phenomenon of ‘pinning’ of simple threephase contact lines on macroscopically flat surfaces. In consequence, the interface between a liquid foam and a solid surface exhibits a yield stress that is distinct from the bulk yield stress of the foam, and depends on the contact angle of the liquid phase on the surface (Fig. 1). This phenomenon was not included in previous experimental studies, e.g. de Bruyn (2004), where non-slip contact at the interface was assumed. The key question now is the extent to which our existing insight into these interactions can be applied to understanding the drag force on spheres in foams. Since the previous studies dealt with forces in the low-speed limit, where yield forces were strongly predominant

Fig. 1. Relationship between yield tension of a foam film contact line and surface yield stress of a foam.

79

over viscous forces, the present study does likewise. The possibility that the system will display hysteresis – i.e., that the force required for incipient motion of the particle will be different to the drag force at the low-speed limit when approached from above – is recognized, but not considered in this study. The relative size of the particle and bubbles will dictate the type of model used to calculate the drag force. If the particle is substantially larger than the bubbles, the foam can be modeled as a homogeneous continuum, with an appropriate slip condition at the particle–foam interface. If, on the other hand, the particle is smaller than the bubbles, its motion through the foam can be modeled as a series of escapes from and captures by the nodes in the foam structure, with the forces calculated from mechanical deformation of the foam films. This study deals with particles that are intermediate in size between these extremes, typical of the ‘coarse flotation’ regime mentioned earlier. In this regime, the bubbles are too large for the foam to be thought of unequivocally as a continuous medium. On the other hand, the particles are large enough to be in contact with more than one bubble or foam film simultaneously. The drag force cannot therefore be described exclusively in terms of film or node attachments and detachments. As the particle comes into contact with more bubbles simultaneously, the geometric complexity of the film contact lines, as they migrate around the particle surface from the front to the rear, increases rapidly. The pressure difference between the front and rear of the particle also becomes important. The complex interaction and combination of the pressure and film network forces preclude analytic structural model forces, as was possible in the small-particle regime. There are two obvious alternatives for modeling the system in this intermediate size regime. The first is to construct a detailed three-dimensional numerical model, analogous to that of Raufaste et al. (2007). The second alternative is to develop a ‘hybrid’ model that takes elements from both a homogeneous continuum model and a single node attachment/detachment model, and combines them to ‘bridge’ the difficult intermediate regime. We have adopted the latter strategy in this work. The key assumption to be tested is that a continuum model can be used for the mean drag force, while transient variations from this mean can be modeled using a simple structural node attachment/detachment model. It is important to be clear which drag-causing mechanisms contribute to which component of the model. The force on the particle due to pressure variations in the surrounding foam and shear forces at the interface clearly falls within the continuum model (see below for an introduction to the concept of film–wall contact line yield tension and foam–wall interface yield stress). It is assumed that the transients in the drag force are the same as they would be for sequential attachment and detachment of single foam nodes from the particle. Deformation of the foam around a moving particle results in a larger density of contact lines at the rear of the particle than that at the front, resulting in a net drag force (Raufaste et al., 2007). For a small particle, this contact line distribution is unequivocally part of the film deformation associated with node attachment and detachment, and the force can be calculated using an analytic structural model. For a large particle, the surface contact line density gradient plays the same role as the surface distribution of normal stress in a continuum model, and is unequivocally part of the continuum drag component. In the intermediate size range, the status of this force component is ambiguous. It remains to be seen whether this ambiguity invalidates the ‘hybrid’ modeling approach outlined above. The mean drag force is modeled by assuming that the foam can be treated as a continuous yield-stress material, and that the effect of tangential stress at the body's surface varies with the contact angle. The fluctuations about this mean drag force are modeled in terms of attachment and detachment of the sphere from Plateau border nodes in the foam (Fig. 2), and may also be dependent on the contact angle. Topologically, the attachment process is the reverse of the detachment process, and the attachment and detachment forces are therefore

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through a foam. Probes with a number of different surface coatings were used to measure the effect of the contact angle. The probe particles were 1.5–3.0 times the radius of the bubbles, i.e., in the intermediate range mentioned earlier. In contrast, de Bruyn (2004) used a probe many orders of magnitude larger than the bubbles, whereas Cantat and Pitois (2005) used a probe that was approximately half the bubble diameter. Finally, we evaluate whether these data are consistent with the ‘hybrid’ force model introduced above, in which the time-averaged drag force is calculated using the continuum model, and the fluctuations around this average are calculated using the node attachment/ detachment model. 2. Models 2.1. Continuum model

Fig. 2. Mechanisms involved in movement of a spherical particle through a foam.

