Bubble Electrostatic Interaction

Journal of Colloid and Interface Science 237, 208–223 (2001) doi:10.1006/jcis.2000.7376, available online at http://www.idealibrary.com on

Microflotation Suppression and Enhancement Caused by Particle/Bubble Electrostatic Interaction N. A. Mishchuk,∗ L. K. Koopal,†,1 and S. S. Dukhin∗ Institute of Colloid and Water Chemistry, Ukrainian National Academy of Sciences, 42 Vernadsky pr., Kyiv 03142, Ukraine; and †Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherlands Received July 6, 2000; accepted December 4, 2000

1. INTRODUCTION The processes of attachment and detachment of small or mediumsized particles to relatively large bubbles during microflotation are considered in terms of the heterocoagulation theory. Calculations are made for the conditions that the surface potentials are of similar sign and constant, that one of the surface potentials is small, that hydrophobic attraction is absent, and that there are no surface deformations. Under these conditions bubble–particle aggregates may form as a result of an electrostatic attraction which exceeds the repulsive van der Waals force at intermediate distances. Next to electrostatic and van der Waals forces, hydrodynamic and gravitational forces are considered. These forces may overcome the electrostatic repulsion at large distances and promote particle bubble attachment. Strong electrostatic attraction at small distances, arising at a large difference of the surface potentials of the bubble and the particle and of low electrolyte concentrations, can prevent subsequent detachment by hydrodynamic and gravitational forces. With increasing electrolyte concentration the electrostatic barrier increases and the attractive electrostatic force diminishes. As a result, a critical electrolyte concentration for microflotation exists. Above this concentration attachment may still occur but it is followed by detachment. At lower electrolyte concentrations the electrostatic attractive force prevents the detachment. The dependence of the critical electrolyte concentration on the values of the bubble and particle potentials and the Hamaker constant is calculated. The critical concentration does not depend on particle or bubble size if the absolute values of the total detachment force and the total pressing force coincide, which is the case for Stokes and potential flow. For every electrolyte concentration lower than the critical value there are two critical particle sizes that limit the flotation possibility. For small particle sizes attachment is impossible because the pressing force is smaller than the electrostatic barrier. For large particle sizes detachment cannot be prevented because the detachment force exceeds the maximum electrostatic attraction. A microflotation domain of intermediate particle sizes exists in which ° C 2001 Academic Press irreversible heterocoagulation occurs. Key Words: particle/bubble attachment; particle/bubble detachment; heterocoagulation; DLVO forces at constant potential; electrostatic attraction; molecular repulsion; hydrodynamic pressing force; hydrodynamic detachment force; microflotation.

1

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It was suggested almost three decades ago (1) that the mechanism of both stages of flotation—approach and attachment— changes qualitatively from large to small particles. In the case of large particles, the positive disjoining pressure of the double layer can be overcome by an inertial impact on the bubble surface. In contrast, small particles (microflotation) do not undergo such an impact; approach occurs without inertia and can be hampered by electrostatic repulsion. Initial experimental proof of the microflotation theory was provided by Derjaguin and Shukakidze (2). Subsequently, the interpretation of the microflotation process in terms of the heterocoagulation theory was considered in a number of review articles (3–5) and hampering of the flotation by electrostatic repulsion has been confirmed in several studies (6–8). Jaycock and Ottewill (6) detected the maximum floatability at the isoelectric point by varying the electrokinetic potential of silver iodide particles through the adsorption of a cationic surfactant. In Refs. (7, 8), it is shown that the flotation rate is high within a narrow pH range and very low beyond this range. In the former case the pH values correspond to very small ζ potential of particles, i.e., those that approach the isoelectric point. The authors interpret their data as a proof of the decisive influence of the electrostatic repulsion on the process of particle/bubble (p/b) attachment. Besides the importance of the electrostatic repulsion, the existence of the repulsive character of the molecular forces between a mineral particle and a bubble has been stressed (9–11). However, it is also known from the heterocoagulation theory that the electrostatic repulsion at large distances may change into an electrostatic attraction at small ones (12–15). This applies if the approach occurs at constant potentials and the surface potentials of the interacting particle and bubble have the same sign but different values. Therefore, flotation results can be explained also by attractive electrostatic forces that exceed the repulsive molecular forces. In general, p/b electrostatic interaction can either suppress or enhance microflotation. This conclusion is important, both for theory and for practice of microflotation. Electrostatic attraction at small distances has been neglected in a number of more recent papers that otherwise have contributed to our understanding of the microflotation process. The

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MICROFLOTATION SUPPRESSION AND ENHANCEMENT

DLVO theory (16, 17) was applied in (18) to quantify the conditions under which electrostatic repulsion can prevent p/b attachment, but only the simplified case of identical surface potentials was considered and the role of molecular repulsion was underestimated. Papers (19–21) that took p/b electrostatic interactions into account also ignored this type of electrostatic attraction. Renewed emphasis on the mechanism of electrostatic attraction at small distances between a bubble and a particle with potentials of the same sign was given in (11), in which the interaction between silica particles and bubbles was studied by atomic force microscopy. Microflotation was discussed in (9), but here the main emphasis is on the role of hydrophobic attraction in flotation. Both the experiments and the theory in (9) are concerned with the case in which electrostatic attraction is probably impossible because the potentials of the bubbles and the particles are rather high. The present paper deals with microflotation under conditions that differ from those in (9); namely, the surface potentials have the same sign and one is sufficiently small. As a result, microflotation occurs due to electrostatic attraction. Although it is not excluded that hydrophobic forces often play a role, they can be suppressed under certain conditions by surfactant adsorption (22) and in this case the mechanism of electrostatic attraction can dominate in microflotation. In the first part of the paper, surface forces and pressing forces will be discussed. Subsequently, their joint analysis will enable us to specify the conditions under which microflotation is possible, i.e., to specify the conditions where p/b attachment will occur and where p/b detachment is prevented. In the last section the results will be discussed in relation to other recent papers regarding heterocoagulation and p/b interaction. 2. MODEL ASSUMPTIONS AND QUALITATIVE CONSIDERATIONS

The movement of a particle to a bubble surface is realized along the bubble axis where the tangential flow velocity is equal or close to zero. The time of deposition of the particle on the bubble is therefore probably long enough for charge regulation and this allows us to apply the constant potential model. Under conditions of constant potential a barrier for microflotation may occur if, together with repulsive molecular forces, electrostatic attraction takes place at short distances and electrostatic repulsion takes place at larger distances (12–15). There are three mechanisms to overcome this barrier. Rather small particles can cross the barrier by Brownian diffusion and very large particles can overcome the barrier by their high kinetic energy (23). In the intermediate range of particle sizes, pressing hydrodynamic forces (PHF) and gravity forces can overcome the repulsive barrier and both inertial forces and Brownian diffusion can be neglected (24). Initially the PHF was specified for the cases of Stokes and potential flow around a rising bubble (25, 26). Afterward, the hydrodynamic field for a rising bubble was quantified for the important case of strong retardation of the bubble sur-

face and large Reynolds numbers (27, 28). In the present paper, which concentrates on intermediate particle sizes, we will use these results. In principle it is also possible to describe particle penetration through the electrostatic barrier using the well-known Fuchs equation for the stability ratio (29). However, we will not consider the diffusion transport of a particle through the electrostatic barrier because the measured values of the stability ratio are systematically 10 to 1000 or more times lower than the calculated ones (30–33). Surface deformation, which can take place under the action of a PHF (34) or an attractive total surface force, has been neglected in the present p/b interaction model. Although this restricts the applicability of the model, it allows a considerable simplification of the analysis of surface and hydrodynamic forces. The model of a nondeformable surface has been applied before to the analysis of surface force measurements (11). The phenomenon of spontaneous surface deformation has been described for droplet/droplet interaction in emulsions (35) and for p/b interaction in the presence of attractive molecular and shortrange repulsive steric forces (36). When the Stokes number is not very small (large particles), inertial forces are strong and largely exceed the electrostatic forces. In that case surface deformation rather than electrostatic force variation deserves attention (10). This has been confirmed recently in an experimental study (37). However, at rather small Stokes numbers (small particles), the inertial forces are less important and the electrostatic forces may suppress or enhance flotation. Although in this case there is some surface deformation due to the PHF (18), the deformation is weak and its role can be neglected relative to the role of the electrostatic interaction. Hydrodynamic and gravity forces cannot only promote p/b attachment, but also lead to p/b detachment. When the detachment force can overcome the attractive well due to the electrostatic force, particles and bubbles will detach and flotation will not occur. Incorporation of the conditions for both attachment and detachment into this theory therefore leads to a “flotation domain,” in which microflotation is possible due to the simultaneous action of DLVO and hydrodynamic and gravity forces. The DLVO theory can be expressed in either interaction energies or interaction forces. In the present paper interaction forces will be considered because the hydrodynamic interaction is expressed as force. A positive value of the force corresponds to repulsion, a negative value to attraction. Although the heterocoagulation theory (13–15) is rather complicated, the main qualitative regularities of the electrostatic interactions can be understood by using a simple equation that characterizes the boundary between the repulsive electrostatic force at large distance and the attractive electrostatic force at smaller distance. When h is the shortest distance between particle and bubble, the change of the direction of electrostatic force occurs at the distance h ∗ (15),

h ∗ = κ −1 ln

ψ12 + ψ22 , 2ψ1 ψ2

[2.1]

