Drainage and rupture of thin foam films in the presence of ionic and non-ionic surfactants

International Journal of Mineral Processing 102–103 (2012) 58–68

Contents lists available at SciVerse ScienceDirect

International Journal of Mineral Processing journal homepage: www.elsevier.com/locate/ijminpro

Drainage and rupture of thin foam films in the presence of ionic and non-ionic surfactants Liguang Wang ⁎ The University of Queensland, School of Chemical Engineering, Brisbane, Qld 4072, Australia

a r t i c l e

i n f o

Article history: Received 8 April 2011 Received in revised form 12 September 2011 Accepted 25 September 2011 Available online 1 October 2011 Keywords: Bubble coalescence Surface forces Film thinning Film rupture Surface mobility

a b s t r a c t The thin film pressure balance technique was used to determine the overall magnitude of the surface forces in foam films stabilized by flotation reagents such as sodium dodecyl sulfate and polypropylene glycol at high NaCl concentrations. The Stefan–Reynolds lubrication approximation was used to estimate the forces from measured film thinning rates while the capillary wave model of Valkovska, Danov and Ivanov was used to calculate the forces from measured critical rupture thicknesses. It was found that at very low surfactant concentrations commensurate with typical flotation reagent dosage, the overall forces were attractive and up to one order of magnitude stronger than the Lifshitz–van der Waals forces. The forces became smaller with increasing surfactant concentration. It was also found that the forces obtained from the capillary wave theory were indifferent to changes in film radii and surface mobility, in contrast to the Reynolds lubrication approximation. For comparison, other film drainage models considering film surface mobility and hydrodynamic corrugation were used to fit the present experimental thinning curves obtained at very low surfactant concentrations. They also showed that the overall attraction forces were several times stronger than the Lifshitz–van der Waals forces. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Control of froth stability plays an important role in determining the product grade, throughput and recovery achieved from a flotation operation (Mathe et al., 1998; Neethling and Cilliers, 2003; Ata et al., 2003; Pugh, 2007). However, the froth phase in flotation has received little attention and has been recognized as “probably the most neglected phase in flotation research in stark contrast to its importance” (Nagaraj and Ravishankar, 2007). In modeling froth behavior, prediction of bubble coalescence and surface busting remains one of the most challenging tasks (Cillers, 2006). Flotation froth is often generated using very low concentrations (b10 − 4 M) of surfactants (frothers), in contrast to bubbles and foams (or froth) generated in many other industries using high surfactant concentrations close to the critical micelle concentrations (cmc). One of the most widely used non-ionic surfactants in flotation is methyl isobutyl carbinol (MIBC), and its dosage in flotation practice is low, usually in the range of 0.5- to 1.5 × 10 − 4 M. Besides MIBC, other commonly used flotation frothers include water-soluble polymers such as polypropylene glycols with molecular weights of 200 to 800. Polypropylene glycols are considered more powerful frothing agents than MIBC, and therefore lower dosages are usually applied (Klimpel, 1995). Ionic surfactants such as sodium dodecyl sulfate are used as both collectors and frothers in non-sulfide mineral flotation.

⁎ Tel.: + 61 7 3365 7942; fax: + 61 7 3365 4199. E-mail address: [email protected]. 0301-7516/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.minpro.2011.09.012

These ionic surfactants also have frothing capability and can stabilize air bubbles in the froth phase of flotation. In addition to surfactants and polymers, salt water (or seawater) has also been used as frothing agent (Klassen and Mokrousov, 1963; Castro et al., 2010). Shortage of fresh water has driven some flotation plants to use salt water as frothing agent. Nevertheless, it is beneficial to use combined addition of inorganic electrolytes and surfactants (Yarar and Dogan, 1987). A fundamental understanding to this phenomenon is limited. In this communication, we studied the drainage and rupture of single foam films containing sodium dodecyl sulfate in the presence of 0.3 M NaCl and polypropylene glycol with an average molecular weight of 400 (Dowfroth 400) in the presence of 0.1 M NaCl. The main component of the driving forces for film drainage and rupture, the inter-bubble attraction, was determined from film thinning kinetics by applying the film drainage model of Scheludko and Platikanov (1961) and from critical rupture thicknesses by applying the capillary wave model of Valkovska et al. (2002), respectively. For a given film, we compared the magnitudes of the overall surface forces obtained from film drainage and rupture. The effect of surface mobility on the magnitudes of the forces was also evaluated. The implication of these results on flotation was discussed. 2. Materials and experimental methods Specially pure sodium dodecyl sulfate (SDS) was obtained from Fluka and recrystallized from ethanol. The polypropylene glycol (PPG) with an average molecular weight of 400 was also obtained from Fluka. Sodium chloride (99.5%, Sigma Aldrich, USA) was purified

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

by roasting at 700 °C for 5 h. De-ionized (DI) water (18.2 MΩ∙cm − 1) used was produced by a Milli-Q unit (Millipore, USA) which was combined by a Reverse Osmosis system. All the glassware and the film holder were cleaned with concentrated sulfuric acid and vigorously rinsed with deionized water. The static surface tension isotherm was measured at 22 °C using Wilhelmy plate method. The TFPB technique developed by Scheludko and Exerowa (1959) was used to study the surface forces in the foam films. Inside a closed and vapor-saturated vessel, a single horizontal foam film was formed using a film holder, so-called Scheludko Cell. Fig. 1 demonstrated the schematic diagram of the Scheludko Cell, which was essentially a glass capillary tube with a small orifice connecting a side glass tube. The side glass tube, whose upper end was connected to a piston, was used to transport solution (or water). A single horizontal foam film is formed by sucking out the aqueous solution in the Scheludko Cell. Prior to experiments, it is important to make the inner wall of the film holder hydrophilic. Therefore, much effort has been made to clean the film holder. The inner radius (Rc) of the Scheludko Cell is 1.90 mm. The film radii (Rf) were controlled at 28–33 μm with measuring uncertainty of ±2.5 μm. It was observed that Rf remained unchanged during the course of film thinning. Special care was taken to ensure that no air bubbles are seen in the side glass tube filled with liquid, which is important for keeping Rf unchanged during film thinning. With a normal incidence light of wave length (λ), the thickness (H) of a homogeneous film with a refractive index (nf) was determined interferometrically from the reflected light intensities using an optical equation (Scheludko, 1967):

H¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u λ Δ u arcsint ð1−ΔÞ 2πnf 1 þ 4R ð1−RÞ2

59

curve. Although it is desirable to use a photomultiplier to directly measure the reflectance of the illuminated central zone of the films (Wang and Yoon, 2005), with the present microscope system this approach cannot be taken for very small foam films with radii below 50 μm. The calibration curve was made using several metastable foam films (Rf ~ 100 μm) at quasi-equilibrium. During calibration, the reflected lights from the film surfaces passed through a bandpass filter of 546 nm before reaching a photomultiplier, and the intensity of the monochromic (green) light was recorded. In what follows, the image of each film was taken on a different light path (free of bandpass filter) and processed offline to obtain the gray levels using the Vision Program of National Instruments. Fig. 2 shows a typical calibration curve. It demonstrates an excellent linear relationship (R 2 = 0.9966) between the digitized pixel gray level on the green plane of the images and the logarithm of the photomultiplier output, which follows the Weber's law (Weber, 1834). The calibration curve holds for films with different radii, i.e. 28–100 μm, where the gray levels of the images of a given film were observed to remain unchanged. 3. Results When a single horizontal foam film is formed in the Scheludko Cell (see Fig. 1), the driving force for the initial stage of film drainage is the capillary pressure (Pc). In the present work, the film radii (Rf) were controlled at 28–33 μm, which is significantly smaller than the inner radius, Rc, of the film holder, so the capillary pressure is given by the following equation (Scheludko, 1967; Exerowa and Kruglyakov, 1998):

