The StefanâReynolds model and the modified StefanâReynolds

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The Stefan–Reynolds model and the modified Stefan–Reynolds model for studying bubble–particle attachment interactions in the context of flotation You Zhou, Boris Albijanic, Bogale Tadesse, Yuling Wang, Jianguo Yang, and Xiang-nan Zhu Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.9b00397 • Publication Date (Web): 20 Feb 2019 Downloaded from http://pubs.acs.org on February 25, 2019

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The Stefan–Reynolds model and the modified Stefan–Reynolds model for studying bubble– particle attachment interactions in the context of flotation You Zhoua,b, Boris Albijanicb,*, Bogale Tadesseb, Yuling Wanga, Jianguo Yanga, Xiangnan Zhuc

aKey

Laboratory of Coal Processing and Efficient Utilization of Ministry of Education, School of Chemical Engineering and Technology, China University of Mining and Technology, 1 University Road, Xuzhou 221116, Jiangsu, China bWestern

Australian School of Mines: Minerals, Energy and Chemical Engineering, Curtin University, 95 Egan Street, Kalgoorlie, WA 6430, Australia cCollege

of Chemical and Environmental Engineering, Shandong University of Science and Technology, 579 Qianwangang Road, Qingdao, Shandong 266590, China

*Corresponding Author’s E-mail: [email protected] ABSTRACT Bubble–particle attachment is the key step for successful flotation. Modelling of attachment interactions between air bubbles and particles after their collision can be analysed using the Stefan–Reynolds model (immobile bubble surfaces) and the modified Stefan-Reynolds model (mobile bubble surfaces). However, these models have been rarely used, and the limitations of these models have not yet been reported. The objective of this paper is to address this matter under a wide range of experimental flotation conditions. It was found that the Stefan–Reynolds model can be used to determine the real bubble–particle hydrophobic constants at low surfactant concentration. However, at high surfactant concentration, the real bubble– particle hydrophobic constants cannot be determined but the fictive bubble–particle hydrophobic constants can be obtained by using the linear extrapolation method. The same analysis was also performed using the modified Stefan–Reynolds model. The results showed that the attachment of quartz particles to air bubbles in the presence of DAH is accelerated due to the mobility of the air-water interface. This paper

demonstrated that the limitations of the Stefan–Reynolds model and the modified Stefan-Reynolds model to analyse the bubble–particle attachment interactions can be addressed by introducing the fictive bubble– particle hydrophobic constants. Key words: Flotation theory, induction time, modelling, hydrophobic constants, critcal film thickness quartz INTRODUCTION 1

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Flotation has been widely used for the upgrading of fine mineral particles and wastewater treatment [13]. Successful flotation is controlled by bubble–particle attachment interactions which may be quantified using attachment time measurements. The attachment time represents the time for the liquid film drainage, the rupture of the liquid film, and the expansion of the gas-liquid-solid contact lines [4]. Attachment times between bubbles and particles were first determined using the Glembotsky device [5]. However, it was found that using a high speed camera for the attachment time, measured by the Glembotsky device, can be approximated as the liquid film drainage time, i.e. the induction time [6]. Various factors may affect induction times such as mineral liberation [7, 8], bubble approach velocity [9], flotation collector concentration [10, 11], particle size [12] and etc. For example, the lower the degree of mineral liberation, the higher the induction time. Although the experimental determination of induction times have been routinely performed, modelling of induction times remains a challenge due to the complex interactions between air bubbles and particles. Given that the Glembotsky device is used for induction time measurements, drainage of liquid films occur due to collision interactions between air bubbles and particles, and therefore the Stefan–Reynolds model can be used. The Stefan–Reynolds model has been experimentally validated for hydrophilic particles but not for hydrophobic particles [13] because the contribution of hydrophobic force was not considered. Bubble–particle hydrophobic interactions may be quantified as the bubble–particle hydrophobic constant [14]. Determination of hydrophobic force constants between air bubbles and particles is challenging in spite of using atomic force microscopy which is an advanced experimental technique [13]. The main reason is due to non-spherical particles, making atomic force microscopy measurements highly irreproducible. Certainly, atomic force microscopy can be used to study forces between a bubble and a spherical particle [15]. However, in the case of a non-spherical particle, atomic force microscopy measurements have to be conducted by changing the position of the non-spherical particle because different positions of the nonspherical particle may affect liquid film drainage process; it should be noted that the non-spherical particle has to be glued to the tip of the cantilever. This is a very complicated experimental task and thus there is no study investigating the influence of particle shape on forces between a particle and a bubble. Bubble–particle hydrophobic constants can also be determined as the geometric mean of the hydrophobic constant for particle–particle interactions and the hydrophobic constant for bubble–bubble interactions [16]. In this method, the atomic force microscopy measurements were used to determine the hydrophobic constants for particle–particle interactions by studying the forces acting between a flat surface and a sphere 2