in opposite directions. The force predicted by the continuum model is of course always in a direction opposite to the particle motion. In this paper, we initially develop these two drag force models separately. The ‘continuum’ model for relatively large particles is presented first. Aqueous foams with a liquid volume fraction of less than ~0.7 display a measurable bulk yield stress (Gardiner et al., 1998), and we therefore adopt the simplest yield stress fluid model, that of Bingham (Chhabra, 2006). The interface slip condition is derived from foam-surface yield study of Ireland and Jameson (2009). In the second ‘node attachment/detachment’ model, for particles that are smaller than the bubbles, we calculate the deformation of a foam film junction when an embedded particle is perturbed from its equilibrium position, and use this to calculate the associated escape and capture forces. In both models, the dependence of the force on the contact angle is emphasized, since surface wettability traditionally forms the physical basis of selectivity in flotation. The node attachment/ detachment model is also able to be expressed directly in terms of the liquid fraction, although that variable was kept at a constant low value in the experiments. Data are then presented from a series of experiments in which a force transducer was used to measure the time-dependent drag force on a spherical probe particle moving

We deal initially with the average force on the body, and assume temporarily that the foam can be treated as a homogeneous medium. Substantial work has been done on the creeping motion of a solid sphere through a yield-stress fluid. The simplest and most commonlyused model for these fluids is the Bingham body (Chhabra, 2006), deformation of which beyond the yield stress τy is characterized by a bulk viscosity μ. This is the linear case of the more general Herschel– Bulkley fluid, where the shear-rate-dependent term is proportional to a nonlinear power of the shear rate. Since the shape of the plastic and rigid regions of the fluid is generally not known beforehand, analytic calculations of flows of viscoplastic fluids are generally not feasible, and numerical models tend to be used instead. A good example is the study by Blackery and Mitsoulis (1997), who modeled the creeping motion of a sphere through a Bingham fluid contained within a solid cylinder of varying diameter. As in almost all similar studies, it was assumed that no slip occurred at the surface of the sphere. The model was formulated in terms of a number of dimensionless groups, including the Stokes drag coefficient CS and the ‘Bingham number’ Bn. This latter parameter characterizes the importance of the yield stress relative to the viscous stress. In this context, CS ¼

F ; 6πμVR

ð1Þ

2τy R ; μV

ð2Þ

and Bn ¼

where R is the radius of the sphere, V is its velocity, and F is the drag force. In the simulations of Blackery and Mitsoulis, the solutions for all ratios of the sphere diameter to that of the containing cylinder approached the limit CS = 1.17 Bn as the Bingham number became large. This applies to Herschel–Bulkley fluids in general, since the entire shear-rate-dependent term vanishes in this regime. Eqs. (1) and (2) imply that in this regime (where the yield stress strongly predominates over the viscous stress), 2

F ¼ 14:04πτy R :

Fig. 3. Rigid and plastic zones in a viscoplastic fluid around a moving sphere: (a) No interfacial slip (Beris et al., 1985); (b) No slip and (c) full slip (de Besses et al., 2004).

ð3Þ

A very large Bingham number is assumed in this work, and this assumption is tested experimentally. A recent study (Ireland and Jameson, 2009) indicated that the interface between an aqueous foam and a solid surface has its own yield stress whose value is strongly dependent on the wettability of the surface. Once this surface yield stress τys reaches a value on par with that of the bulk foam, τyb, a non-slip boundary condition can be assumed. In models of the flow of Newtonian liquids around objects, the total drag force is usually resolved as a sum of ‘form drag’ and ‘skin drag’ components. Large differences in the drag coefficients of

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We denote the full non-slip drag value derived from Blackery and Mitsoulis (1997), (see Eq. (3)), as Fns: b 2

Fns ¼ 14:04πτy R :

ð5Þ

This result and those of de Besses et al. (2004) and Liu et al. (2002) for this case are in good agreement. From this, if the average drag force was simply the superposition of form and skin drag components, and the skin drag was equal to Fref, we would expect the average net drag force on the sphere to be hFi ¼

8

 þ π 2 R 2 τsy −τ by ;

:F

;

ns ns

Fig. 4. Deformation of a Plateau border junction with an externally-loaded spherical particle embedded.

rough and smooth spheres moving through viscoplastic fluids have been measured by Merkak et al. (2006) and Jossic and Magnin (2001), among others. In this study, we test whether a similar approach can be adopted to the drag force on a sphere moving through a foam. In a Newtonian fluid at low Reynolds numbers, the form drag is the result of a pressure difference between the leading and trailing sides of the body, and is therefore only non-zero if boundary layer separation occurs. However, a material with a yield stress can sustain normal and shear stresses, and even at very low deformation velocities. A sphere moving through a yield-stress fluid will therefore experience non-zero form drag even at low velocities and in the absence of boundary layer separation. Without a detailed numerical model, it is difficult to precisely calculate the skin drag component for τys b τyb. Simulations, (Beris et al., 1985; de Besses et al., 2004) indicate that a viscoplastic fluid around a moving sphere, with no slip at the interface, features an inner, liquefied zone enclosed by the outer, rigid material (Fig. 3). Small unyielded ‘caps’ of material are also found next to the sphere surface in the stagnation zones at the poles. If the interface between the liquid and the sphere has a lower yield stress than the bulk fluid, this is likely to have an effect on the pattern of plastic and rigid zones. For the planar surface experiments of Ireland and Jameson (2009), the interaction between bulk and surface yield was very straightforward. For a spherical surface, this interaction is potentially very complicated. De Besses et al. (2004) modeled both the non-slip and full-slip cases, and found that the rigid and plastic zones changed substantially between the two cases. A key question is now whether the drag force can be modeled as a simple sum of foam and skin drag components; if so, the drag force in cases intermediate between no slip and full slip can be calculated quite simply. To confirm whether this is true, we now calculate a reference value of the skin drag, corresponding to the situation where the shear stress parallel to the interface is equal to τys everywhere. The component of τys acting in the z-direction will be τys sin φ (refer to Fig. 3). To obtain the net force on the sphere, we integrate this over the entire surface: π
 s Fref ¼ ∫ τy sinφ ð2πR sinφÞRdφ 0