ð1Þ Pc ¼ 2

2

with where Δ = (I − Imin) / (Imax − Imin), R = (nf − 1) / (nf + 1) nf = 1.335. In the above equation, λ is the wave length (equal to 546 nm in our case) and Imax and Imin are respectively the last maximum and minimum intensities of the reflected light from the film, while I is the instantaneous value of the reflected intensity during the thinning of the film. The reflected light intensities (I) in Eq. (1) were estimated from the gray levels of the images of the films, according to a calibration

2γ Rc

ð2Þ

where γ is the surface tension of the film-forming solution. Fig. 3 shows the surface tension isotherms for the frothers studied in the present work. The solid lines represent the Langmuir–Szyszkowski equation, γ = γ0−RTΓm ln(1 + KLc)which has been fitted to the surface tension data, where γ0 is the surface tension of pure water, R gas constant, T absolute temperature, Γm a maximum adsorption density, KL the Langmuir equilibrium adsorption constant, and c the bulk concentration. For SDS, it gives 3.8 × 10 − 6 mol/m 2 for Γm, and 1.0 × 10 5 M − 1 for KL. The value of Γm for PPG is 1.0 × 10 − 6 mol/m 2, which is much lower than that of SDS. As is known, PPG has a larger molecular weight and hence a larger parking area than SDS. On the other hand, the value of KL for PPG is 6.6 × 10 5 M − 1, which is higher than that of SDS. 140

Film Water

2Rf

Gray level (a.u.)

120 Air

100 80 60 Gray level = -112.95 + 134.18 log(I)

Air

R2=0.9966

40

2Rc

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

log(I ) Fig. 1. Schematic diagram of the Scheludko Cell, a glass capillary tube with a small orifice connecting a side glass tube. A single horizontal foam film is formed in the capillary tube by sucking out the aqueous solution through the orifice.

Fig. 2. Typical calibration curve between gray level of images and photomultiplier output (I).

60

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

75

where the electrostatic double layer repulsion was effectively suppressed (Πel ≈ 0). So the first term of Eq. (4) can be ignored to obtain:

70

(mN/m)

65

Π¼−

60

PPG400 with 0.1 M NaCl

50 45

  2 2 2     n −n 2 3 3 ε2 −ε3 2 3h ν −2κH A232 ðH Þ ¼ kT ð2κH Þe þ Ppffiffiffie  F H˜  3=2 4 ε2 þ ε3 16 2 n22 þ n23

SDS with 0.3 M NaCl

35 10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

in which H˜ is the dimensionless distance, given by

Fig. 3. Surface tensions of SDS with 0.3 M NaCl (Wang and Yoon, 2005) and PPG with 0.1 M NaCl. The solid lines represent the best fits of Langmuir–Szyszkowski equation. a) 1 × 10 − 5 M SDS + 0.3 M NaCl. b) 1 × 10 − 7 M PPG + 0.1 M NaCl.

As the film thickness (H) is reduced, the film drainage process is controlled by surface forces. The driving force is given by the following relation: ΔP ¼ Pc −Π

ð3Þ

where the capillary pressure Pc is given by Eq. (2), and Π is the disjoining pressure. The classical Derjaguin–Landau–Verwey–Overbeek (DLVO) theory is commonly used to explain the stability of foams and foam films with varying degrees of success. It consists of the electrostatic double layer force (Πel) and van der Waals force (Πvw). For single soap films, discrepancies between DLVO theory and experiment have been repeatedly found (Lyklema and Mysels, 1965; Exerowa et al., 1987; Bergeron, 1999; Tchaliovska et al., 1994), indicating that the DLVO forces alone could not explain the results using the classical DLVO theory. It is now recognized that non-DLVO forces such as steric force, hydration force and hydrophobic force may also be important in determining the stability of thin water films (Rabinovich and Derjaguin, 1988; Bergeron, 1999; Angarska et al., 2004; Churaev, 2005). It is customary to assume that various contributions to the disjoining pressure are additive and one can use the following equation to express the surface forces in soap films: Π ¼ Πel þ Πvw þ Πst

ð4Þ

in which the non-DLVO force (Πst) may be represented by the following relation (Rabinovich and Derjaguin, 1988; Tabakova and Danov, 2009; Yoon and Aksoy, 1999): K232 : 6πH 3

ð5Þ

Eq. (5) is of the same form as the van der Waals pressure: A232 6πH3

  2 2 1=2 2πνe H H˜ ¼ n3 n2 þ n3 c and !3=2 !−2=3 πH˜ pffiffiffi 4 2

  F H˜ ≈ 1 þ

3.1. Kinetics of film thinning

Πvw ¼ −

ð8Þ

-1

Surfactant concentration (M)

Πst ¼ −

ð7Þ

With considering the electromagnetic retardation and electrolyte screening effects, the Hamaker constants were calculated on the basis of the Lifshitz–Hamaker theory using the following equation (Russel et al., 1989):

55

40

1 ½A232 ðH Þ þ K232 : 6πH 3

ð6Þ

so that K232 can be directly compared to the Hamaker constant (A232) for the interaction between two gas phases (or bubbles) 2 interacting in water 3 at a separation distance of H. In the present work, we measured film thinning rates and rupture thicknesses at high electrolyte concentrations (i.e., 0.1 or 0.3 M NaCl),

where k is the Boltzmann's constant, hP the Planck's constant (6.63 × 10 − 34 J·s), νe the main electronic adsorption frequency (3 × 10 15 Hz), c the speed of light in vacuum (3.0 × 10 8 m/s), n2 the refractive index of air, n3 the refractive index of solution, ε2 the dielectric constant of air, ε3 is the dielectric constant of water, and κ the inverse Debye length for a 1:1 electrolyte is given by: κ¼

2e2 NA Cel εr ε0 kT

!1=2

where ε0 and εr are the permittivity of the vacuum and the dielectric constant of water, respectively, e is the electronic charge, NA the Avogadro's number, and Cel is the concentration of electrolyte. We fitted the experimental film thinning data to the film drainage model of Scheludko, namely the Stefan–Reynolds lubrication approximation (Stefan, 1874; Reynolds, 1886; Scheludko and Platikanov, 1961), which gives the Reynolds thinning velocity, VRe: −

dH 2H 3 ΔP ¼ dt 3μRf 2

ð9Þ

where t is drainage time, μ is dynamic viscosity, and ΔP is the driving force for film thinning. In principle, the Reynolds equation is applicable to plane-parallel foam films with tangentially immobile surfaces. It has been shown that this condition can be met by using very small films stabilized by a surfactant (Exerowa, and Kruglyakov, 1998; Langevin, 2000; Coons et al., 2005). Specially, some researchers stressed that the Reynolds equation is only applicable to foam films with Rf below 50 μm (Manev et al., 1997; Karakashev et al., 2010). Fig. 4 shows some typical images of the thinning foam films containing 1 × 10 − 5 M SDS with 0.3 M NaCl and 1 × 10 − 7 M PPG with 0.1 M NaCl, respectively. On each row, four images were placed side by side to indicate the changes in the thickness profile of a thinning film at a given radius. As shown, the surfaces of these small films with Rf below 40 μm were prevalently flat, while relatively large films with Rf above 50 μm had pronounced dimples or surface corrugations. Similar profiles were observed for other films studied in the present work. Our experiments were performed at low surfactant concentrations commensurate with typical flotation reagent dosage. It has also been shown that Eq. (9) can be used even at low surfactant concentrations where the Gibbs elasticity (EG) is higher than 1 mN/m (Ivanov et al., 2005). With using relatively strong frothing reagents