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MISHCHUK, KOOPAL, AND DUKHIN

FIG. 1. Qualitative scheme of DLVO forces for a bubble and particle with the same sign of surface charge as a function of shortest distance between bubble and particle h.

where the particle and bubble potentials ψ1 and ψ2 have the same sign and κ is the reciprocal Debye length. At ψ1 = ψ2 , electrostatic repulsion takes place at any distance since h ∗ = 0. The larger the difference between ψ1 and ψ2 and the lower the electrolyte concentration or, correspondingly, the larger, the Debye length, the thicker the zone of the attractive electrostatic force. Thus, a decrease of the electrolyte concentration and an increase of the potential ratio are favorable for microflotation caused by electrostatic attraction. The zone of the attractive total surface force for p/b interaction is even narrower than estimated by [2.1] because of the addition of the repulsive molecular force Fm (h). At large distances, the total surface force Fme (h) = Fm (h) + Fe (h) is always repulsive because both its components are repulsive. In other words, there is a force barmax at relatively large distances and a force rier with height Fme min well with depth Fme at relatively small ones. Qualitatively the situation is illustrated in Fig. 1. Two different cases appear. In the first case molecular forces exceed electrostatic ones at all distances, and even in the region of electrostatic attraction the total force is repulsive (curves 1, 2) or equal to zero (curve 3). In the other case, the electrostatic attractive force dominates at certain distances and causes a negative well in the total surface force (curve 4). The notion of the pressing hydrodynamic force (PHF) was introduced in the theories of Spielman (38) and Goren (25). It is worth mentioning that mainly these theories were used when particle capture from laminar flow was considered in the recent monograph of Russel et al. (39). With regard to the purpose of this paper, it is interesting to note that, considering capture with electrostatic repulsion, the authors wrote that “interactions may change from attraction to repulsion as the separation changes” (39). In (40) the theories of Goren and Spielman were applied to quantify particle capture in a packed bed during filtration. In view of the large similarities in the processes of particle capture under conditions of filtration and flotation, the theories were also applied to the elementary act of flotation (24). As the time re-

quired for deposition of a particle on a bubble can be indefinitely long, the viscous resistance of the liquid film between a particle and a bubble can be neglected in the balance of acting forces (15). At small distances between a particle and a bubble, the PHF is almost independent of the distance (25, 26, 41). The action of the PHF, Fhd , should be combined with the action of the gravitational p force, Fg , that is caused by the weight of the particle. Near the upper pole of a bubble, the gravitational force is directed to the bubble surface and is also independent of distance. The p total pressing force, Fpr = Fhd + Fg , is negative; however, to compare it with surface forces it is convenient to show its value with the opposite sign (see Fig. 1, curve 5). In each case that is shown in Fig. 1, the total pressing force draws a particle nearer to the bubble surface. However, it can be insufficient for the formation of a p/b aggregate, as seen from the comparison of curve 5 and curves 1 and 2, where the barrier height is larger than the total pressing force. It is necessary to underline that the existence of a well with a positive or even small negative value of the minimal force (curves 2 and 3) does not lead to a p/b aggregate. A particle, initially present at this minimum, will be repulsed if the pressing force is insufficiently large. Even when the pressing force is large enough and the particle is pressed to the bubble surface, this does not mean that irreversible attachment occurs. As the particle moves along the bubble surface, the direction of the normal component of the liquid velocity changes. When it is directed from the surface to the bulk, the PHF changes its direction and turns into a detachment hydrodynamic force. Similarly the gravity force changes direction. If both the total surface force and total detachment force are positive, detachment occurs. Thus all curves above the curve 3 correspond to detachment. The possibility of detachment for curve 4 depends on the correlation between the depth of the min , and the value of the total detachment force, Fdet . If the well, Fme absolute value of the detachment force (Fig. 1, curve 6, which is again presented with the opposite sign) is smaller than the depth of the well, the aggregate is stable and the heterocoagulation of the particle with the bubble is irreversible. The condition necessary to overcome the force barrier and to have attachment can be presented (36) as max + Fpr ≤ 0. Fme

[2.2]

Similarly, the condition to prevent detachment can be written as min + Fdet ≤ 0. Fme

[2.3]

In both cases the sum of all forces should be negative to provide irreversible heterocoagulation. Since the electrostatic force strongly depends on the electrolyte concentration, it is useful to analyze the required conditions [2.2] and [2.3] from this point of view. At fixed surface potentials of particle and bubble, the existence of the minimum of the negative force depends on the electrolyte

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

concentration. The electrolyte concentration corresponding to curve 3 will be referred to as the first characteristic concentration, C1cr . At concentrations larger than C1cr detachment occurs. Detachment will take place at electrolyte concentrations lower than C1cr if the detachment force exceeds the attractive surface force. The concentration corresponding to this condition is introduced as the second critical electrolyte concentration, C2cr . Detachment occurs at concentrations larger than C2cr . At first glance, there is no need to introduce concentration C1cr because concentration C2cr characterizes the transition from attachment to detachment. Moreover, the difference between concentrations C1cr and C2cr cannot be very large. Yet, concentration C1cr is more convenient for the analysis of results because the set of parameters that affects C2cr is considerably larger than the set that affects C1cr . Among the latter are the parameters that affect the surface forces, namely the Hamaker constant, the potential ratio ψ2 /ψ1 , and the particle/bubble radius ratio a/R0 . In addition to these factors, concentration C2cr depends on the set of parameters that influence the total detachment force.

Derjaguin has, carried out the theoretical analysis of the interaction of dissimilar double layers in terms of the DLVO theory (16). Based on this work a more convenient equation for low surface potentials was derived by Derjaguin (12) and by Hogg et al. (15). The approach is known in literature as the HHF approximation. According to the exact theory of interaction between flat plates (42–44), the HHF approximation can also be used under the condition that ψ1,2 À 1. As can be seen in Figs. 1 and 2 of (44), the difference between the values of the electrostatic barrier estimated by the full theory and the HHF theory does not exceed 10% even for potentials equal to 50 and 75 mV. With the decrease of one of the potentials under the condition that ψ2 − ψ1 À ψ1 , the difference between values of electrostatic barrier decreases. It is this case that corresponds to the appearance of the region of electrostatic attraction and it is the most interesting for us. Combined with the approximation of Derjaguin (43), the HHF approximation is sufficiently exact to evaluate the interaction of dissimilar spheres (44). Certain approximations can be made when molecular forces are described. It will be shown that a sufficiently large attractive force exists at rather small distances, i.e., smaller than 3 nm. At these distances electromagnetic retardation and screening of the molecular forces are always insignificant (45–47). Also for the calculation of the repulsive barrier a simple form of the molecular forces suffices because the barrier is situated in the region where the electrostatic force dominates. Thus, the molecular forces can be presented in the simplest form, Fm = −K

A , 12π h 2

distance, and the parameter K = 2π R0 a/(R0 + a)

[3.1]

where A is the effective Hamaker constant, h is the interparticle

[3.2]

takes into account the spherical form and size of the interacting particles, a, and bubbles, R0 . It is assumed that A < 0, and in general, this is the case for the interaction between mineral particles and bubbles (45). The interaction force can be obtained as a derivative of the potential energy of interaction between dissimilar spherical colloidal particles given in the HHF approximation (15). The result is Fe = K εε0

£ ¡ 2 ¢ ¤ κ − ψ1 + ψ22 e−2κh + 2ψ1 ψ2 e−κh . −2κh 1−e [3.3]

It is seen that [2.1] follows from [3.3]. The equation for the total surface force is obtained as a combination of [3.1] and [3.3]: Fme = Fm + Fe .