¼ π2 R2 τsy :

ð4Þ

s

τ sy b τby

b

τ y ≥τy

:

ð6Þ

If we were to assume conditions of perfect slip at the interface, τys = 0, and thus (from Eqs. (5) and (6)), the drag force would be ~0.78 of the non-slip value. The modeling of de Besses et al. (2004) predicts a ratio of ~ 0.73. The discrepancy between these values will be considered further later in this paper. In previous studies, Ireland (2008) and Ireland and Jameson (2009) measured the force per unit length on foam film contact lines during very slow motion across flat surfaces of varying wettability, and then measured τys as a function of this ‘yield tension’ for these same flat surfaces. It is worth noting that this yield tension, and the associated interfacial yield stress, will be zero if the surface is continuously wetted by the liquid. However, in the present experiments, involving relatively dry foam, such a wetting film was neither expected nor observed, even at the smallest contact angles (see ‘Results’, below). It is worth noting that the presence or absence of a continuous wetting film should also be important in determining the viscous interfacial stress at higher speeds. In the absence of continuous wetting, the viscous force at the interface will be due to the confined circulation of liquid in the film–surface junctions. On the other hand, a continuous wetting film will undergo plane shear, forming a lubricating layer between the foam and the surface, while some circulation in the junctions will still occur at the same time (Denkov et al., 2006; Saugey et al., 2006; Tisne et al., 2004). The previous studies (Ireland, 2008; Ireland and Jameson, 2009) were carried out with an identical surfactant solution and in the same foam column as the experiments in the present study, so the resultant foam has the same properties in this case. The bulk yield tension τyb was also estimated for this foam. Both of the previous studies included analytical models of the yield tension and surface yield stress. However, for the purposes of the present study, we simply combine their experimental results to obtain an empirical relationship between the wetting contact angle θc and the surface yield stress, which can be summarized as s

τy ≅ 44:8 ½ sinð0:858 θc −1:51Þ þ 1

−2

N:m

ð7Þ

(for a contact angle expressed in radians). For this particular foam, the surface yield stress rose from zero for perfect wetting to a value of ~ 0.54 N.m − 2 at a contact angle of ~ 50°. This latter value was the bulk yield stress of the foam, and the surface yield stress could therefore not be measured for larger contact angles. The bulk yield stress estimate was τyb ≅ 8.7 N.m − 2. Combining these empirical results with Eqs. (5) and (6), the drag force is found as a function of θc and R, and can be compared with experimental values. 2.2. Node attachment/detachment model The simple ‘continuum model’ outlined above was based on the assumption that the foam could be treated as a homogeneous medium. In reality, movement of an object through a foam necessitates a series of discrete topological changes in the foam structure as foam

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Fig. 7. Dimensionless restoring force vs. angle α for a sphere embedded in a Plateau border junction. In the undeformed state, α = 54.7∘. Increased deformation corresponds to a decrease in α.

Fig. 5. 2D section through ‘foot’ of a liquid film intersecting a curved solid surface. The two lamellae diverge to join the surface at θc.

nodes attach to the front and detach from the rear of the object. The ‘homogeneous fluid’ model can therefore do no more than predict the average drag force on the object. Variations about this mean due to the structure of the foam require a model of node attachment and detachment. The spherical probes used in the present study were ~1.5–3.0 times the radius of the mean bubble diameter in the foam, so we expect that in many cases the variations in the drag force due to the foam structure will be nearly as large as the mean drag force. Raufaste et al. (2007) measured and modeled the force on circular disks in a flow of two-dimensional foam. Full slip at the interface and a zero contact angle were assumed. Attachment and detachment of nodes were governed by a ‘wall cut-off length’, where two film contact points became close enough that the spreading lamellar ‘feet’ of two films met and joined (see also Cox et al., 2006).