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

61

a) 1x10-5 M SDS+ 0.3 M NaCl Rf = 37 µm

Rf = 79 µm

b) 1x10-7 M PPG+ 0.1 M NaCl Rf = 28 µm

Rf = 50 µm

Fig. 4. Typical images of the thinning foam films at different film radii Rf and concentrations: a) 1 × 10− 5 M SDS and 0.3 M NaCl; b) 1 × 10− 7 M PPG and 0.1 M NaCl. The leftmost images represent the films with thickness being approximately 100 nm, while the rightmost images are the last ones immediately before rupture (not shown to scale).

where H0 and H represent the film thicknesses at time 0 and t, respectively. For each film thinning curve, we chose H0 = 102.5 nm, corresponding to the film with the highest brightness and digitalized gray level. It was assumed that Rf is independent of H during the film thinning, which was found to be the case in the present work. Eq. (10) was used to obtain an H vs. t curve by integrating it at 0.1 nm intervals of H using the central scheme of the Euler integration

such as SDS and PPG, the above Rf and EG requirements have been fulfilled in the present work. By separating the variables of Eq. (9) and integrating it, one can obtain, H

t¼−∫ H0

3μR2f 2H ðPc −ΠÞ 3

ð10Þ

dH

a)

100

SDS 1x10

-7

80

-20

K 232 = 8.0x10

H (nm)

-5

K 232 = 0 K 232 = 5.0x10

60

J

40

0

2

c)

100

4

PPG 1x10

-7

6

8

10

R f = 32.3 um

M

20

0

2

d)

100

4

PPG 1x10

-4

80

80 -19

60

K 232 = 2.0x10

8

10

R f = 31.2 um

M

-20

K 232 = 2.5x10

60

J

6

K 232 = 0

K 232 = 0

J

40

40

20

R f = 28.0 um

M

-20

J

40

20

SDS 1x10

80

K 232 = 0

60

b)

100

R f = 28.0 um

M

0

2

4

6

8

10

20

0

2

4

6

8

10

Time (sec) Fig. 5. Kinetics of film thinning at a) 1 × 10− 7 M SDS and 0.3 M NaCl; b) 1 × 10− 5 M SDS and 0.3 M NaCl; c) 1 × 10− 7 M PPG and 0.1 M NaCl; d) 1 × 10− 4 M PPG and 0.1 M NaCl. The radii of the foam film (Rf) are in the range of 28–33 μm. The solid lines represent the Reynolds equation (Eq. (9)) by considering contributions from the non-DLVO attraction force with fitted values of K232. The contributions from double-layer force are set to 0, while those from the van der Waals force were calculated using the Hamaker constant obtained using the Lifshitz theory (Eq. (8)).

62

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

method. Note that an interval of 0.1 nm is sufficiently small for the integration of the Reynolds equation (Wang, 2006). The integration was carried out to fit the film thinning kinetics data with K232 being the only adjustable parameter. Fig. 5 shows the film thickness (H) versus time (t) plots obtained at different concentrations of SDS and PPG in the presence of 0.3 and 0.1 M NaCl, respectively. At such high electrolyte concentrations, the electrostatic double layer force was effectively screened so that Пel ≈ 0, and as a consequence the films became unstable. The film ruptured in 4.0 s at H = 29.1 nm at 1 × 10 − 7 M SDS. At 1 × 10 − 5 M SDS, the film became more stable, with the rupture occurring in 6.0 s at 23.9 nm. Likewise, the film ruptured in 3.2 s at H = 32.8 nm at 1 × 10 − 7 M PPG. At 1 × 10 − 4 M PPG, film rupture occurred in 9.0 s at 24.4 nm. Also shown in Fig. 5 are theoretical fits by Eq. (9) with the extended DLVO theory (K232 ≠ 0) and the DLVO theory (K232 = 0). It was found that the experimental curve can be fitted better with considering an additional attractive force. Specially, at 10 − 7 M SDS, the kinetics was considerably faster than predicted (dotted line) without considering the non-DLVO force, i.e., K232 = 0. The experimental data could be fitted to the Reynolds equation (solid line) with K232 = 8.0 × 10 − 20 J. At a higher SDS concentration, 1 × 10 − 5 M, the non-DLVO attraction forces in foam films became smaller, being 5.0 × 10 − 20 J. Likewise, increasing PPG concentration from 1 × 10 − 7 to 1 × 10 − 4 M considerably dampened the non-DLVO force. Fig. 6 shows the K232 values determined as such as a function of SDS and PPG concentrations. For SDS films over the concentration range studied, the K232 values determined from the kinetics data are approximately 5 to 2.7 times larger than the Hamaker constants calculated at Hcr using Eq. (8). For PPG films, the K232 values determined from the kinetics data are approximately 13 to 0.8 times larger than the Hamaker constants calculated at Hcr.

3.2. Critical rupture thickness As the thickness of a foam film is reduced by drainage, the film ruptures catastrophically when the thickness reaches a critical thickness (Hcr). It is believed that a film surface is always in thermally- or mechanically-induced oscillation, causing the instantaneous distance between the two interfaces in some regions of a foam film to be smaller than the measured distance. The amplitude of the oscillation increases, when the instantaneous distance between the two surfaces is within the range of an attractive force, e.g., van der Waals force.

When the distance between the two surfaces reaches Hcr, the film ruptures spontaneously. Recently, Angarska et al. (2004) showed that the predictions of the capillary wave model developed by Valkovska et al. (2002) were in good agreement with the experimental data obtained by Manev et al. (1984) at high surfactant concentrations, under which condition the surface adsorption density was close to its maximum and the air/ water interface was hydrophilic. At low concentrations of surfactants, however, Angarska et al. found that the capillary wave mechanism driven solely by the van der Waals force failed to predict Hcr. Fig. 7 shows the changes in measured Hcr as a function of SDS concentration (a) and PPG concentration (b), in the presence of 0.3 and 0.1 M NaCl, respectively. At a given surfactant concentration, a set of six Hcr measurements was made and averaged. As shown, Hcr of the films gradually decreases with increasing surfactant concentration. Also shown in Fig. 7 for comparison are the predictions (solid and dashed lines) from the capillary wave model of Valkovska et al. (2002) with considering the van der Waals force alone. The dashed lines represent the Hcr values predicted using the capillary wave model under immobile surface condition (Eqs. (A1–A4)), while the solid lines represent the Hcr values predicted by using the capillary wave model under fully mobile surface condition (Eqs. (A2, A5– A7)). The film radii (Rf) were controlled within a narrow range (i.e., 28–33 μm). The input parameters are Rf, A232 and γ, and there are no adjustable parameters. As shown, the predicted Hcr gradually increased with increasing surfactant concentration. This trend is largely due to the role of surface tension (or capillary pressure) in determining the driving force (ΔP) and dampening the growth rate of surface waves (Valkovska et al., 2002). It appears that the model predicts a trend which is opposite to the experimental curve. The predicted Hcr values appear to be closer to the experimental ones at higher surfactant concentrations. At very low surfactant concentrations, the experimental Hcr values were much larger than the predicted ones, suggesting that the overall inter-bubble attraction should be much stronger than the van der Waals force predicted by the classical Lifshitz–Hamaker theory. A similar conclusion was drawn by Angarska et al. (2004), but these investigators neither reported Hcr for very small foam films (with Rf below 50 μm), nor did they compare the strength of the non-DLVO attraction force to that of the van der Waals force. 35

a)

0.3 M NaCl

b)

0.1 M NaCl

30

-18

10

K232 (J)

Hcr (nm)

SDS with 0.3 M NaCl PPG with 0.1 M NaCl

25

-19

10

20

15

A 232

10-7

10-6

SDS (M)

10-5

10-7

10-6

10-5

10-4

10-3

PPG (M)

-20

10

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

Surfactant concentration (M) Fig. 6. Effect of surfactant concentration on K232. The results were obtained by fitting the film thinning data to the Reynolds equation (Eq. (9)). The Hamaker constants calculated at Hcr using the Lifshitz theory (Eq. (8)) are in a narrow range of 1.50- to 1.87 × 10− 20 J. The radii of the foam film (Rf) are in the range of 28–33 μm.