3. HETEROCOAGULATION THEORY FOR MICROFLOTATION

211

[3.4]

The influences of the electrolyte concentration, the surface potentials, and the effective Hamaker constant on the total surface force, calculated according to [3.4], are shown in Figs. 2, 3, and 4, respectively. In accordance with the qualitative considerations of the preceding section, repulsion can dominate at all distances at relatively large elecrolyte concentrations, while attraction arises at relatively small concentrations and small distances (Fig. 2). The larger the difference between the two potentials, the stronger the manifestation of the attractive force (Fig. 3). An increase of the effective Hamaker constant shifts the force minimum to larger distances and, correspondingly, decreases its depth (Fig. 4). The effect of A on the force maximum is relatively weak.

FIG. 2. DLVO forces ([3.1] to [3.4]) as function of shortest distance h between the bubble and the particle at A = −1 × 10−20 J, Fψ1 /RT ≡ ψ˜ 1 = 1.5, and Fψ2 /RT ≡ ψ˜ 2 = 6 at C0 (mol/L) = 0.1 (curve 1); 0.01 (curve 2); 0.001 (curve 3); 0.0001 (curve 4).

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MISHCHUK, KOOPAL, AND DUKHIN

the abscissa axis twice; i.e, there are two roots H1,2 of the equation Fme (h) = 0,

[3.7]

where the force is expressed by [3.6]. Under the condition that ¡ ¢ 3π εε0 (ψ2 − ψ1 )4 + 4Aκ ψ12 + ψ22 − ψ1 ψ2 > 0

[3.8]

the roots q ¡ ¢± 1 ± 1 + 4Aκ ψ12 + ψ22 − ψ1 ψ2 3π εε0 (ψ2 − ψ1 )4 ¡ ¢ H1,2 = 4κ ψ12 + ψ22 − ψ1 ψ2 /(ψ2 − ψ1 )2 FIG. 3. DLVO forces as a function of shortest distance h between the bubble and the particle at A = −1 × 10−20 J, C0 (mol/L) = 0.01, Fψ1 /RT ≡ ψ˜ 1 = 1, and different values of Fψ2 /RT ≡ ψ˜ 2 . Curve 1: ψ˜ 2 = 2; curve 2: ψ˜ 2 = 4; curve 3: ψ˜ 2 = 6. Curves 20 and 200 demonstrate the role of the small potential: curve 20 : ψ˜ 1 = 0.5, ψ˜ 2 = 4; curve 200 : ψ˜ 1 = 1.5 ψ˜ 2 = 4.

Let us now pay attention to the position and the depth of the force minimum. It can be shown that the total force becomes attractive at κh < 1.

[3.5]

Under this condition, [3.3] can be simplified using the expansion in a Taylor series and accounting for the first two terms only. This simplification yields the expression for the total force: Fme

· A (ψ2 − ψ1 )2 = Fm + Fe = −K − K εε 0 12π h 2 2h ¸ ¡ ¢ − κ ψ12 + ψ22 − ψ1 ψ2 . [3.6]

The zone of the attractive force exists if the curve Fme (h) crosses

[3.9] of [3.7] are real. Attraction occurs at H1 < h < H2 , while repulsion takes place at h < H1 and h > H2 . When the Hamaker constant is large or the difference between the potentials is small, condition [3.8] is not satisfied and the total force is repulsive at all distances. If H1 and H2 approach each other, the zone of attraction shrinks. It disappears if the l.h.s. of [3.8] equals zero. For this situation the Debye length, or correspondingly, the electrolyte concentration, C1cr , can be expressed as function of the Hamaker constant and the surface potentials: C1cr =

9π 2 ε3 ε03 RT (ψ2 − ψ1 )8 ¡ ¢ . 32F 2 A2 ψ 2 + ψ 2 − ψ1 ψ2 2 1 2

Figure 5 illustrates this relation. Figure 5a shows the dependence of C1cr on the value of the large potential at two values of the Hamaker constant. The influence of the Hamaker constant on the critical concentration is plotted in Fig. 5b. An increase of the large potential leads, in general, to a progressive increase in C1cr . Figure 5a also shows that ψ2 has to become larger than ψ1 before C1cr starts to become noticeable. Increasing the negative value of the effective Hamaker constant requires a larger value of ψ2 in order to obtain the same value of C1cr (Fig. 5b). Or, alternatively, an increase in the negative Hamaker constant strongly decreases C1cr for a given value of ψ2 . Figure 5b also shows some results based on [2.2] and the full expressions for Fme . Qualitatively, the results are very similar to the approximate result, but quantitatively some differences exist. The coordinates of the minimum of the total force can be obtained on the basis of d Fme =0 dh

FIG. 4. DLVO forces as a function of shortest distance h between the bubble and the particle at Fψ1 /RT ≡ ψ˜ 1 = 1, C0 (mol/L) = 0.01: A = −1 × 10−20 J, Fψ2 /RT ≡ ψ˜ 2 = 4 (curve 1); A = −1 × 10−20 J, ψ˜ 2 = 6 (curve 10 ); A = −2.5 × 10−20 J, ψ˜ 2 = 4 (curve 2); A = −5 × 10−20 J, ψ˜ 2 = 4 (curve 3); A = −5 × 10−20 J, ψ˜ 2 = 6 (curve 30 ).

[3.10]

[3.11]

together with the additional condition that d 2 Fme > 0. dh 2

[3.12]

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MICROFLOTATION SUPPRESSION AND ENHANCEMENT

minimum: h min

1 = 4κ

s

Ã

8Aκ 1− 1+ 3π εε0 (ψ1 − ψ2 )2

! .

[3.15]

Thus, the depth of the force well is equal to ·

min Fme

A 12π h 2min · ¸¸ ¡ 2 ¢ (ψ2 − φ1 )2 2 − εε0 − κ ψ1 + ψ2 − ψ1 ψ2 . [3.16] 2h min

=K −

Due to the negative sign of the Hamaker constant, the solution [3.15] takes real values only under the condition that −

8Aκ < 1. 3π εε0 (ψ2 − ψ1 )2

[3.17]

Thus, the larger the difference between the potentials ψ2 − ψ1 , the larger the range of the Hamaker constants that satisfy the condition [3.17]. Taking into account the inequality ψ2 /ψ1 À 1, it follows that conditions [3.17] and [3.8] differ only by a multiplier of 2. Using condition [3.17], it follows from [3.15] that κh min < 0.25. This may also be expressed as κh(C1cr ) < 0.25. This justifies the derivation of [3.10] and [3.16]. At the more severe restriction κh min ¿ 1, FIG. 5. (a) Critial value of the electrolyte concentration C1cr ([3.10]) as a function of the large potential ψ˜ 2 for two values of the Hamaker constant and different values of the small potential ψ˜ 1 . Curves 1 to 4: A = −1 × 10−20 J; curves 10 to 40 : A = −2 × 10−20 J. Values of the small potential: ψ˜ 1 = 0.5 (curves 1 and 10 ); ψ˜ 1 = 1 (curves 2 and 20 ); ψ˜ 1 = 1.5 (curves 3 and 30 ); ψ˜ 1 = 2 (curves 4 and 40 ). (b) Critical value of the electrolyte concentration C1cr ([3.10]) as a function of the large potential ψ˜ 2 at the fixed value of the small potential, ψ˜ 1 = 0.5, and different values of the Hamaker constant: A = −0.5 × 10−20 J (curve 1), A = −1 × 10−20 J (curve 2), A = −1.5 × 10−20 J (curve 3), A = −2 × 10−20 J (curve 4), A = −3 × 10−20 J (curve 5), A = −5 × 10−20 J (curve 6). The curves 20 (A = −1 × 10−20 J) and 60 (A = −5 × 10−20 J) show the results of exact numerical calculations according to [2] and the exact expressions for Fme .

[3.18]

only the two first terms in the Taylor series for [3.15] have to be taken into account and the last term in [3.16] can be omitted. This results in A 3π εε0 (ψ2 − ψ1 )2

[3.19]

3π ε2 ε02 (ψ2 − ψ1 )4 . 4A

[3.20]

h min = − and min =K Fme

Taking into account [3.4], [3.11] can be rewritten as ·

¢ A 2κ 2 εε0 £¡ 2 K + ψ1 + ψ22 e−2κh 2 3 6π h (1 − e−2κh ) ¸ ¤ − ψ1 ψ2 (e−κh + e−3κh ) = 0.