Since the Plateau border curvature is dependent on the liquid fraction of the foam, this cut-off length is governed by the liquid fraction. We do not attempt to model the foam structure in detail. However, since this study concerns spherical bodies and the effect of wettability, we adapt the ‘cut-off length’ concept accordingly. Consider a spherical body embedded in an ‘ideal’ Plateau border vertex (Fig. 4), and subjected to a downward external force. If this external force increases quasi-statically, the film structure deforms as shown, until a topological transformation occurs where the triangular face at the top of the sphere (as shown in Fig. 4) becomes a Plateau border junction, which detaches from the sphere. This happens when either (a) the lamellar ‘feet’ touch and join at the top of the sphere (an occurrence that we call ‘crowning’); (b) a maximum in the restoring force curve (a ‘tipping point’) is passed, and the system passes dynamically to (a). In both instances, a topological change corresponding to node detachment occurs. Fig. 5 shows a section through the lamellar ‘foot’ for a contact angle θc, sphere radius R and Plateau border curvature radius r. The

Fig. 6. The arrangement of films, Plateau borders and forces just prior to ‘crowning’. (a) The relationship between α and X (refer to Fig. 5). Crowning occurs when α = X. (b) Position of contact lines, Plateau borders (P.B.) and ‘baseline’ chords (S1 and S2), and direction of film forces. (c) Geometric construction showing the relationship between α, S1 and S2.

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angle subtended at the center of the sphere by the ‘foot’ from its symmetry line, ∠ CBD, is denoted X, and is given by cosX ¼

Q¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2 1−Q 2 1−Q 2 sin2 θc þ Q sinθc ;

R R2 1 þ 2 cosθc þ 2 r r

ð8Þ

!−1 2

:

ð9Þ

Fig. 6 shows how the value of angle X calculated above determines the force required for detachment. In Fig. 6(a), we see that the entire triangular upper face will vanish when the contact line midpoint angle, α, is equal to the foot angle, X. Fig. 6(b) shows the direction of the forces due to six films intersecting the sphere surface (indicated by arrows). The force normal to the surface due to each of the contact lines is equal to the film tension, denoted Tn (= 2γlg), multiplied by the length of the chord joining the endpoints of the contact line (i.e., the ‘baseline’ along which the film tension acts). We might expect an extra force to be produced by the lower pressure in the Plateau borders, acting over the lamellar ‘footprint’. However, in a dry foam, Laplace–Young equilibrium requires that this pressure force be precisely counteracted by the reduction in vertical lamellar forces due to the lamellae intersecting the surface at an oblique angle θc (Ireland, 2008). For angles α and β, as shown in Fig. 6(c), the ‘baseline’ chord lengths of the contact lines are as follows: S1 ¼

pffiffiffi 3R sinβ

S2 ¼

pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R 1 þ cosβ;

ð10Þ

Fig. 9. Apparatus used to measure the force on a spherical particle moving through a foam.

where tanβ ¼ 2 tan α:

ð11Þ

The force at each of the three ‘S1’ contact lines acts at an angle α to the z-axis, while that at each of the ‘S2’ contact lines acts at an angle (π−β)/2 from the z-axis. The net restoring force in the z-direction is thus the sum of the z-component of all six contact line forces: Fres ¼ 6γ lg R

hpffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 3 sin β cos α− ð1 þ cos βÞð1− cos βÞ :

ð12Þ

Fig. 7 shows the ‘S1’ and ‘S2’ force components and the net force on the sphere in dimensionless form, plotted against the angle α. As

Fig. 8. Crowning angle vs. wetting contact angle for different ratios of the Plateau border/particle surface curvature ratio. The angle of maximum restoring force is also shown. The greater of the two angles governs detachment of the node from the particle.

expected, the net force is zero where α = 54.7 ∘, its equilibrium value according to the Laws of Plateau (Weaire and Hutzler, 2001). An important feature of the restoring force curve is a maximum at α = 25.6 ∘, corresponding to Fres/(γlgR) = 2.333, and the system cannot continue to deform quasi-statically if the external force on the sphere is greater than this value. It therefore represents a ‘tipping point’ beyond which the sphere automatically escapes from the node. ‘Crowning’, on the other hand, will occur when α is equal to the angular half-width X of the lamellar ‘footprint’, as given by Eq. (8) (Fig. 6). This is the three-dimensional equivalent of the twodimensional ‘cut-off length’ concept of Raufaste et al. (2007). The Plateau border node will detach from the sphere surface when either crowning occurs or the maximum in the force curve is exceeded, whichever requires less deformation. Fig. 8 shows X as a function of the contact angle θc for a variety of ratios of the Plateau border curvature to the sphere radius. The maximum-force value of α is also shown. This maximum occurs because the force in the z-direction depends not only on the orientation of the incident films, but also on the length of the contact lines; when α decreases below 25.6°, the length of the contact lines decreases more rapidly than the increase in the zcomponent of the film tension. (Interestingly, in two dimensions there is no such ‘tipping point’, since there are no contact lines as such, merely contact points. The force will increase steadily as the films migrate around to the trailing edge of the object.) Referring to Fig. 8, we see that if r/R is less than ~0.75, the maximum in the force curve will be exceeded before crowning occurs, at all contact angles. The crowning angle depends on the contact angle and the film curvature, and thus on the liquid fraction. We would therefore expect the detachment force to be independent of the contact angle and liquid fraction in the regime where r/R b 0.75. Topologically, attachment of a node to the sphere is effectively the same process in reverse. The attachment process must begin with the Plateau border junction splitting into a triangular arrangement of contact lines, the reverse of the process shown in Fig. 4. This also