Fig. 7. Changes in the critical rupture thickness (Hcr) as a function of SDS concentration (a) and PPG concentration (b), respectively. The error bars represent the 95% confidence limits. The dashed lines represent the Hcr values predicted using the capillary wave model under immobile surface condition (Eqs. (A1–A4)) with considering the van der Waals force alone. The solid lines represent the Hcr values predicted by using the capillary wave model under fully mobile surface condition (Eqs. (A2, A5–A7)) with considering the van der Waals force alone (K232 = 0).

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

It is relatively easy, however, to use Eq. (5), a power law to obtain K232 values, which may then be compared to the Hamaker constant A232 values directly. Tabakova and Danov (2009) also used a power law to represent non-DLVO forces in their theoretical film thinning and rupture studies. In the present work, we used the capillary wave theory coupled with Eq. (5) to fit the measured critical rupture thicknesses of SDS and PPG. We followed Angarska et al.'s (2004) numerical procedure with input parameters being Hcr, Rf and γ. The goal is to allow the calculated Hcr to approach the experimental Hcr by adjusting K232. The detailed numerical procedure is given in Appendix A. In using the capillary wave theory, we followed Angarska et al. to assume that the film surfaces are tangentially immobile. In addition, A232 values calculated using Eq. (8) at corresponding Hcr values were used as fixed values when integrating the Eqs. (A2 and A4) from Hcr to Htr. Under these conditions, analytical expression for the integrals can be obtained (Wolfram Mathematica Online Integrator) and thus K232 values can be readily fitted from experimental Hcr data and are presented in Fig. 8 and the central columns of Table 1 and Table 2. The simplification in calculating the integrals in Eqs. (A2 and A4) gave a satisfactory accuracy and it would not overpredict the K232 values. Without this simplification in calculating the integrals, that is, if A232 were allowed to vary as a function of film thickness from Hcr to Htr by following Eq. (8), the numerical integration will make the fitted K232 values slightly larger. At 10 − 7 M SDS with 0.3 M NaCl, for instance, the numerical calculations for the integrals in Eqs. (A2) and (A4) with A232 (H) varying with H gave K232 = 1.0 × 10− 19 J. When applying the analytical expression to integrate Eqs. (A2) and (A4) with fixed A232 (at Hcr = 29.1 nm) = 1.62 × 10− 20 J, we obtained K232 = 9.8 × 10− 20 J. In contrast, the experimental uncertainty of Rf (i.e. ±2.5 μm) in the present work is a bigger source of error. For example, it could result in the fitted K232 values for 10− 7 M SDS with 0.3 M NaCl to vary from 0.82- to 1.20 × 10− 19 J. The effect of surface mobility on Hcr predictions was also examined. If one assumes that the SDS and PPG films are fully tangentially mobile with a mobility parameter d being infinitely large, so the term (3Hst + 2 d)/(Hst + d) in the work of Valkovska et al. (2002) would approach 2 for mobile interfaces. Thus, we used a different numerical procedure presented in Appendix A to determine K232. By using Eqs. (A2, A5–A7), we obtained the fitted K232 values for MIBC and PPG films whose surfaces are assumed to be fully tangentially mobile, and the results are presented in the right columns of Tables 1 and 2. The fitted K232 values with fully mobile surfaces were slightly smaller than those with immobile surfaces. It appears that the above

-18

10

K232 (J)

SDS with 0.3 M NaCl PPG with 0.1 M NaCl

63

Table 1 Effect of SDS on the values of K232 (in unit of J) as calculated using the capillary wave theory for immobile and fully mobile surfaces. SDS (M)

Immobile

Fully mobile

1 × 10− 7 5 × 10− 7 1 × 10− 6 1 × 10− 5

1.0 × 10− 19 6.2 × 10− 20 5.4 × 10− 20 3.4 × 10− 20

9.9 × 10− 20 5.9 × 10− 20 5.2 × 10− 20 3.2 × 10− 20

calculations on K232 values from experimental critical thickness data are insensitive to mobility parameter. A similar conclusion was drawn by Angarska et al. (2004).

4. Discussion Further comparison was made for the K232 values obtained from film thinning kinetics and critical rupture thickness by using the Reynolds lubrication approximation and capillary wave model, respectively. The K232 values shown in Figs. 6 and 8 were replotted in Fig. 9. Fig. 9a shows the K232 values as a function of SDS concentration in the presence of 0.3 M NaCl. As shown, at a given concentration, the K232 values obtained from these two different models are very close. Likewise, Fig. 9b shows the K232 values as a function of PPG concentration in the presence of 0.1 M NaCl. The capillary wave model gave slightly smaller K232 values than the Reynolds equation at PPG concentrations below 5 × 10 − 5 M. At higher PPG concentrations, the K232 values from film thinning kinetics and critical rupture thickness are in good agreement. In Figs. 6 and 8, the K232 values of SDS films are systematically smaller than those of PPG films. This finding suggests that the head group of PPG may be less hydrophilic than that of the SDS group, which is likely in view of the fact that negatively charged head group of SDS is more strongly hydrated than the OH-group of PPG. The difference between the two sets of K232 data may also be due to the difference in NaCl concentration. The SDS films have an ionic strength higher than the PPG films. It has been reported that increasing NaCl concentration has a dampening effect on the non-DLVO attraction force in foam films (Wang and Yoon, 2004, 2008). Note that the K232 values of SDS films obtained in the present work are three to four times smaller than those of larger SDS films with Rf being in the range of 55–80 μm (Wang and Yoon, 2005). This discrepancy appears noticeably large, so we further performed a series of similar experiments for films of different radii, and the effect of film radii on the K232 value results can be seen in Fig. 10. As shown, when changing film radii from 28 to 80 μm, the K232 values obtained from the Reynolds lubrication approximation kept becoming larger, while the K232 values obtained from the capillary wave theory were indifferent to Rf. The latter was consistent with the finding of Angarska et al. (2004). These investigators also found that with using the capillary wave theory, the fitted hydrophobic force constant is insensitive to changes in film radii from 62 to 155 μm. It is,

-19

10

Table 2 Effect of PPG concentration on the values of K232 (in unit of J) as calculated using the capillary wave theory for immobile and fully mobile surfaces. PPG (M)

A 232

−7

-20

10

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

Surfactant concentration (M) Fig. 8. Effects of surfactant concentration on K232, determined from critical rupture thicknesses and the capillary wave model.