[3.13]

Under condition [3.5] this can be further simplified to εε0 (ψ1 − ψ2 )2 (1 − 2κh) A + = 0. 3 6π h 2h 2

[3.14]

The solution of this equation gives the position of the force

According to [3.20], the depth of the well increases with a decrease of the Hamaker constant and with an increase of the largest potential, as qualitatively predicted in Section 2. Now we return to the force barrier at large distances. As is seen in Fig. 4, the influence of the Hamaker constant on the position of force barrier is weak. This allows us to identify the coordinate of the total force barrier with the one of the electrostatic force barrier. As has been shown in (15), the maximum of the electrostatic force for spherical bodies exists at distance h max =

¢ 1 ¡ ln ψ2 /ψ1 κ

[3.21]

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MISHCHUK, KOOPAL, AND DUKHIN

and at this point the value of force depends mainly on the value of small potential ψ1 :

Fhd = 6πaηUr (h) f (h/a),

·

max Fme

Aκ 2 =K − 12π ln2 (ψ2 /ψ1 )

distance equal to particle radius,

¸

+ 8εε0 κ(RT /F)2 sinh2 (Fψ1 /4RT ) .

[3.22]

In general the molecular force (first term in [3.22]) is noticeably less than the electrostatic force (second term). The first term can max only at large values of A or essentially change the value of Fme at low values of the electrolyte concentration. 4. HYDRODYNAMIC FIELD AND THE DYNAMIC ADSORPTION LAYER OF A RISING BUBBLE

In this section we consider the pressing and detachment forces. In Section 5 these forces are compared with the surface forces to establish the conditions for the possibility of flotation. Before considering the pressing and detachment forces, we should remark that, according to Derjaguin et al. (48), the possiblity of flotation is the first requirement for flotation, but for the description of the flotation efficiency it is also nessesary to consider the collection efficiency. The latter depends on the viscous forces, i.e., the draining rate. For the description of the critical conditions that make flotation possible, the viscous forces can be neglected (48), because under these conditions the pressing force just equals the repulsive surface force and, correspondingly, the normal component of the particle velocity is negligible. Pressing Forces In the area above the equatorial plane, lines of liquid flow approach the bubble surface, which means that the radial component of the liquid velocity is directed towards the bubble. The approach of the particle to the bubble is accompanied by drainage from the liquid film between them. This is possible due to the increase of pressure in the film, retarding the approach of the particle to the bubble surface. Since the motion of the particle toward the surface is obstructed, the radial velocity of liquid is higher than that of the particle. As the radial flow of liquid envelops the particle, the approach of the particle to the bubble retards and presses the particle against the bubble. As a first approximation, this hydrodynamic force can be obtained from the Stokes formula by substituting the particle radius and the difference in local values between the velocity of the liquid and the particle. An important distinction is that, in the case of large particles, thinning of liquid interlayer is accomplished through impact, while in the case of small particles it is due to the effect of hydrodynamic pressing forces. At very small distance from the bubble surface, the particle radial velocity can be neglected; thus the hydrodynamic force is proportional to the radial component of liquid velocity at a

[4.1]

where η denotes the viscosity of the liquid, Ur (h) is the radial component of the liquid velocity near the upper pole of the bubble, f (h/a) is a function that takes into account the hydrodynamic interaction between the particle and bubble and, according to (24, 25), for h/a ¿ 1, this function is approximately equal to 3.23 for the Stokes regime and to 2.0 for the potential regime. If the bubble floats under the Stokes regime (Re ¿ 1, with Re = U0 R0 /ν the Reynolds number, U0 the rate of floating, and ν the kinematic viscosity of the liquid) and its surface is completely retarded by surface active substances, as it usually happens, Ur (h) is given by the expression (41) Ur (h) = −

¶ µ 3U0 a 2 3U0 a 2 h 2 cos θ ≈ − , 1 + 2 a 2R0 2R02

[4.2]

where θ is the angle that characterizes the position of the particle at the bubble surface: θ = 0 at the front pole of the bubble surface. At small Re, the bubble velocity is equal to U0 =

2gρw R02 . 9η

[4.3]

If the bubble surface is mobile and the bubble floats under the potential regime (Re À 1), the expression for U (h) is (41) Ur (h) = −3U0

¶ µ h a a cos θ ≈ −3U0 , 1+ R0 a R0

[4.4]

where the theoretical value of velocity for a bubble with mobile surface is equal to U0 =

gρw R02 . 9η

[4.5]

The maximum magnitude of the normal component of the liquid velocity and the PHF correspond to the front pole of a rising bubble and further calculations will be accomplished for cos θ ∼ 1 as in [4.2] and [4.4]. In this case, the PHF has its maximum; hence one can obtain a necessary and sufficient condition for the deposition of particles on the surface of a rising bubble. g The gravitational force Fp is directed along the bubble axis and is equal to Fgp = −

4π 1ρga 3 , 3

[4.6]

where g is the acceleration of gravity, and 4ρ is the difference between the densities of the particle and liquid.

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

Taking into account [4.1]–[4.6], the total pressing force for Re ¿ 1 and a retarded surface can be expressed as Fpr = Fgp + Fhd = 2πa 3 g(21ρ/3 + 3.23ρw )

[4.7]

and for Re À 1 and a mobile surface as Fpr = Fgp + Fh = 4πa 3 g1ρ/3 + 36πη

a2 U0 . R0

[4.8]

In [4.8] the velocity U0 is used because at Re À 1 the discrepancy between theoretical and experimental values can be rather large. In the calculations experimental data will be used. In microflotation the size of a particle is usually very small in comparison with the size of a bubble; therefore a/R0 ¿ 1.

[4.9]

This was assumed in the derivation of [4.2] and [4.4]. After the substitution of the bubble velocity into the second term of [4.8], it is seen that the ratio of the first term to the second one is proportional to the ratio a/R0 (¿1). Thus the gravity does not influence attachment of a particle to a mobile surface of a rising bubble. On the contrary, as it is seen from [4.7], its influence is important in the case in which the surface is strongly retarded by a surfactant. In both regimes the total pressing force increases very rapidly with increasing particle size; i.e., it is very small for small particles and very large for large ones. The DLVO forces are only linear functions of the particle size. Therefore electrostatic repulsion may suppress the attachment of small particles, but its influence on the attachment of large particles is weak (neglecting the effect of the electrostatic force on the contact angle). So far the total pressing force has been discussed in the two extreme cases. In practice, however, the bubble surface is often not fully retarded or fully mobile, and in this case, the PHF depends on the dynamics of the surface layer. At Re > 100, a bubble surface is mobile if the impurity level or the surfactant concentration is not very high. Under this condition [4.4] and [4.8] can still be used. However, at Re < 100, even very small concentrations of surface active components retard the rising bubble surface almost completely. For Re > 1, but a retarded or immobile surface, an empirical stream function ψ(r, θ ) has been derived by an analysis of experimental results reported in the literature (27). The result is that for the upper part of the bubble surface the radial velocity of the liquid can be described as · 1 3 2Re0.72 + 3+ Ur = U0 cos θ 1 − 2x 2x 15 ¶¸ µ 1 1 1 1 − 3+ − 2 , × 4 x x x x

[4.10]

where x = 1 + a/R0 . To obtain the expression for the maximum

215

PHF, [4.10] with cos θ = 1 is substituted in [4.1]. According to [4.1], the function f must still be specified for conditions of intermediate flow. Theoretically, this hydrodynamic problem cannot be solved easily. However, there is no need to solve it because the empirical stream function and [4.10] are derived through interpolation, i.e., by combining the Stokes and potential flow equations. This allows us to express f as the average value of the two extreme regimes; i.e., f = 2.5. This value of f and [4.10] make possible the calculation of the maximum PHF with [4.1]. Detachment Forces Similarly to the pressing force, the detachment force contains two components: the gravitational force and the hydrodynamic force (DHF). Accounting for the gravitational force, Fgd occurs by [4.6], taking into account that near the rear pole the gravitational force is directed away from the bubble and this changes p the sign (Fgd = −Fg ). The analysis of the hydrodynamic force is more complicated. In the two extreme cases of Stokes and potential flow it is easy to evaluate the total detachment force, because the following equality holds for the radial velocity of the liquid at the bubble surface: Ur (π − θ ) = −Ur (θ ).