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attached to the tip of the cantilever. The Scheludko technique [17] was used to determine the hydrophobic constants for bubble–bubble interactions. Considering the challenges in studying attachment interactions using a non-spherical particle and an air bubble, one of the possibility for determination of bubble–particle hydrophobic constants is by using the Stefan–Reynolds model and the induction times obtained using the Glembotsky device. It is very important to highlight that the Glembotsky device can be used to experimentally determine the induction time for non-spherical particles [8, 10]. The literature provides limited information about the determination of hydrophobic constants between air bubbles and particles as well as critical film thicknesses (i.e. the liquid film thickness when the liquid film ruptures). For example, Albijanic et al [9] found that the lower the bubble approach velocities, the lower the hydrophobic constants. However, there is no study investigating the application of this methodology under a wide range of flotation conditions. This is very important considering that the limitations of the Stefan–Reynolds model (immobile bubble surfaces) and the modified Stefan Stefan–Reynolds model (mobile bubble surfaces) to determine hydrophobic constants have not yet been reported. Therefore, the objective of this study is to address this issue. The knowledge about hydrophobic constants is essential not only to better understand the liquid film drainage mechanism between air bubbles and particles during flotation but also to predict flotation rate constants, and thus mineral recovery by flotation [14].

THEORY The Stefan–Reynolds equation describes a liquid film drainage process between two immobile surfaces, [13]: dh dt

=-

2h3 3μR2

(Pσ - Π)

(1)

while a liquid film drainage process between one immobile surface (i.e. a particle) and one mobile surface (i.e. an air bubble) is given by the modified Stefan-Reynolds model [4]. dh dt

=-

8h3 3μR2

(Pσ - Π)

(2)

Where h is the wetting film thickness, t is time of film thinning, μ is the liquid viscosity, P  2 / R p is the capillary pressure (σ is the surface tension and Rp is the particle size), R is the film radius which can be approximated as Rp [18]; Π is the total disjoining pressure.

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The disjoining pressures and the energies for the electrostatic double layer (EDL), van der Waals, and hydrophobic contribution are given in Table 1. The sum of the energies, shown in Table 1, represents the total energy. Similarly, the sum of the disjoining pressures, given in Table 1, denotes the total disjoining pressure. Table 1. Energies and disjoining pressures [4] Type of interactions

Π (Pa)

 edl  Electrostatic double layer

 0 k 2 1 2 cosh(kh)   2  0 Rb R p  4 1 2 arctan h(e kh )  ( 12  22 ) ln(1  e2 kh )  Eedl  2 2 sinh (kh) Rb  R p 2

E (J) 2 1

2

Van der Waals

 vdw  

A132 6 h3

Evdw  

Hydrophobic

 hyd  

K132 6 h3

Ehyd  

Rb R p A132 6( Rb  R p ) h

Rb R p K132 6( Rb  R p )h

Where Rb is the bubble radius and A132 is the bubble–particle Hamaker constant. It should be noted that the Hamaker constant will be neglected since it is very small i.e. around 1×10-20 J.

1

and  2 are the

surface potential of a particle and a bubble, respectively; K132 represents the bubble–particle hydrophobic constant; 0 is the vacuum permittivity (8.854×10-12 C2J-1m-1);  is the dielectric constant (81 for water);

κ-1 is the Debye length which can be defined by the following equation [19, 20]: κ ―1 =

(

0𝑘𝑇 2

2𝑒 𝑁𝐴𝑐𝑧

)

0.5

(3)

2

Where e is the electron charge (1.602× 10−19 C), k is the Boltzmann constant (1.381× 10−23J K−1), NA is the Avogadro number (6.023×1023 mol−1), z is the valence of the ion, c is the concentration (mol m−3), T is the temperature (K). EXPERIMENTAL METHOD AND MATERIALS Materials. The quartz samples (Western Australia, Kalgoorlie) were dry-ground in a laboratory rod mill to obtain two size fractions: -300+212 µm and -105+53 µm for the induction time measurements. Distilled water was used in all experiments. Dodecyl amine hydrochloride (DAH) with 97% purity was purchased from Alfa Aesar. All experiments were conducted at 22±1°C. Induction time measurements. The induction time experiments were conducted using an Induction Timer (University of Alberta, Canada) [21,22]. It should the noted that the Indution timer determines the 4