1 × 10 1 × 10− 6 1 × 10− 5 5 × 10− 5 1 × 10− 4 7.5 × 10− 4

Immobile − 19

1.6 × 10 6.2 × 10− 20 3.2 × 10− 20 3.1 × 10− 20 2.9 × 10− 20 1.7 × 10− 20

Fully mobile 1.5 × 10− 19 6.0 × 10− 20 3.1 × 10− 20 3.0 × 10− 20 2.8 × 10− 20 1.6 × 10− 20

64

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68 -18

Eq. (9)) for diffusion and surface mobility. The expression for the velocity of thinning is given as

10

0.3 M NaCl

a)

b)

0.1 M NaCl

K232 (J)

Reynolds Capillary wave

Reynolds Capillary wave

−

  dH 2H 3 ΔP hs ¼ 1 þ b þ dt H 3μRf 2

ð11Þ

where -19

10

b¼

-20

10

A 232 10

-7

A 232 10

-6

10

-5

10

-7

10

-6

SDS (M)

10

-5

10

-4

10

-3

PPG (M)

Fig. 9. Effect of SDS and PPG concentration on the K232 values in foam films with 0.3 and 0.1 M NaCl, respectively. For comparison, the filled symbols represent the K232 values obtained by using Eq. (9) (filled symbols) to fit experimental film thinning curves, and the empty symbols represent those obtained by using Eqs. (A1–A4) to fit experimental Hcr values. The film surfaces are assumed to be tangentially immobile.

therefore, suggested that the capillary wave theory is more advanced in studying surface forces in foam films in view of the dynamic nature of these films and, at Rf above 33 μm the non-DLVO attraction force obtained from film thinning kinetics by using the Reynolds lubrication approximation may have been overestimated. Next, we shall discuss some potential factors such as tangential mobility (see Figs. 11 and 12) and surface deformation (see Fig. 13), which might cause the overestimation of the non-DLVO force when applying the Reynolds lubrication approximation to relative large films. The film drainage model of Radoev et al. (1974) is a method of correcting the Reynolds film thinning velocities (VRe, given by

a)

SDS 1x10

-7

3μD 6μDs andhs ¼ EG ha kTΓ

in which D and Ds are coefficients of the bulk and surface diffusion of the surfactant molecules, respectively, EG [= − Г(∂γ/∂Г)] is Gibbs surface elasticity, Г is surfactant adsorption at the film surfaces, ha (=∂Г/∂c) is the adsorption length, and c is surfactant concentration. Note that in Eq. (12), D and Ds are coefficients of the bulk and surface diffusion of the surfactant molecules. For ionic surfactants like SDS, D and Ds may be estimated from a series of surface tension isotherms at varying electrolyte concentrations (Angarska et al., 2004). For very low SDS concentration at 0.3 M NaCl, b values in Eq. (11) are in the order of 10 − 6 and thus, can be neglected. In integrating Eq. (11) to obtain H versus t curve, we also took the central scheme of the Euler integration method at an interval of 0.1 nm. When the film surfaces were assumed to be tangentially immobile (hs = 0), Eq. (11) reduces to the Reynolds lubrication approximation. Fig. 11 shows an experimental film thinning curve obtained at 5 × 10 − 7 M SDS and 0.3 M NaCl. Also shown are theoretical fits by Eq. (10) with the extended DLVO theory (K232 = 2.7 × 10 − 19 J) and immobile surfaces (hs = 0) and Eq. (10) with the DLVO theory (K232 = 0) and immobile surfaces (hs = 0). It was found that the experimental curve can be fitted better with the Extended DLVO theory. With surface mobility taken into account, according to Angarska et al. (2004) the value of surface mobility parameter (hs) for 5 × 10 − 7 M SDS was 4.68 nm, which was obtained by the theoretical fit of a series of SDS surface tension isotherms at various NaCl concentrations. In Fig. 11, the red dash-dotted lines represent Eq. (11) with the

b)

M Reynolds

-18

10

ð12Þ

10

SDS 1x10

-5

M

-18

Reynolds -19

10

10

-19

10

-20

Capillary wave

K232 (J)

Capillary wave -20

10

30

c)

40

PPG 1x10

50 -7

60

M

70

80

30

d)

Reynolds

-18

10

10

40

PPG 1x10

50 -4

60

70

80

M

-18

Reynolds

-19

10

10

-19

Capillary wave Capillary wave -20

10

30

40

50

60

70

80

10

-20

30

40

50

60

70

80

Fig. 10. Effect of film radius (Rf) on the K232 values of foam films at a) 1 × 10− 7 M SDS and 0.3 M NaCl; b) 1 × 10− 5 M SDS and 0.3 M NaCl; c) 1 × 10− 7 M PPG and 0.1 M NaCl; d) 1 × 10− 4 M PPG and 0.1 M NaCl. The straight lines were drawn to guide the eye. The filled symbols represent the K232 values obtained by using the Reynolds lubrication approximation (Eq. (9)), and the open symbols represent the K232 values obtained by applying the capillary wave theory (Eqs. (A1–A4)).

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

a) SDS 5×10-7 M + NaCl 0.3M

100

Rf =0.063mm

K 232 = 4.0x10

K 232 =2.7×10

-19

J (h s =0)

K 232 =2.3×10

-19

J (h s =4.68 nm)

H (nm)

H (nm)

K 232 = 0

80

K 232 =0 ( h s =4.68 nm) 60

-7

SDS 1×10 M + NaCl 0.3M R f =76 µm

Experimental K 232 =0 ( h s =0)

80

a)

100

65

-20

J

Experimental

60

40

40 20 20

5

0

10

15

20

25

0

30

1

0

2

3

4

t (s)

b) SDS 5×10-7 M + NaCl 0.3M

100

Experimental K 232 =0 ( h s =0)

80

7

8

9

10

-7

SDS 1×10 M + NaCl 0.3M

Experimental

K 232 =0 ( h s =4.68 nm) 60

6

R f =28 µm

80

K 232 =7.0×10

-20

K 232 =5.4×10

-20

-20

H (nm)

H (nm)

b)

100

Rf =0.028mm

5

t (s)

J (h s =0) J (h s =4.68 nm)

K 232 = 3.0x10 60

J

K 232 = 0 Eq.(9) R f < R f,Re

40 40 20 20

0

2

4

6

Eq.(14) R f > R f,Re

8

0

t (s)

0

1

2

3

4

t (s) Fig. 11. Kinetics of film thinning for 5 × 10− 7 M SDS with 0.3 M NaCl at film radius of a) 63 μm and b) 28 μm. The solid line represents Eq. (9) with the extended DLVO theory, while the dotted line represents the equation with the DLVO theory. The red dashdotted line represents Eq. (11) with the extended DLVO theory and surface mobility (hs ≠ 0), while the green dashed line represents the equation with the DLVO theory with surface mobility.

extended DLVO theory (K232 = 2.3 × 10 − 19 and 5.43 × 10 − 20 J at Rf = 63 and 28 μm, respectively) and surface mobility (hs = 4.68 nm), while the green dashed lines represent Eq. (11) with the DLVO theory (K232 = 0) with the same hs value. It was found that with considering the surface mobility parameter, the fitted value of K232 was slightly smaller than without considering the -17

10

no surface mobility Wang and Yoon, 2005 with surface mobility -18

K232 (J)

10

-19

10

A 232

-20

10

0.3 M NaCl

-21

10

-7

10

-6

10

-5

10

-4

10

-3

10

-2

10

SDS (M) Fig. 12. Comparison of K232 values calculated from Eq. (11) without considering surface mobility (Wang and Yoon, 2005) and with considering surface mobility parameter (hs). The radii of the various films were in the range of 55 to 80 μm.