[4.11]

In combination with [4.1], this means that the DHF at angle π − θ equals the PHF at angle θ . Moreover, the maximum DHF arises near the rear pole since the maximum PHF exists near the front pole. Hence, the total detachment force can be easily calculated for Stokes and potential flow, covering small Re and retarded surfaces and large Re and mobile surfaces, respectively. It should be remarked that potential flow is the main term only in the description of the hydrodynamic field of a rising bubble with a mobile free surface. In the Levich theory (41), the condition of zero viscous stress on the mobile surface is satisfied due to the rise of the hydrodynamic boundary layer. However, close to the rear pole of the bubble a correction to potential flow is essential. This correction will lead to a decrease of the DHF near the rear pole. Consequenctly the maximum DHF exists not far from the rear pole and its value will only be approximately equal to the maximum PHF. At a retarded surface and Re > 1, [4.10] cannot be used to analyze the situation at the rear pole of the bubble; [4.10] applies to the situation at the front pole only. However, in this case there are no reasons for the existence of a DHF that is large in comparison with the PHF. The most interesting case appears when the top part of the bubble is mobile and the bottom part is retarded. With an increase of the surfactant concentration, the dynamic adsorption layer passes from the state of weak retardation into the transient state, characterized by the appearance of a “rear stagnant cap” (RSC) which increases with further increase of surfactant concentration. In this case, weakly and strongly retarded parts of the bubble surface coexist. For mathematical reasons, the RSC

216

MISHCHUK, KOOPAL, AND DUKHIN

model has been elaborated mainly for the case of Re ¿ 1 (49). However, in the Stokes regime the surface of small bubbles is usually very strongly retarded by small concentration of impurities (10, 50) and the RSC model is not very relevant. For these conditions [4.11] is a good approximation. An attempt to extend the RSC model to the interesting case of large Reynolds numbers was made only recently and valuable information about the properties of the dynamic adsorption layer was obtained (51). It must be emphasized that, for many reasons, these results are only the first step in this important direction. Since the theory is numerical, results are available for only a few conditions. In order to use these results for the present analysis, it is worthwhile to consider first the expression for the velocity divergence, dUr d 1 (Uθ sin θ) + = 0, r sin θ dθ dr

[4.12]

where r is the radial distance to the centre of the bubble, Uθ is the tangential component of the liquid velocity, and Ur is the radial component. By integrating from R0 to r under the assumption that r − R0 ¿ R0 , it should be realized that the tangential velocity hardly changes. For the radial velocity we thus find µ ¶ dUθ a . Ur (R0 + a) = − Uθ ctg θ + dθ R0

[4.13]

Equation [4.13] shows that the radial velocity is proportional to both the tangential velocity and its derivative, dUθ /dθ . This result is general and can be used to analyze the effect of surface retardation on the radial velocity. The most important implication of [4.13] is that large values of Ur result if either or both Uθ and/or dUθ /dθ are large. Equation [4.13] can be used to analyze the angular dependence of the bubble surface velocities reported in Fig. 6 of (52) and in Fig. 14 of (50). These results apply for different sizes of the RSC at Re = 200. The velocity equals zero at the front pole, increases along the bubble surface, attains its maximum value, and goes to zero very rapidly near the boundary of the RSC. Because of this very rapid decrease, the tangential derivative of the surface velocity is very large and this will lead to a value of the DHF that considerably exceeds the maximum PHF. This result is a consequence of the discontinuity under boundary conditions (51); the shear stress equals zero on the clean upper part of the bubble surface and the nonslipping condition is fulfilled on the lower part of the bubble covered with surfactant. In reality, the boundary between these parts is not a line but a strip, across which surface concentration and velocity gradually change. Thus the existing RSC model must be supplemented with this transition zone in order to make the calculation of the radial velocity and the DHF possible. At present only qualitative conclusions based on figures presented in Refs. (50), (52) can be drawn.

5. MICROFLOTATION DOMAIN

The notion of flotation domain was first introduced by Crawford and Ralston (52). However, in the present paper the domain will be determined for a different condition. First, we will analyze the condition [2.2] for attachment. It indicates that near the front pole of a bubble for all values of h, a particle is subjected to a force directed toward the bubble surface; otherwise deposition cannot occur. The same expression imposes limitations on the values of the parameters at which the disjoining pressure can be overcome. The substitution of the maximum electrostatic repulsive force and the pressing forces into [2.2] yields the value of the critical radius of a particle aatcr . At a > aatcr , attachment occurs, while at a < aatcr , the total pressing force does not lead to attachment. For the Stokes regime, the substitution of [3.22] and [4.7] into [2.2] yields an equation that allows us to express the critical radius aatcr through the electrolyte concentration and the small potential: v u Aκ 2 2 2 u− t 12π ln2 (ψ2 /ψ1 ) + 8εε0 κ(RT /F) sinh (Fψ1 /4RT ) cr . aat ≈ g(21ρ/3 + 3.23ρw ) [5.1a] By neglecting the molecular forces and assuming that ψ1 is small, [5.1a] reduces to s ¡ aatcr

≈

± ¢1/2 p 2εε0 F 2 RT 4 C 0 ψ1 . 2g(21ρ/3 + 3.23ρw )

[5.1b]

The most interesting case for practical purposes is a low electrolyte concentration for which the maximum repulsion takes place at large distances where molecular forces are negligible. This situation is described by [5.1b]. In deriving [5.1b], the Debye length was expressed through the electrolyte concentration, C0 ; ψ1 is assumed to be sufficiently small. According to [5.1b] √ the critical radius is proportional to the small potential ψ1 and 4 C0 . The same procedure with the use of [4.8] for the PHF and neglecting the gravity force results in the expression for aatcr in the case of potential flow,

aatcr

≈

− 12π

Aκ 2 ln2 (ψ2 /ψ1 )

+ 8εε0 κ(RT /F)2 sinh2 (Fψ1 /4RT ) 18ηU0 /R0

,

[5.2a] or, by neglecting the molecular forces and at small values of the small potential, ¡

aatcr

± ¢1/2 p 2εε0 F 2 RT ≈ · C0 · ψ12 . 36ηU0 /R0

[5.2b]

217

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

FIG. 6. (a) Critical particle radius (numerical calculations based on [2.2] and the exact expressions for Fme and Fpr ) as functions of the electrolyte concentration at A = −1 × 10−20 J, ψ˜ 2 = 6, and different values of ψ˜ 1 . Curves 1 to 3 apply to the Stokes regime ([4.7]) with bubble radius R0 = 75 µm); curve 1: ψ˜ 1 = 0.5; curve 2: ψ˜ 1 = 1; curve 3: ψ˜ 1 = 1.5. Curves 10 and 100 apply to the intermediate regime, [4.10], with R0 = 700 µm, U0 = 14 cm/s (curve 10 ), and R0 = 350 µm; U0 = 7 cm/s (curve 100 ). (b) Critical particle radius (numerical calculations based on [2.2] and the exact expressions for Fme and Fpr ) as a function of the electrolyte concentration for the potential regime at A = −1 · 10−20 J, ψ˜ 2 = 6, and different values of ψ˜ 1 . Curves 1 and 10 : ψ˜ 1 = 0.5; curves 2 and 20 ; ψ˜ 1 = 1; curves 3 and 30 : ψ˜ 1 = 1.5; curves 4 and 40 : ψ˜ 1 = 2. Bubble radius R0 = 375 µm, U0 = 20 cm/s (curves 1 to 4); R0 = 750 µm, U0 = 30 cm/s (curves 10 to 40 ). Curve 1∗ shows the result of a calculation on the basis of approximation [5.2b] using the same parameter values as for curve 1.