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involves the foam films near the surface changing from their Plateau law orientations to normal incidence at the sphere surface. The system is now at the far left of the graph in Fig. 7; i.e. at α = 0. This is an unstable configuration, since an incremental increase in α will result in a net force on the sphere toward the node. The next stable configuration corresponds to the sphere being fully embedded in the node, i.e., where α = 54.7 ∘. Let us assume that the films maintain normal incidence on the surface despite the dynamic character of the attachment process — a reasonable assumption, since the time required for film rearrangement is presumably much shorter than that for motion of the sphere. In that case, the force on the sphere will have the same maximum value of Fres/(γlgR) = 2.333during attachment. Since Fres is always directed toward the node, this will be in the opposite direction to the maximum detachment force. If the sphere is smaller than the bubbles, its progress through the foam will consist of a series of attachments and detachments of successive nodes. The drag force would be expected to display positive and negative of approximate amplitude F/(γlgR) = 2.333. As already noted, while node attachment and detachment are topologically the reverse of each other, there is also an important difference. While the deformation of the films prior to detachment is quasistatic, there is no equivalent quasistatic stage in the attachment process, which is entirely dynamic. As a result, attachment may occur over a shorter time-scale than detachment. The time-averaged ‘anti-drag’ force due to node attachment will therefore be smaller than the time-averaged drag force due to detachment, resulting in a distinct drag component with a non-zero mean value. Interestingly, in the two-dimensional simulations of Raufaste et al. (2007) and Davies and Cox (2009), the film network was allowed to equilibrate fully between positional increments; this meant that the node attachment force vanished entirely. If the attachment time-scale is significantly less than the detachment time-scale, this should be visible in the drag force vs. time profile — in that case, we would expect the minima in the drag force to be narrower than the maxima. If the sphere is larger than the bubbles, the film geometry during attachments and detachments will be much more complex than assumed in the model above. In addition, multiple node attachments and detachments may overlap or occur simultaneously. For instance, in the two-dimensional models of Raufaste et al. (2007), the ‘cut-off length’ determined the density of films that could remain attached to the trailing edge of the object (there being no ‘tipping point’ in the force curve). Since the cut-off length was dependent on the liquid fraction, so was the detachment force component. We would also expect it to depend on the contact angle, for the same reason, although this was not explored by Raufaste et al. Similarly, for a sphere with r/R N 0.75 , the density of films able to remain attached to the trailing side of the sphere will depend on the crowning angle, and we would expect the ‘detachment’ component of the drag force to be dependent on the liquid fraction and contact angle. It is uncertain how useful the simple node attachment/detachment model is for predicting the amplitude of drag force transients in this regime. It is useful to express the above model explicitly in terms of the liquid fraction, even though the present study did not examine the effect of varying the liquid content of the foam. This can be done directly by expressing the Plateau border radius r in terms of the liquid fraction εL. We must first make a few assumptions about the structure of the foam. In the dry limit, the liquid can be assumed to reside primarily in the Plateau borders, and in this case 2

r ¼

  Vb εL δL 1−εL

researchers have simulated random foam structure, and we use their results to determine L. Kraynik et al. (2004) modeled polydisperse foam structures, and defined a ‘polydispersity parameter’, p, as 32

p ¼ Rb

.

3 1=3

hRb i

−1;

ð14Þ

where Rb is the bubble radius in an ensemble and Rb32is the Sauter mean bubble radius. In these simulations, the average (redundant) number of faces per bubble, 〈F〉agreed well with the empirical relation 2

hFi ≅ 10:4p −10:2p þ 13:8:

ð15Þ

Cox and Graner (2004) found that the mean total edge length (redundant) per bubble, which equates to the total Plateau border length, has the following empirical relationship to 〈F〉: Lred ≅ 4:35 Vb

1=3

1=2

hFi

:

ð16Þ

Now, given that each edge is shared between three bubbles, the nonredundant Plateau border length per bubble (i.e. the total length for the ensemble of bubbles, divided by the number of bubbles), is L ¼ Lred =3:

ð17Þ

Eqs. (13)–(16) allow the Plateau border radius, and hence the whole node attachment/detachment model, to be expressed in terms of the liquid fraction. Calculation of the polydispersity parameter requires the bubble size distribution to be known in addition to the Sauter mean bubble size. However, 〈F〉 is not acutely sensitive to polydispersity (as p varies from 0 to 0.5, 〈F〉 decreases from 13.8 to 11.3). Thus, in cases where the bubble size distribution is not known in detail, 〈F〉 can be approximated by its polydisperse value of 13.8. 3. Materials and methods The experimental techniques used in this study are very similar to those described by Ireland and Jameson (2009). Fig. 9 shows the apparatus used to measure the drag force. A sensitive force transducer (Aurora Scientific 403A) was mounted on a platform which could be raised or lowered by screw shaft drive controlled by a computer. The transducer had a range of −5 to 5 mN and a resolution of 0.1 μN, and its output was logged as a function of time, at intervals of 1 ms. The entire apparatus was enclosed against air currents, and mounted on rubber feet on a heavy steel table-top to minimize vibrations. The surfactant solution was Triton X-100 (Dow Corp.) at a concentration of 1000 ppm, with a manufacturer-specified nominal

ð13Þ

where Vb is the mean bubble volume; δis a dimensionless geometric constant (=0.161) giving the Plateau border cross-section as a proportion of r 2; and L is the mean non-redundant length of Plateau borders per bubble (Bhakta and Ruckenstein, 1995). A number of

Fig. 10. Typical trace of drag force vs. time for a spherical probe in a foam. Movement starts at t = 0. In the case shown, the surface was paraffin wax, R = 1.5 mm, and the velocity was 0.5 cm.s− 1.