Fig. 13. Kinetics of film thinning for 1 × 10− 7 M SDS with 0.3 M NaCl at film radius of a) 76 μm and b) 28 μm. In sub-figure a), the solid line represents Eq. (14) with the extended DLVO theory, while the dashed lines represent the equation with the DLVO theory. In sub-figure b), the solid line represents Eq. (9) at the early drainage stage and Eq. (14) at the later drainage stage with the extended DLVO theory, while the dashed line represents the equations with the DLVO theory.

mobility, but the K232 value is still much larger than the A232 value (the values of A232 at the final drainage stage of these SDS films fell in the range of 1.6–1.8 × 10 − 20 J). The K232 values at the SDS concentration range of 10− 7 − 5× 10− 5 M with 0.3 M NaCl have been plotted in Fig. 12. Also shown are the K232 values calculated using Eq. (11) with hs =4.68 nm for 5 ×10− 7 M SDS, hs = 1.27 nm for 1× 10− 6 M SDS and hs =0.586 nm for 1× 10− 5 M SDS. It was found that the K232 values obtained at 5 ×10− 7 and 1× 10− 6 M with considering surface mobility parameter (hs) were slightly smaller than those obtained without considering surface mobility parameter, while the K232 values obtained at 1 ×10− 5 M SDS with and without considering surface mobility parameter were coincident. It appears, therefore, that hs may be neglected and the foam film surfaces may be regarded as tangentially immobile, which corroborates with the treatment of Angarska et al. (2004). In other words, the considerable overestimation of the K232 values obtained from the drainage curves of larger films (see Fig. 10) cannot be attributed to surface mobility. Manev et al. (1997) proposed that for relatively large films the acceleration of film thinning which cannot be accounted for by the Reynolds lubrication approximation may be due to hydrodynamic corrugation. They noted that the Reynolds lubrication approximation (Eq. (9)) is applicable to only planar surfactant films, whose film radii are smaller than a transition radius defined as:

Rf;Re ¼ 4

 1 Hγ 2 : ΔP

ð13Þ

66

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

If film radii are larger than this transition radius, the drainage of the films should follow the MTR equation: dH Rf ¼ VRe ⋅ − dt Rf;Re

¼

1 6μ

12

!6 5

8

H ΔP 4γ3 Rf 4

!1

ð14Þ

5

:

Eqs. (13) and (14) were used to fit the experimental thinning data for the films of different radii at 1 × 10 − 7 M SDS in the presence of 0.3 M NaCl, and results are shown in Fig. 13. We calculated the Rf,Re values without considering the non-DLVO force (i.e., K232 = 0). It was found that Rf,Re changed from 39 to 23 μm when decreasing H from H0 (102 nm) to Hcr (40.1 nm). With considering additional non-DLVO force, the driving force ΔP would be even larger, so at a given H the Rf,Re values would be lower. Therefore, for a film with film radius of 76 μm, being always larger than Rf,Re so Eq. (14) may be used to fit the entire film thinning process until rupture. In Fig. 13a, the dashed line shows the predicted H versus t curve obtained by using Eq. (14) with the DLVO theory. Obviously, there is a discrepancy between the experiment and theory. When using the extended DLVO theory, the fit was much better. The solid line represents MTR equation with fitted K232 = 4.0 × 10 − 20 J. Eq. (14) was also used to fit the experimental thinning curve of a very small film, whose radii was 28 μm. The calculated Rf,Re changed from 39 to 17 μm throughout the film thinning until rupture. This means that Eq. (14) is applicable only at the later drainage stage, while at the initial stage the Reynolds lubrication approximation should be used. In Fig. 13b, the dotted line represents the predicted thinning curve obtained by using Eq. (9) for H = 102 − 54 nm and by using Eq. (14) for H = 54 − 29.1 nm, coupled with the DLVO theory. Again, there is a discrepancy between the experiment and theory. When using the extended DLVO theory, the fit was better. The solid line represents Eq. (9) (H N 58.7 nm) and Eq. (14) (H b 58.7 nm), with fitted K232 = 3.0 × 10 − 20 J. By comparing Fig. 13a and b, the K232 values obtained with the MTR equation appear to be less sensitive to Rf than with the Reynolds lubrication approximation. The results presented in Fig. 13 suggest that MTR equation can give K232 values at least ten times lower than the Reynolds lubrication approximation, or two to three times lower than the capillary wave theory. Despite the discrepancies present among the K232 values obtained from different approaches, one can see that K232 values at very low surfactant concentrations are positive and larger than the A232 values, indicating the presence of additional attraction forces which are stronger than the van der Waals force. The results presented hitherto strongly support that the overall inter-bubble attractions (represented by Eq. (7)) determined from our experimental work were considerably larger than the van der Waals force predicted by the Lifshitz–Hamaker theory. To provide a single, exclusive explanation of this discrepancy is a formidable task and is out of the scope of the present communication. However, some main arguments may be summarized as follows: i) A non-DLVO attraction force is present in foam films at low surfactant concentrations. Hydrophobic interactions appear to universally exist between hydrophobic surfaces in aqueous media (Israelachvili and Pashley, 1982; Pashley et al., 1985; Rabinovich and Yoon, 1994; Christenson and Claesson, 2001; Meyer et al., 2005). The rupture of thin wetting films confined between hydrophobized solid surface and air bubble is induced by the long-range hydrophobic influence (Laskowski and Kitchener, 1969) or by the hydrophobic force (Blake and Kitchener, 1972). There is accumulating evidence that air bubbles in water are hydrophobic (Israelachvili, 1992; Ducker et al., 1994; Du et al.,

1994; Yoon and Pazhianur, 1998; van Oss et al., 2005). However, whether or not a new force (referred to as hydrophobic force) exists in foam films and its origin is still under debate (Stubenrauch et al., 2007; Kralchevsky et al., 2008; Yoon and Wang, 2008). ii) Only the van der Waals force exists in the foam films at very low surfactant concentrations. If this is the case, the Lifshitz– Hamaker constants might have been underestimated so K232 in Eq. (7) may be deemed as correction for the Hamaker constant. Thus, one can use the effective Hamaker constant, A232 + K232. iii) Both an enhanced van der Waals force and a non-DLVO attraction force of different nature exist. Recent molecular simulation of thin free films of pure water conducted by Bhatt et al. (2002) found that the Hamaker constant obtained from their simulations is about one order of magnitude larger than that from classical Hamaker theory which is based on dispersion forces but assumes a slab geometry for the density profile and completely neglects fluid structure and entropy. However, recent work by Wang and Qu (submitted to publication) showed that the inter-bubble attraction in pure water was two orders of magnitude stronger than the Lifshitz–van der Waals force. These arguments and conjectures point to the need for more work to examine how the surface forces in foam films could be affected by other factors which have not been considered in the framework of DLVO theory. For instance, considerable amount of dissolved gas molecules is present in foam films, but their impact on surface forces in thin liquid films is not clear (Ninham, 2006). Our knowledge on how surfactant adsorption affects surface forces in foam films is also limited. 5. Conclusions The thin film pressure balance (TFPB) technique was used to measure the thickness of very small foam films stabilized with sodium dodecyl sulfate and polypropylene glycol at high NaCl concentrations. The film radii were controlled primarily in the range of 28 to 33 μm. The Stefan–Reynolds lubrication approximation was used to estimate the contribution of non-DLVO force to the disjoining pressure from measured film thinning rates while the capillary wave model of Valkovska, Danov and Ivanov was used to calculate the hydrophobic force from measured critical rupture thicknesses. The non-DLVO force was expressed by a power law in the same form as the van der Waals force, which makes it possible to directly compare these two forces by means of the force constant (K232) and the Hamaker constant. Non-DLVO forces of comparable magnitudes were detected both from film drainage and from rupture thickness analyses. And the forces were considerably stronger than the Lifshitz–van der Waals forces. At low surfactant concentrate range (1 × 10 − 7 − 1 × 10 − 5 M SDS and 1 × 10 − 7 − 7.5 × 10 − 4 M PPG), both models found a monotonic decrease in K232 with increasing surfactant concentration. The effect of film radii and film surface mobility on the fitted K232 values was also studied. The K232 values obtained from the capillary wave theory were indifferent to both film radii and film surface mobility, in contrast to the Stefan–Reynolds lubrication approximation. The K232 values obtained from the Stefan–Reynolds lubrication approximation increased with increasing film radii. For relatively large films (with radii over 70 μm), the film drainage model gave K232 values one order of magnitude larger than those obtained from the capillary wave model. Other film drainage models such as the Radoev–Dimitrov–Ivanov model coupled with film surface mobility and Manev–Tsekov–Radoev model incorporating hydrodynamic corrugation gave lower K232 values. The assumptions of immobile and mobile surfaces found 10–20% changes in K232 obtained from the film drainage model of Radoev,

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

Dimitrov and Ivanov. The Manev–Tskov–Radoev model applied to foam films at very low surfactant concentrations gave K232 values several times lower than the Stefan–Reynolds lubrication approximation. Despite the discrepancies among the fitted K232 values obtained from these film drainage models and the capillary wave model, the K232 values at very low surfactant concentrations were larger than the Hamaker constants, indicating the inter-bubble attractions in flotation system are much stronger than the van der Waals force as predicted by the Lifshitz–Hamaker theory.