The mobility of the bubble surface and its spherical shape are preserved within a rather narrow range of bubble radii, namely 150–500 µm. Correspondingly, within this range the ratio of the bubble velocity to its radius, U0 /R0 can be considered invariant potential and approximately equal to 500 s−1 . Thus, in the case of√ flow the critical radius of a particle is proportional to C0 and the square of the small potential ψ1 . Although expressions [5.1b] and [5.2b] for aatcr are approximations, they are important for a qualitative understanding of aatcr . The radius aatcr as a function of the electrolyte concentration is shown in Figs. 6a and 6b for Stokes and potential flow, respectively. The results shown are based on [2.2] with the full expressions for the different functions, rather than on the approximate

expressions [5.1b] and [5.2b]. In general, aatcr increases with increasing salt concentration. For Stokes flow and high values of the small potential the domain in which attachment occurs is very small. For potential flow the domains at which attachment will occur are larger than for Stokes flow, but also here the domain decreases with increasing value of ψ1 . Comparison of the solid and dotted curves in Fig. 6b gives an indication of the effect of the bubble size and bubble velocity. Curve 1∗ applies under the same conditions as curve 1, but is calculated with the approximate expression [5.2b]. The calculations were carried out for electrolyte concentrations that satisfy the condition C0 < C1cr , because at higher concentrations the attraction force is absent and attachment is impossible. If the concentration is lower than C1cr but the difference between C0 and C1cr is insufficiently large, attachment on the upper part of the bubble surface is accompanied by detachment on the lower part. The smaller the difference between C0 and C1cr , the smaller the attractive force, because the latter is zero at C1cr . Thus, near C1cr , detachment takes place. Because of the complicated expressions for the regime of intermediate Reynolds numbers, analytical calculation of the critical radius is impossible and we cannot provide an equation analogous to [5.1] or [5.2]. Results of some numerical calculations, based on [2.2], [4.1], and [4.10] and a set of Re values, are included in Fig. 6a as curves 10 and 100 . As before, [2.2] is used with the exact expressions for all functions. The observed trends for the intermediate region correspond very well with those for Stokes flow. The interrelationship between attachment and detachment depends on the correlation between the maximum repulsive force and the maximum attractive force and can be analyzed, using [3.20] and [3.22], when the molecular forces are neglected. For the extreme case in which there is a large difference between the surface potentials and a sufficiently low electrolyte concentration, i.e., condition [3.18] is satisfied, we obtain min Fme = max Fme

µ

ψ2 − ψ1 ψ1

¶2

1 > 2κh min

µ

ψ2 ψ1

¶2 .

[5.3]

Let us now specify the situation for the case in which the total pressing force equals (or slightly exceeds) the repulsive force max . Taking into account that the maximum debarrier: Fpr ≈ Fme tachment force equals the maximum pressing force, we may max | is conclude that the detachment force |Fdet | = |Fpr | ≈ |Fme min , because, very small in comparison with the attractive force Fme min max considerably exceeds Fme . Therefore, according to [5.3], Fme at sufficiently low electrolyte concentration, a pressing force that exceeds the force barrier is sufficient to provide attachment without further detachment. Let us now consider the condition [2.3] to prevent detachment in the more general case that is not restricted by condition [3.18]. In order to indicate the flotation domain an additional curve corresponding to [2.3] must be calculated. The analysis accomplished for the attachment is restricted by the condition

218

MISHCHUK, KOOPAL, AND DUKHIN

C0 < C1cr ; however, these concentrations are not necessarily sufficient for the determination of the maximum electrolyte concentration at which detachment is prevented, i.e., C2cr . In order to evaluate C2cr , it is necessary to analyze the case with equal values of maximum repulsive and maximum attractive forces because, as indicated in Section 4, the maximum pressing force will be about equal to the maximum detachment force. For definite values of the potentials and the effective Hamaker constant, this situation is achieved at the critical concentration C2cr : ¡ ¡ ¢¢ ¡ ¡ ¢¢ min max h min C2cr = Fme h max C2cr . −Fme

[5.4]

At concentrations slightly lower than C2cr , the force barrier decreases. This ensures attachment and, at the same time, the attractive surface force increases and this prevents detachment. Therefore, [5.4] expresses the necessary and sufficient condition for microflotation, because under this condition, attachment is provided and detachment is prevented. At fixed values of the potentials and the Hamaker constant, this important statement is also valid for any concentration lower than C2cr . Therefore, C2cr is the maximum electrolyte concentration under which microflotation takes place. Some results of calculations of C2cr are shown in Fig. 7 for different values of the parameters. In general C2cr increases with increasing values of the large potential. An increase of the value of the small potential decreases C2cr substantially (Fig. 7a). More strongly negative values of the Hamaker constant also decrease C2cr considerably (Figs. 7a and 7b). Curve 20 in Fig. 7b shows the result of calculations based on the full Eqs. [3.1] to [3.4], whereas curve 2 represents results obtained with [3.16] and [3.22]. Now we will show that the width of the microflotation domain is very small at concentration C2cr . This is a consequence of the coincidence of the barrier height and potential well depth with the attachment force and the detachment force at this concentration, respectively, ¡ ¢ ¡ ¢ max cr min cr a , C2cr = Fme a , Cccr . F pr (a cr ) = −Fdet (a cr ) = −Fme [5.5] This equality is true only for a single radius a cr . A small increase in particle size leads to an increase of the detachment force and the latter becomes larger than the depth of the force well. A small decrease in particle size leads to a decrease of the pressing force; thus the latter becomes smaller than the barrier height. The decrease of the concentration to a level below C2cr leads to a decrease of the barrier height and an increase in the depth of the force well and this causes an increase of the width of the microflotation domain. Indeed, the smaller the barrier, the smaller the pressing force and, correspondingly, the smaller the minimal particle radius, aatcr , that is necessary to overcome the barrier, ¡ ¢ max (h max , C0 ) C0 < C2cr . Fpr aatcr = Fme

[5.6]

FIG. 7. (a) Critical value of the electrolyte concentration C2cr ([5.4]) as a function of the large potential ψ˜ 2 for two values of the Hamaker constant and different values of the small potential ψ˜ 1 . Curves 1 to 4: A = −1 × 10−20 J; curves 10 to 40 : A = −2 × 10−20 J. Values of the small potential: ψ˜ 1 = 0.5 (curves 1 and 10 ); ψ˜ 1 = 1 (curves 2 and 20 ); ψ˜ 1 = 1.5 (curves 3 and 30 ); ψ˜ 1 = 2 (curves 4 and 40 ). (b) Critical value of the electrolyte concentration C2cr ([5.4] in combination with [3.16] and [3.22]) as a function of large potential ψ˜ 2 at a fixed value of the small potential, ψ˜ 1 = 0.5, and different values of the Hamaker constant: A = −0.5 × 10−20 J (curve 1), A = −1 × 10−20 J (curves 2 and 20 ), A = −1 × 5 · 10−20 J (curve 3), A = −2 × 10−20 J (curve 4), A = −3 × 10−20 J (curve 5), A = −5 × 10−20 J (curve 6). Curve 20 : results are based on the exact equations for Fme and the same condition as for curve 2.

The deeper the potential well, the larger the detachment force cr for which and, correspondingly, the larger the particle size, adet detachment is possible, ¡ cr ¢ min (h min , C0 ) C0 < C2cr . = Fme Fdet adet

[5.7]

Equation [5.6] describes the lower boundary and [5.7] describes the upper boundary of the microflotation domain. Equation [5.6], which applies to the Stokes and potential regime, is mathematically equivalent to [5.1] and [5.2]. However, application of [5.6] is restricted by the condition C0 < C2cr . Consequently, for a given concentration C0 satisfying this condition, [5.6] yields

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

219

the minimal radius for which attachment is provided without further detachment. For concentrations C0 smaller than C1cr , but larger than C2cr , [5.1] and [5.2] describe the situation in which attachment is followed by detachment. The critical particle radius, above which the particle is detached from the bubble surface, can be calculated on the basis of [3.15], [4.7] or [4.8] and [5.7]. In the case of Stokes flow (Re ¿ 1) the result is s cr adet

1 g(21ρ/3 + 3.23ρw )

= s

×

¸ · ¡ 2 ¢ A (ψ2 − ψ1 )2 2 + εε − κ ψ + ψ − ψ ψ 0 1 2 1 2 2h min 12π h 2min [5.8]

and for potential flow (Re À 1) cr adet

µ · 1 A (ψ2 − ψ1 )2 = + εε 0 18ηU0 /R0 12π h 2min 2h min ¸¶ ¡ ¢ − κ ψ12 + ψ22 − ψ1 ψ2 .