P.M. Ireland, G.J. Jameson / International Journal of Mineral Processing 102–103 (2012) 78–88 Table 1 Wetting contact angle of probe surfaces by 1000 ppm. Triton X-100 aqueous solution. Surface

Equilibrium contact angle, θc (± 5°)

Lead Lead/Tin solder Epoxy resin PVA Ethyl cyanoacrylate PVC Sealing wax Paraffin wax

10° 12° 13° 22° 23° 27° 41° 43°

surface tension of 0.031 Nm − 1. This value was confirmed experimentally using the capillary rise method (Rusanov and Prokhorov, 1996). An extremely stable foam was produced: visible coalescence and coarsening took ~5 min to become evident, and the foam often took hours to vanish completely. It is important to note that the surfactant solution and the foam generation method and apparatus were deliberately identical to those used in Ireland and Jameson (2009). All relevant parameters of the foam were therefore the same in the two studies. Probe tips were manufactured by heating an appropriate quantity of lead or solder with a blow-torch until it was pulled into a nearspherical drop by surface tension. Tips manufactured in this way had a sphericity better than 90%. Each probe tip was connected to the transducer input by a length of 250 μm diameter tungsten wire, which had been inserted into the drop of molten metal. Table 1 shows the surface coatings used, and their contact angles with the surfactant solution, measured using a technique described by Ireland (2008). These angles are smaller than the equivalent values for water, as a result of the smaller gas–liquid surface tension of the surfactant solution. With the exception of PVC, the coatings were applied in liquid form and allowed to harden. The PVC was applied as a thin solid film (cling wrap) and gently heated until it annealed to the metal surface. To measure the drag force, the transducer head was positioned so that the attached probe was well inside a 45 mm internal diameter foam column. Before the probe was immersed in the foam, a baseline force measurement was taken. Air was sparged into the feed chamber via a sintered glass frit at 1 L/min, corresponding to a superficial gas velocity in the upper part of the column of 1.04 cm/s. The resultant foam was allowed to rise to the top of the column and overflow for 20 s (immersing the probe), before the air supply was cut off, and a ruler was used to carefully scrape off any foam above the overflow lip. The foam was then allowed to sit for 60 s, until quite dry. The

Fig. 11. Mean drag force vs. probe velocity, for large and small contact angles and four values of the probe radius.

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volumetric liquid fraction at this stage had previously been measured by the following method: the upper part of the column, whose volume was known, could be detached while full of foam, and weighed. In addition photographs of the foam were taken through the column wall, and used to estimate both the bubble diameter and the Plateau border curvature. After sitting, the transducer signal was invariably the same as the initial (un-immersed) baseline reading. This was taken as an indication that all deformations resulting from the immersion process, resulting in residual forces, had relaxed. It also indicated that the buoyancy force on the probe was unimportant. Once the foam was prepared, the transducer head was moved upward at the desired speed, dragging the probe through the foam.

4. Results Fig. 10 shows a typical profile of drag force vs. time for a probe moving through a foam. The constant signal representing the weight of the probe has been subtracted. This profile is strikingly similar to that shown by Cantat and Pitois (2005) — the main difference being a much greater predominance of the mean drag over the transient variations. This difference can be explained by the larger size of the probe, relative to the bubbles, in the present study. A constant drag force with transient fluctuations had been expected after the initial load phase, as seen in Fig. 10. However, in some cases, a slight linear drift was observed over time. It was eventually determined that this was due to small cumulative horizontal forces on the probe, to which the transducer was somewhat sensitive. These horizontal translations were the result of the mechanism of ‘geometric dispersion’ (Meloy et al., 2007). This term describes the stochastic cumulative motion of a particle in a direction normal to the principal direction of motion, due to its having to negotiate the network of Plateau borders in the foam. In the present experiments, the particle was confined to a near-vertical path, so the tendency to geometric dispersion manifested itself as a small cumulative horizontal force on the transducer. To eliminate the small errors that may otherwise have arisen, a linear fit was performed on the profile once the initial loading phase was complete (in the case shown in Fig. 10, after ~ 1 s). The difference between this linear fit when extrapolated back to t = 0 and the initial baseline force was taken to be the mean drag force. The liquid fraction of the foam after sitting for 60 s was measured at 0.01–0.03. The error in these values from the relatively crude liquid fraction measuring technique was probably at least as large as the measured range of the values. A more accurate determination was not possible, but given the identical foam generation method for all experiments, the liquid fraction was regarded as constant. The Sauter mean bubble radius, calculated from photographs of bubbles visible through the column wall, was 0.62 mm, with a polydispersity parameter p ≃ 0.39. The Plateau border curvature at the junctions was ~0.25 mm, although most of the foam cell wall was flat, consistent with the low liquid content of the foam. The maximum horizontal extent of the foam film ‘foot’ on a flat surface with a contact angle approaching zero would be expected to be twice this, i.e., 0.5 mm. Since this is smaller than the bubble diameter, we would not expect continuous wetting films to form on surfaces in contact with the foam. As a consequence, the foam–particle interface was assumed to have a yield stress in all of these experiments, as proposed in the ‘continuum model’. To check for consistency between the measured Plateau border curvature and liquid fraction, Eqs. (13)–(16) are used in reverse order. The measured polydispersity corresponds to 〈F〉 ≃ 11.4, giving L ≃ 4.9 mm/bubble, corresponding to a liquid fraction of 0.047. The discrepancy between this and the directly-measured liquid fraction can probably be attributed to inaccuracies in the latter, as discussed earlier. In any case, a dry foam (εL b 0.05) is indicated by both measures.