67

approach 2 for mobile interfaces. Thus, we used the following numerical procedure to determine K232 (Valkovska et al., 2002): Step 1′. Solve the following equation F ðHst Þ ¼ 1−

2 2 Hst Rf Π′ ðHst Þ ¼0 32γ½Pc −ΠðHst Þ

ðA5Þ

to determine Hst. Step 2′. Choose a tentative value of Hcr which is in between 0 and Hst. Then choose a tentative value of Htr which is in between Hcr and Hst. Solve Eq. (A2) to determine the dimensionless wave number kcr. The surface mobility has little effect on kcr

Appendix A Immobile surfaces By assuming that MIBC and PPG films are tangentially mobile, we started with a tentative value of K232 and went through the following steps (Valkovska et al., 2002; Angarska et al., 2004) to determine the value of K232:

Step 3′. Solve the following equation

GðHtr Þ ¼ 1 þ

h i 4 3 2 3 Htr γHtr kcr −2R2f Hcr Π′ ðHtr Þ

Step 1. Solve the following equation

6 2 Rf ½Pc −ΠðHtr Þ 32Hcr

¼0

ðA6Þ

to determine Htr.

2 2 H R Π′ ðHst Þ ¼0 F ðHst Þ ¼ 1− st f 48γ½Pc −ΠðHst Þ

ðA1Þ

to determine Hst, which is an upper limit for the critical rupture thickness Hcr. Note that Π′(H) ≡ ∂Π/∂H. Step 2. Choose a tentative value of Hcr which is in between 0 and Hst. Then choose a tentative value of Htr which is in between Hcr and Hst. Solve the following equation 3 ′ k2cr γ Htr H 6 H H Π dH ¼ ∫Htr dH ∫H 2 3 cr P −Π Rf Hcr cr Pc −Π c

ðA2Þ

Step 3. Solve the following equation

GðHtr Þ ¼ 1 þ

h

6 2 Rf ½Pc −ΠðHtr Þ 48Hcr

¼0

ðA3Þ

to determine Htr. Step 4. By using the values of kcr and Htr determined from Eqs. (A2) and (A3), respectively, we solved the following equation Y ðHcr Þ ¼ 1−

Htr Hcr

2 γHtr kB T

!1=4 exp −

2 γHtr kB T

!1=2

3 ′ k2 h H Π dH exp − cr 3 ∫htr 16Hcr cr Pc −Π

! ¼0

ðA7Þ

to determine Hcr. Step 5′. The Hcr value was then used as the tentative value of Hcr in Step 2′, and repeat Steps 2′–4′ and the iteration will end if convergence on Hcr is achieved.

Acknowledgment

i

3 2 3 γHtr kcr −2R2f Hcr Π′ ðHtr Þ

H Y ðHcr Þ ¼ 1− tr Hcr

Step 6′. Adjust K232 value and repeat Steps 1′–5′ to allow the calculated Hcr value to be approximately equal to the experimental Hcr value, with the difference being less than 0.1 nm.

to determine the dimensionless wave number kcr.

4 Htr

Step 4′. By using the values of kcr and Htr determined from Eqs. (A2) and (A6), respectively, we solved the following equation

3 ′ k2cr h H Π dH ∫htr 3 32Hcr cr Pc −Π

Financial support for this study, provided by a Discovery Grant from the Australian Research Council, is gratefully acknowledged. The author thanks Xuan Qu and Karri Berg for experimental assistance, Prof. Peter Kralchevsky and Prof. Krassimir D. Danov for confirming Eqs. (A5–A7) on the capillary wave model, and Prof. Anh V. Nguyen for giving permission to use the surface tension meter in his laboratory.

! ¼0

ðA4Þ

to determine Hcr. Step 5. This value was then used as the tentative value of Hcr in Step 2, and repeat Steps 2–4 until the iteration results in convergence of Hcr. Step 6. Adjust K232 value and repeat Steps 1–5 to allow the calculated Hcr value to approach the experimental Hcr value, with the difference being less than 0.1 nm. Fully mobile surfaces We also examined the effect of surface mobility on Hcr predictions. If one assumes that the MIBC and PPG films are fully tangentially mobile with a mobility parameter d being infinitely large, so the term (3Hst + 2d)/(Hst + d) in the work of Valkovska et al., 2002 would

References Angarska, J.K., Dimitrova, B.S., Danov, K.D., Kralchevsky, P.A., Ananthapadmanabhan, K.P., Lips, A., 2004. Detection of the hydrophobic surface force in foam films by measurements of the critical thickness of the film rupture. Langmuir 20, 1799–1806. Ata, S., Ahmed, N., Jameson, G.J., 2003. A study of bubble coalescence in flotation froths. Int. J. Miner. Process. 72, 255–266. Bergeron, V., 1999. Forces and structure in thin liquid soap films. J. Phys. Condens. Matter 11, R215–R238. Bhatt, D., Newman, J., Radke, C.J., 2002. Molecular simulation of disjoining-pressure isotherms for free liquid, Lennard–Jones thin films. J. Phys. Chem. B 106, 6529–6537. Blake, T.D., Kitchener, J.A., 1972. Stability of aqueous films on hydrophobic methylated silica. J. Chem. Soc., Faraday Trans. 68, 1435–1442. Castro, S., Venegas, I., Landero, A., Laskowski, J.S., 2010. Frothing in seawater flotation systems. Proceedings of XXV International Mineral Processing Congress, Brisbane, Australia, pp. 4039–4047. Christenson, H.K., Claesson, P.M., 2001. Direct measurements of the force between hydrophobic surfaces in water. Adv. Colloid Interface Sci. 91, 391–436. Churaev, N.V., 2005. Aqueous wetting films in contact with a solid phase. Adv. Colloid Interface Sci. 114–115, 3. Cillers, J., 2006. Understanding froth behaviour with CFD. Fifth International Conference on CFD in the Process Industries. CSIRO, Melbourne, Australia (13–15 December 2006).