[5.9]

Substitution of the expression for the Debye length into these equations leads to s cr adet

=

1 g(21ρ/3 + 3.23ρw ) v s " # u ´ u (ψ2 − ψ1 )2 A 2F 2 C0 ³ 2 t 2 × ψ1 + ψ2 − ψ1 ψ2 + εε0 − 2h min εε0 RT 12π h 2min

[5.10] or cr adet =

µ · 1 A (ψ2 − ψ1 )2 + εε 0 18ηU0 /R0 12π h 2min 2h min s ¸¶ ¢ 2F 2 C0 ¡ 2 ψ1 + ψ22 − ψ1 ψ2 − [5.11] εε0 RT

cr a function of the electrolyte concenFIG. 8. (a) Critical particle radius adet tration for the Stokes regime ([5:10], R0 = 75 µm, curves (1) to (5) and the intermediate regime ([3.15], [4.10], and [5.7], R0 = 350 µm, curves 40 and 50 ) for a series of Hamaker constants. The surface potentials are kept constant: ψ˜ 1 = 0.5, ψ˜ 2 = 2. Hamaker constants: A = −0.5 × 10−20 J (curve 1), A = −1 × 10−20 J (curve 2), A = −1.5 × 10−20 J (curve 3), A = −2 × 10−20 J (curves 4 and 40 ), A = −3 × 10−20 J (curves 5 and 50 ). The calculations are based on [2.3] and the exact expressions for Fme and Fdet (= −Fpr ). (b) Dependence of the critical parcr on the electrolyte concentration for the potential regime at differticle radius adet ent values of the Hamaker constant. The surface potentials are fixed at ψ˜ 1 = 0.5 and ψ˜ 2 = 2. Two bubble sizes are used: R0 = 375 µm (curves 1 to 5) and R0 = 750 µm (curves 10 to 50 ). The Hamaker constants are A = −0.5 × 10−20 J, curves1 and 10 ; A = −1 × 10−20 J, curves 2 and 20 ; A = −1.5 × 10−20 J, curves 3 and 30 ; A = −2 × 10−20 J, curves 4 and 40 ; A = −3 × 10−20 J, curves 5 and 50 .

or for Stokes and potential flow respectively. If we neglect for the moment the weak dependence of h min on the electrolyte concentration, one can easily see that the critical radius decreases with an increase of the electrolyte concentration. In the case of a very low electrolyte concentration, where [3.18] and [3.20] apply, the critical radius for detachment becomes independent of the electrolyte concentration, s cr adet ≈

9πε2 ε02 (ψ2 − ψ1 )2 4g A(21ρ/3 + 3.23ρw )

[5.12]

cr adet ≈

π ε 2 ε02 (ψ2 − ψ1 )4 , 8η AU0 /R0

[5.13]

for Stokes and potential flow respectively. cr as a function of log C0 are shown in Fig. 8. Some results for adet The solid curves in Figs. 8a and 8b are calculated with [5.10] and [5.11], respectively. The dashed curves in Fig. 8a are concerned with the intermediate regime (Ur given by [4.10]) assuming that the detachment force equals the pressing force (as explained in Section 4, this is a rough approximation). The solid and dashed

220

MISHCHUK, KOOPAL, AND DUKHIN

FIG. 9. Microflotation domain for the potential regime at A = −1 × 10−20 J and ψ˜ 1 = 1.5. Curve 1 limits the microflotation domain for ψ˜ 2 ≥ 4; curves 1 (attachment) and 2 (detachment) limit the microflotation domain for ψ˜ 2 = 3. At ψ˜ 2 ≤ 2.5 all particles are detached.

curves in Fig. 8b apply to two different values of the bubble radii. Figures 8a and 8b both indicate that, for C0 > 10−4 mol/L cr strongly decreases with inand a given set of parameters, adet creasing electrolyte concentration. For more negative values of cr shifts to lower salt concentrations. For the Hamaker constant adet cr −4 C0 ¿ 10 mol/L, adet becomes independent of the electrolyte concentration, as indicated by [5.12] and [5.13]. However, such low concentrations are hardly met in practice. Examples of a flotation microdomain in the case of potential flow are shown in Fig. 9 for a given value of the Hamaker constant and various values of the small potential. The lowest electrolyte concentration that may occur in practice is assumed to be 10−4 mol/L. The region between 10−4 mol/L and curve 1 is the flotation domain for Fψ2 /RT ≥ 4. Curves 1 and 2 limit the flotation domain for Fψ2 /RT = 3. For Fψ2 /RT ≤ 2.5 particle/bubble aggregates detach by the detachment force. 6. DISCUSSION

Microflotation of Particles and Bubbles with Potentials with the Same Sign Primary attention is paid to the determination of the necessary and sufficient condition, for microflotation, i.e., the calculation of concentration C2cr . At concentration C2cr , attachment is provided and detachment is prevented for one particle size only. At concentrations higher than C2cr detachment cannot be prevented because the detachment force exceeds the attractive force. At concentrations lower than C2cr , a microflotation domain exists. For particles that are larger than the value of the upper boundary detachment occurs. For particles that are smaller than the value of the lower boundary, the electrostatic barrier exceeds the pressing force and this excludes the possibility of attachment. Attachment without further detachment takes place only when the particle size is between these boundaries.

The equations for C2cr and the boundaries of the microflotation domain are specified for Stokes and potential flow, accounting for the approximate equality of the maximum pressing force and the maximum detachment force. There is no theory for the detachment force in the case of intermediate Re and retarded surfaces. For this case, important from the practical point of view, only the lower boundary of microflotation has been quantified. Nevertheless, the conclusion can be made that the microflotation domain is rather wide at low concentrations, because the attractive force considerably exceeds the repulsive force in this case and the DHF is not too different from the PHF. The exactness of the analytical expression [3.10] for C1cr is reasonably good, as can be seen in Fig. 5b. The same applies to C2cr (see Fig. 7b). According to condition [3.18], the agreement will improve when the electrolyte concentration increases and h min decreases. A decrease of h min can be realized when A decreases and the potential ψ2 increases. In any case, analytical solutions seem to be useful because many parameters affect microflotation and the generalization of results obtained by numerical computation is rather difficult. The results obtained show that the electrostatic attraction and the pressing force can cause microflotation at sufficiently low electrolyte concentrations and certain additional conditions concerning the potentials and the particle size. This does not mean that at higher electrolyte concentration, namely 10−2 – 10−1 mol/L, this mechanism is inefficient. The larger the electrolyte concentration, the smaller h min . According to [3.14] and [3.18], h min should be smaller than 0.25 κ −1 , i.e., smaller than 1 nm in 0.01 mol/L and smaller than 0.25 nm in 0.1 mol/L. For such small distances, the discreteness of interacting molecules and ions affects not only the electrostatic part of the interaction energy but also the Van der Waals part. Accounting for all the effects considered in a recent review (53), we have to realize that conclusions concerning C0 > 0.01 mol/L should be avoided. When the pressing force is the only force that can overcome the repulsive barrier, the concentration C2cr and the boundary of the domain characterize the impossibility of microflotation. Yet microflotation may still be possible due to particle penetration through the electrostatic barrier by Brownian diffusion. In this case, the flotation will be considerably retarded as can be concluded on the basis of experimental data (20, 21). At sufficiently low electrolyte concentration (C0 < C2cr ), microflotation is not retarded for rather large smooth particles, as is seen in Figs. 6a and 6b. However, the electrostatic repulsion is weaker for rough particles and for particles with sharp edges that occur in practice. Therefore, the microflotation mechanism considered in this paper is already effective for particles smaller than the one that are characterized by Figs. 6a and 6b. Paper (54) presents a way to model the Van der Waals and electrostatic interactions of nonideal particles. For the examination of the present theory and the existence of the microflotation domain systematic measurements of the

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

electrokinetic potentials of both particles and bubbles and of the collection efficiencies for particles and bubbles of different sizes are necessary. Microflotation of Particles and Bubbles with Potentials with a Different Sign The above-mentioned model deals with particles and bubbles with potentials that are of the same sign. The idea of improving the flotation by application of cationic surfactants to create opposite potentials was proposed in (24) and also experimentally verified (10). In this case the electrostatic force becomes attractive at all distances. As a result, attachment is possible without a pressing force. However, the detachment force can still suppress the flotation, and the determination of the maximum attractive surface force is still relevant. To estimate this maximum, it is sufficient to change the sign of ψ1 in [3.3]. As a result, the difference of the potentials in [3.13]–[3.17] should be replaced by their sum. This change does not cause an essential difference in comparison with the case of potentials of the same sign, provided the difference in the values of the potentials is large (this case was considered in Section 5). Replacement of the difference with the sum causes a small change only in the maximum attractive force and concentration C2cr . When the difference between the potentials is not large, the attractive force is very small when the potentials are of the same sign, but it increases greatly if the signs become opposite. Thin Film Rupture Another mechanism that may speed up microflotation is the spontaneous breaking of the thin film between a particle and a bubble. Spontaneous rupture occurs at a critical thickness h cr and two cases should be considered separately, h cr < h max