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component. If this effect were significant, we would expect the troughs in the force vs. time profile (associated with attachment events) to be noticeably narrower than the peaks (associated with detachment events). There was no discernible difference between the width of the minima and the maxima in any of these profiles. This means that either the attachment events were so rapid as to be undetectable by our measurement technique (this seems extremely unlikely, given the 1 millisecond sampling interval) or the attachment and detachment events were making equal and opposite transient contributions to the drag force, as assumed in our model. It is difficult to extract a numerical value from Fig. 10 corresponding directly to Fres in Eq. (12). In this particular instance, the speed of the probe was 0.5 cm.s − 1, and each second of the force-time profile thus corresponds to a distance of 5 mm traveled. The largest-scale fluctuations in the force are therefore on approximately the same scale as the characteristic period of the foam structure. As a measure of the vertical scale of these fluctuations, we used the root mean squared value of F−〈F〉 (F denoting the instantaneous measured force). This RMS value is denoted σF. 5. Discussion

In Section 2.2, the possibility was raised that attachment events were occurring over a shorter time-scale than detachment events, resulting in a non-zero time-averaged attachment/detachment force

Fig. 11 shows the drag force on the probe as a function of translation speed through the foam. Data are shown for all four sphere radii and for a low contact angle material (solder) and a high contact angle material (sealing wax). In all but one case, over a four-fold increase in speed, there was no significant change in the drag force with velocity relative to the variation in the data. In the remaining case (the 1.5 mm radius wax probe), the trend was actually slightly negative. We conclude from this that all experiments in this study took place in a high-Bingham-number regime. Eq. (3) is thus expected to hold throughout. Fig. 12 shows the measured mean drag force on the probe as a function of the contact angle, for all four values of the probe radius, and a probe velocity of 0.5 cm.s − 1. Assuming that the bulk and surface yield stresses are as measured by Ireland (2008) and Ireland and Jameson (2009) (τyb ≈ 8.7 N.m − 2, τys given by Eq. 7), the drag force should reach its maximum value at θc ~ 38 ∘ and remain constant at higher contact angles. The dashed lines on Fig. 12 indicate the maximum drag force (no slip) associated with τyb = 8.7 N.m − 2 and Eq. (3); these are broadly consistent with the measured drag force at large contact angles. The dotted curves on Fig. 12(a) correspond to the predicted drag force, given by Eq. (6); agreement between model and data is poor. While the ratio of non-slip to full-slip drag

Fig. 13. σF (RMS value of F−〈F〉) vs. contact angle, for four different values of the probe radius.

Fig. 14. Average of σF overall contact angles, vs. probe radius, with best linear leastsquares fit through origin. Error bars indicate the largest and smallest measured values.

Fig. 12. Mean drag force vs. contact angle, for four values of the probe radius. Dashed lines: theoretical maximum value for non-slip contact. (a) Dotted lines: theoretical drag force (Eq. (6)). (b) Theoretical drag force with optimal value of adjustable constant (Eq. (18)).

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force predicted by the model of de Besses et al. (2004) is slightly smaller than the one used here, it is also much too large to explain the measured force. The solid curves in Fig. 12(b) correspond to a modified version of Eq. (6), in which the skin drag component includes a dimensionless constant multiplier K:

hFi ¼

8
ns

:F

ns


 þ K ðπRÞ2 τsy −τby ; ;

τsy ≥τby

τsy bτby

:

ð18Þ

The displayed curves correspond to K = 2.8, which gives the closest fit with the experimental data; for this value of the adjustable constant, agreement between model and experiment is far better. It seems that Eq. (6) underestimates the effect of skin drag by a factor of 2.8. The reasons for this are not clear. Coalescence and bursting of bubbles in the foam might explain the lower-than-expected drag force, but none was observed, and in any case, the model and experimental drag force agreed well at large contact angles. It may simply be the case that the continuum model was inadequate under these circumstances. Since the probe spheres were only ~1.5–3.0 times the size of the bubbles, the ‘continuum assumption’ seems open to question. For instance, the utility of the concept of ‘plastic zones’ and ‘rigid zones’, as observed in a homogeneous material, becomes doubtful when the equivalent regions in a foam are actually smaller than the bubbles. Nonetheless, the fact that the predicted and measured values of the skin drag differ consistently by a constant multiplier suggests that the basic principle behind the model is correct, but that its exact working-out requires modification. Fig. 13 shows the root mean squared (RMS) deviation of the measured drag force from its mean, σF, as a function of the contact angle, for each probe size. For the photographically estimated Plateau border curvature radius of r ~ 0.25mm, r/R ranges from 0.17 to 0.33. According to the node detachment model (Fig. 7), the detachment force is determined by the force curve maximum for all of this range, and should be independent of the contact angle. The experimental data are consistent with this prediction. The force curve for the node attachment/detachment model (Eq. (12)) predicts a maximum restoring force of 2.33γlgR. For the present study, where γlg = 0.031N.m − 1, the model therefore predicts a characteristic drag force transient amplitude of 0.072RmN. Fig. 14 shows σF (the RMS deviation), averaged over all contact angles, as a function of probe radius. The best linear fit to these data through the origin has a gradient of 0.035. This RMS deviation is 0.49 times the predicted transient amplitude. For comparison, the ratio between the RMS deviation and the pffiffiffi amplitude is 1= 2 for a sinusoid, and 0.58 for a sawtooth wave (the common waveform most closely resembling the observed force profiles). The closeness of these values supports the proposition that the observed transients correspond to the node attachment/detachment process modeled in this paper. The sizes of the probes used in these experiments were in a sense the most difficult, and also the most interesting, to deal with theoretically. As already noted, they were not so much larger than the bubbles that the foam could be treated unequivocally as a continuum, and yet they were too large for the force to be modeled entirely as a series of unrelated node attachments and detachments. A ‘hybrid’ modeling approach – which combines a continuum drag model of the average drag force with a node attachment/detachment model for the transients – seems at least partly successful in characterizing the system. 6. Conclusions A particle's floatability in a flotation froth is in effect determined by whether its weight is sufficient for it to move downward through the froth. Particles that are substantially smaller than the bubbles can simply drain through the Plateau borders. In that case, selectivity is

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achieved because some particles adhere to the bubbles and others do not. For larger particles, whatever their surface wettability, the foam structure must be deformed and permanent topological changes must occur for the particle to move irreversibly through the foam. This study has presented experimental measurements of the lowvelocity drag force on spherical particles in this larger particle size range. Wetting contact angles of 10∘−43∘ and spherical radii of ~ 1.5− 3.0 times the bubble diameter were explored. A ‘hybrid’ model of the force was developed and compared with the experimental data. In this model, the mean drag force was assumed to be the sum of form drag and skin drag components. The form drag was calculated by treating the bulk foam as a homogeneous Bingham fluid with the bulk yield stress value measured for the same foam by Ireland and Jameson (Ireland and Jameson, 2009). The skin drag was calculated by assuming that the stress at the interface between the sphere and the foam was equal to the interfacial yield stress for the corresponding contact angle. This interfacial yield stress was derived from measurements (again, for the same foam and surfactant solution) by Ireland and Jameson (2009) and Ireland (2008). It was assumed that the drag force would approach the large Bingham number limit solution for the non-slip contact of Blackery and Mitsoulis (1997) as the interfacial yield stress approached the bulk yield stress. The measured mean drag force was consistent with the modeled form drag component, whereas the modeled skin drag underestimated the measured value by a constant factor. The precise reason for this discrepancy is not yet known. Transient variations in the drag force were predicted by developing a model of the attachment and detachment of the particle from individual nodes in the foam structure. The predicted attachment/ detachment force was broadly consistent with the measured amplitude of the drag force transients. Nomenclature Roman Bn Bingham number (−) CS Stokes drag coefficient (−) F Drag force 〈F〉 Mean drag force Fns Non-slip drag force Fref Surface yield stress reference force Fres Restoring force 〈F〉 Redundant number of faces per bubble K Constant in Eq. (13) (−) L Non-redundant Plateau border length per bubble Lred Redundant Plateau border length per bubble p Polydispersity parameter (−) Q Parameter in Eqs. (8) and (9) (−) R Particle/sphere radius Rb Bubble radius Rb32 Sauter mean bubble radius r Plateau border curvature radius S1,S2 Chord lengths — see Fig. 6 Tn Foam film tension (N.m − 1) Vb Bubble volume Greek α β γlg δ θc μ σF τy τyb τys

Angle — see Fig. 6 Angle — see Fig. 6 Gas–liquid surface tension (N.m − 1) Geometric constant (−) Wetting contact angle Bulk viscosity (Pa.s) RMS value of F−〈F〉 Yield stress (Pa) Bulk yield stress Surface yield stress

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