68

L. Wang / International Journal of Mineral Processing 102–103 (2012) 58–68

Coons, J.E., Halley, P.J., McGlashan, S.A., Tran-Cong, T., 2005. Bounding film drainage in common thin films. Colloids Surf., A 263, 197–204. Du, Q., Freysz, E., Shen, Y.R., 1994. Surface vibrational spectroscopic studies of hydrogen bonding and hydrophobicity. Science 264, 826–828. Ducker, W.A., Xu, Z., Israelachvili, J.N., 1994. Measurements of hydrophobic and DLVO forces in bubble–surface interactions in aqueous solutions. Langmuir 10, 3279–3289. Exerowa, D., Kruglyakov, P.M., 1998. Foam and Foam Films. Elsevier. Exerowa, D., Kolarov, T., Kristov, K.H.R., 1987. Direct measurement of disjoining pressure in black foam films. I. Films from an ionic surfactant. Colloids Surf. 22, 171–185. Israelachvili, J.N., 1992. Intermolecular and Surface Forces. Academic press, London. Israelachvili, J.N., Pashley, R., 1982. The hydrophobic interaction is long range, decaying exponentially with distance. Nature 300, 341–342. Ivanov, I.B., Danov, K.D., Ananthapadmanabhan, K.P., Lips, A., 2005. Interfacial rheology of adsorbed layers with surface reaction: on the origin of the dilatational surface viscosity. Adv. Colloid Interface Sci. 114–115, 61–92. Karakashev, S.I., Ivanova, D.S., Angarska, Z.K., Manev, E.D., Tsekov, R., Radoev, B., Slavchov, R., Nguyen, A.V., 2010. Comparative validation of the analytical models for the Marangoni effect on foam film drainage. Colloids Surf., A 365, 122–136. Klassen, V.I., Mokrousov, V.A., 1963. An Introduction to the Theory of Flotation. Butterworths, London. Klimpel, R.R., 1995. The influence of frother structure on industrial coal flotation. In: Kawatra (Ed.), High-Efficiency Coal Preparation. Society for Mining, Metallurgy, and Exploration, Littleton, CO, pp. 141–151. Kralchevsky, Peter A., Danov, Krassimir D., Angarska, Jana K., 2008. Reply to comment on “Hydrophobic Forces in the Foam Films Stabilized by Sodium Dodecyl Sulfate: Effect of Electrolyte” and subsequent criticism. Langmuir 24, 2953-2953. Langevin, D., 2000. Influence of interfacial rheology on foam and emulsion properties. Adv. Colloid Interface Sci. 88, 209–222. Laskowski, J.S., Kitchener, J.A., 1969. The hydrophilic–hydrophobic transition on silica. J. Colloid Interface Sci. 29, 670–679. Lyklema, J., Mysels, K.J., 1965. A study of double layer repulsion and van der Waals attraction in soap films. J. Am. Chem. Soc. 87, 2539–2546. Manev, E.D., Sazdanova, S.V., Wasan, O.T., 1984. Emulsion and foam stability — the effect of film size on film drainage. J. Colloid Interface Sci. 97, 591–594. Manev, E., Tsekov, R., Radoev, B., 1997. Effect of thickness non-homogeneity on the kinetic behaviour of microscopic foam films. J. Dispersion Sci. Technol. 18, 769–788. Mathe, Z.T., Harris, M.C., O'Connor, C.T., Franzidis, J.-P., 1998. Review of froth modelling in steady state flotation systems. Miner. Eng. 11, 397–421. Meyer, E.E., Lin, Q., Israelachvili, J.N., 2005. Effects of dissolved gas on the hydrophobic attraction between surfactant-coated surfaces. Langmuir 21, 256–259. Nagaraj, D.R., Ravishankar, S.A., 2007. Flotation reagents — a critical overview from an industry perspective. In: Fuerstenau, M.C., Jameson, G., Yoon, R.-H. (Eds.), Froth Flotation—A Century of Innovation. Society for Mining, Metallurgy, and Exploration, Inc, p. 405. Neethling, S.J., Cilliers, J.J., 2003. Modelling flotation froths. Int. J. Miner. Process. 72, 267–287. Ninham, B.W., 2006. The present state of molecular forces. Prog. Colloid Polym. Sci 133, 65–73. Pashley, R.M., McGuiggan, P.M., Ninham, B.W., Evans, D.F., 1985. Attractive forces between uncharged hydrophobic surfaces: direct measurements in aqueous solution. Science 229, 1088–1089. Pugh, R.J., 2007. The physical and chemistry of frothers. In: Fuerstenau, M.C., Jameson, G., Yoon, R.-H. (Eds.), Froth Flotation—A Century of Innovation. Society for Mining, Metallurgy, and Exploration, Inc., p. 259.

Rabinovich, Y.I., Derjaguin, B.V., 1988. Proceed. 5th Hungarian Conference on Colloid Chemistry, Loránd Eötvös University, Budapest, Hungary. Rabinovich, Y.I., Yoon, R.-H., 1994. Use of atomic force microscope for the measurements of hydrophobic forces between silanated silica plate and glass sphere. Langmuir 10, 1903–1909. Radoev, B.P., Dimitrov, D.S., Ivanov, I.B., 1974. Hydrodynamics of thin liquid films effect of the surfactant on the rate of thinning. Colloid Polym. Sci. 252, 50–55. Reynolds, O., 1886. On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil. Philos. Trans. R. Soc. London 177, 157–234. Russel, W.B., Saville, D.A., Schowalter, W.R., 1989. Colloidal Dispersions. Cambridge University Press, Cambridge. Scheludko, A., 1967. Thin liquid films. Adv. Colloid Interface Sci. 1, 391–464. Scheludko, A., Exerowa, D., 1959. Comm. Dept. Chem., Bul. Acad. Sci. 7, 123. Scheludko, A., Platikanov, D., 1961. Kolloid Zh. 175, 150–158. Stefan, J., 1874. Akad. Wiss. Wien Math.-Naturw. Kl. Abt. II 69, 713. Stubenrauch, C., Langevin, D., Exerowa, D., Manev, E., Claesson, P.M., Boinovich, L.B., Klitzing, R.v., 2007. Comment on “Hydrophobic Forces in the Foam Films Stabilized by Sodium Dodecyl Sulfate: Effect of Electrolyte". Langmuir 23, 12457–12460. Tabakova, S.S., Danov, K.D., 2009. Effect of disjoining pressure on the drainage and relaxation dynamics of liquid films with mobile interfaces. J. Colloid Interface Sci. 336, 273–284. Tchaliovska, S., Manev, E., Radoev, B., Eriksson, J.C., Claesson, P.M., 1994. Interactions in equilibrium free films of aqueous dodecylammonium chloride solutions. J. Colloid Interface Sci. 168, 190–197. Valkovska, D.S., Danov, K.D., Ivanov, I.B., 2002. Stability of draining plane-parallel films containing surfactants. Adv. Colloid Interface Sci. 96, 101–129. van Oss, C.J., Giese, R.F., Docoslis, A., 2005. Hyperhydrophobicity of the water–air interface. J. Dispersion Sci. Technol. 26, 585–590. Wang, L., 2006. Surface Forces in Foam Films. PhD Thesis, Virginia Polytechnic Institute and State University. Appendix 3.B. Wang, L. and Qu, X., submitted to publication. Stability of thin free water films formed at various approach speeds of air–water interfaces. J. Colloid Interface Sci. Wang, L., Yoon, R.-H., 2004. Hydrophobic forces in the foam films stabilized by sodium dodecyl sulfate: effect of electrolyte. Langmuir 20, 11457–11464. Wang, L., Yoon, R.-H., 2005. Hydrophobic forces in thin aqueous films and their role in film thinning. Colloids Surf., A 263, 267–274. Wang, L., Yoon, R.-H., 2008. Effects of surface forces and film elasticity on foam stability. Int. J. Miner. Process. 85, 101–110. Weber, E.H., 1834. De pulsu, resorptione, audita et tactu, Annotationes anatomicae et physiologicae. Koehler, Leipzig. Wolfram Mathematica Online Integrator, 1996. http://integrals.wolfram.com/index.jsp (accessed 18 March 2010). Yarar, Baki, Dogan, Z.M., 1987. Mineral Processing Design, p. 91. Yoon, R.-H., Aksoy, B.S., 1999. Hydrophobic forces in thin water films stabilized by dodecylammonium chloride. J. Colloid Interface Sci. 211, 1–10. Yoon, R.-H., Pazhianur, R., 1998. Direct force measurement between hydrophobic glass sphere and covellite electrode in potassium ethyl xanthate solutions at pH 9.2. Colloids and Surfaces A Physicochem. Colloids Surf., A 144, 59–69. Yoon, R.-H., Wang, L., 2008. A response to the comment on “Hydrophobic Forces in the Foam Films Stabilized by Sodium Dodecyl Sulfate: Effect of Electrolyte”. Langmuir 24, 5194–5196.