[7.1]

h cr > h max ,

[7.2]

and

where h max characterizes the location of the electrostatic barrier. In the case of [7.1], it is necessary to overcome the electrostatic barrier before rupture can take place, whereas [7.2] is applied when film rupture takes place before the barrier is reached. It will be clear that the results of the present work are relevant if condition [7.1] is satisfied. However, h max and the barrier height also have significance in the case of [7.2]. Experiments with aqueous wetting films (55) show a dependence of h cr on the electrostatic interaction. The main trend is that a strong repulsive barrier suppresses spontaneous film rupture, while attraction or weak repulsion considerably intensifies film rupture. Although the theory of spontaneous rupture applies to flat films and the mentioned results relate to emulsions, one may expect that qualitatively the same results hold for a thin film between a particle

221

and a bubble. Therefore, calculations of the force barrier remain important. The role of h cr in p/b interaction and microflotation was investigated systematically in (24). “Spontaneous” and “forced” thinning of the liquid film that separates droplets was studied in (56) and (57) in relation to the DLVO theory (18, 58). The critical thickness of rupture of aqueous films lies between 30 and 50 nm according to experimental data (55–57). These experimental results and the DLVO theory were used for the interpretation of the first reported visual study of coalescence of microscopic oil drops (70–140 µm in diameter) in water under a wide range of pH conditions (59, 60). The authors concluded that if the total interaction energy is close to zero or has a positive slope in this critical thickness range, coalescence of the oil droplets is to be expected. It is also interesting to note that there is a large similarity in the coalescence of large (59) and micrometer-sized (33) droplets. Analysis of Experimental Results of p/b Interaction The most systematic investigation of the influence of surface forces on the interaction of a particle with a rising bubble was accomplished by Yoon and Mao (9). However, the results of this investigation cannot be compared with the findings of the present paper because Yoon and Mao used experimental conditions that intensified the role of hydrophobic forces and these forces are not considered in the present paper. A valuable systematic investigation of the role of electrostatic interactions in p/b encounters with microflotation was accomplished by Okada et al. (20, 21). Their work is notable for its novel experimental investigation. They visualized the trajectory of a particle and its attachment to a bubble surface; p/b detachment did not occur. In order to change the surface charges of the bubbles and/or the particles different types of surfactants or AlCl3 were used. Okada et al. explained their experiments with a theoretical model based on the HHF equation and accounting for the PHF. In their model it was assumed that the Hamaker constant is positive and no attention was paid to a possible reversal of the direction of the electrostatic force at small distances. Hence, flotation enhancement due to electrostatic attraction is excluded in this model. Despite these shortcomings, the calculated predictions are in good agreement with the experimental results. In considering this result, it should be realized that the influence of the shortcomings of Okada’s model on the calculated attachment behavior is small. As discussed above, the value of the Hamaker constant and even its sign are not very important for the calculation of the attachment barrier. The attractive component of the electrostatic force, omitted by Okada et al., decreases with increasing distance more rapidly than the repulsive electrostatic component. Therefore, neglecting the electrostatic attraction does not cause a large error in the evaluation of the height of the barrier. That is, evaluation of the electrostatic barrier for attachment, with Okada’s model or with the

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present model, will not easily lead to crucial differences. The shortcomings of Okada’s model would be mainly important for the description of the detachment. In their experiments Okada et al. did not observe detachment and, possibly as a consequence, detachment prevention by electrostatic interactions was not considered in Okada’s theoretical treatment. In our view, it is crucial to explain the observation that no detachment occurred. The present model can easily explain detachment being excluded in the experiments. For very small particles and low electrolyte concentration Eq. [5.3] predicts that the maximum attractive force considerably exceeds the electrostatic barrier. In general, one may conclude that the present theory is in qualitative agreement with the experiments of Okada et al. This view is further corroborated by the influence of the electrostatic interactions on the collision efficiency. The experiments of Okada et al. (20, 21) show that for large values of the electrostatic potentials, the collection efficiency is low. The collection efficiency increases 30 or more times at the transition from large bubble and particle potentials to the state when at least one potential is small. Moreover, the increase in collection efficiency occurs without reversal of charge. This is in agreement with our model predictions and it corroborates the valuable proposition made in the literature (1, 10, 23) that electrostatic attraction can be provided by adsorption of (cationic) surfactants. The measured collision efficiency is almost invariant in the particle size range from 0.5 to 1.5 µm. This can be explained as follows: when the particle size decreases, the electrostatic barrier also decreases and the retardation of the Brownian diffusion caused by it diminishes. Simultaneously, the pressing force decreases. The simultaneous decrease of the electrostatic barrier and the pressing force together with the enhancement of the Brownian diffusion result in a collision efficiency that is hardly dependent on the particle size. 7. CONCLUSIONS

Strong electrostatic attraction at small interparticle distances, arising at a large difference of the electrostatic potential values for a bubble and a particle, together with a hydrodynamic pressing force lead to microflotation. The pressing force is required to overcome the electrostatic repulsion at large distances. The strong electrostatic attraction at small distances prevents subsequent detachment by the hydrodynamic force. The electrostatic barrier for attachment is proportional to the small potential squared and its dependence on the large potential and the effective Hamaker constant is weak. The shortrange electrostatic attraction increases with a decrease of the electrolyte concentration and the value of the negative effective Hamaker constant and it increases with an increase of the potential ratio. Microflotation is possible when the following conditions hold simultaneously: the small potential is smaller than 10 to 20 mV, the ratio of potentials is larger than 3 to 6, and the particle size

is larger than 10 to 20 µm for the retarded bubble surface and larger than 3 to 5 µm for a mobile surface. The latter conditions lead to a sufficiently large pressing force. Microflotation takes place for particles with a size that is in between two critical values. The two critical sizes overlap at an electrolyte concentration C2cr that depends on the Hamaker constant and the ratio of the potentials. At concentrations higher than C2cr , microflotation due to the mechanism under consideration is impossible. At concentrations lower than C2cr a difference exists between the largest and smallest critical particle sizes (aatcr cr and adet ) and this range represents the microflotation domain. Microflotation due to Brownian diffusion is possible for a particle with a radius smaller than aatcr or at an electrolyte concentration higher than C2cr . However, there must be a strong decrease in the rate of microflotation due to the reduction of the diffusion flux caused by the electrostatic barrier. In the case of particles with sharp edges microflotation may be more efficient than predicted. Microflotation can also be enhanced by spontaneous rupture of the wetting film; i.e., it may take place at larger electrolyte concentrations and the microflotation domain may be larger. Moreover, film rupture may be enhanced by the action of the pressing forces.

APPENDIX: NOMENCLATURE

A a acrat acrdet C0 C1cr C2cr F Fe Fg Fhd Fm Fme g h h max h min K R r R0 T U0 Uθ Ur x

Hamaker constant particle radius critical radius for particle attachment critical radius of particle detachment electrolyte concentration electrolyte concentration that corresponds to zero minimum of DLVO forces the critical value of electrolyte concentration that causes detachment of particle Faraday constant electrostatic force gravitational force hydrodynamic force molecular force sum of molecular and electrostatic force (DLVO forces) acceleration of gravity shortest distance between particle and bubble co-ordinate of maximum of DLVO forces co-ordinate of minimum DLVO forces constant related to spherical form of bubble and particle gas constant distance between bubble and particle centres bubble radius absolute temperature velocity of bubble floating tangential component of liquid velocity radial component of liquid velocity distance between centre of particle and surface of bubble

MICROFLOTATION SUPPRESSION AND ENHANCEMENT

Greek Symbols 1ρ

difference between densities of a particle and liquid η dynamic viscosity of liquid Debye length κ −1 ν kinematic viscosity of liquid ρ density of particle density of liquid ρW θ angle surface potential (small) ψ1 ψ˜ 1 = Fψ1 /RT relative surface potential (small) surface potential (large) ψ2 ψ˜ 2 = Fψ2 /FR relative surface potential (large) Abbreviations PHF DHF RSC

pressing hydrodynamic force detachment hydrodynamic force “rear stagnant cup” model ACKNOWLEDGMENTS

We are grateful to the International Association for the Promotion of Cooperation with Scientists from the Commonwealth of Independent States for financial support of this investigation within the project INTAS 95–IN/UA– 0165.

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