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attachment time (i.e the sum of the liquid film drainge time, the liquid film rupture time and the time required for the three-phase contact line to expand from the critical radius to a stable wetting perimeter) rather than the induction time (i.e. the liquid film drainge time); however, Wang et al [6] used a high speed camera and found that the attachment time was very close to the induction time. The conditioned quartz particles together with the solution were transfered in a small cell under the bubble holder. The induction time apparatus is shown in Fig. 1. Precisely, the bubble was kept in contact with the bed of particles for a defined controlled contact time; the bubble size was 1.5 mm and the bubble approach velosities were 2 mm/s and 50 mm/s. After that, the camera was used to visually observe whether quartz particles were attached onto the bubble. Ten measurements were repeated for a given controlled contact time. The induction time representes the controlled contact time at which the attachment efficiency was 100%; the reason is that 50% attachment was not a suitable option for the measurements in this work because for some systems (eg. CDAH=10-6 mol/L, particle size=-105+53 µm, bubble approach velocity= 50 mm/s) 100% attachment was achieved at the lowest contact time.

Fig. 1. A schematic representation of induction timer [23]. RESULTS AND DISCUSSION Induction time results. Fig. 2 shows the induction time experiments conducted at different DAH concentrations, bubble approach velocities, and particle sizes. For example, when the DAH concentration was 10-6 mol/L and the bubble approach velocity was 50 mm/s for -300+212 µm quartz particles, the measured induction time was 22.5±2.5 ms. It should be noted that the bubble approach velocities were controlled with the speaker’s drum via a capillary tube. As seen in Fig. 2, the lowest induction time was obtained mainly at high DAH concentration probably due to the high hydrophobicity of the particles. By 5


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contrast, the longest induction time was obtained at low DAH concentration (10-6 mol/L), large particle size (-0.300+0.212 mm) and low bubble approach velocity (2 mm/s). The reasons are that at low DAH concentration and large particle size, the particles were slightly hydrophobic [15], and it takes a long time for the displacement of the liquid film on the surface of the large particles [12]. Finally, at low bubble approach velocity, the liquid film drainage is slow due to a low pressure of a bubble on the liquid film [9]. The induction time results, shown in Fig. 2, will be used to determine the bubble–particle hydrophobic constants and the critical film thickness values.

(a) 10-6mol/L

(b) 10-3mol/L Fig. 2. Attachment probabilities for different dwell times in the IT, with different nominal bubble approach velocities, for samples with different particle size in a) 10-6 mol/L and b) 10-3 mol/L DAH solution. Methodology for determination of K132. Two theories were used to compute the hydrophobic force 6


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constants: the Yoon’s theory [16] and the Stefan–Reynolds theory [13]. The Yoon’s theory was used to calculate the critical film thickness (hcr) for different K132 values. The Stefan–Reynolds model was employed to determine the change in liquid film thickness with time. In the Yoon’s theory, the total energies of bubble–particle interactions (i.e. Evdw+Eedl+Ehyd) were determined. The energy of EDL interactions was calculated using the equations given in Table 1; the zeta potential measurements are available in Albijanic et al [15]. The energy of vdW interactions was insignificant. In the Stefan–Reynolds theory (Eq. 1), the change in liquid film thickness with time will be determined. The Stefan–Reynolds model can be solved using the Runge-Kutta method of the fourth order (i.e. the numerical method) because the model is a complex non-linear differential equation. The surface tension measurements used in this calculation are available in [24]. Computation of K132 in the presence of low concentration DAH solution. In the Yoon’s theory, the total energy of the hydrophobic interactions depends on K132. The total energy of bubble–particle interactions as a function of K132 is shown in Fig. 3. The critical film thickness and the maximum total energy can be determined based on the maximum of the curve E vs h. It means that the bubble–particle repulsive force increased slightly with decreasing the distance between the bubble and the particle at the beginning, and after that, the bubble-particle repulsive force decreased rapidly. Fig. 3 also shows hcr values as a function of K132. Yoon [16] successfully validated this theory for the DAH-quartz system.

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(a)

(b) Fig. 3. hcr vs K132 at

10-6 mol/L

of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

The Stefan–Reynolds model shows the linkage between h and time at various K132 as seen in Fig. 4; the reason is that K132 values are not known. Given that the induction time is defined as the time required for the liquid film to drain to a critical liquid film thickness, the induction times (tin) were computed when the liquid film thickness is equal as the critical film thickness (hcr) (see Fig. 4). Finally, the linkage between K132 and tin was obtained by using the results given in Figs. 3 and 4.

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(a)

(b) Fig. 4. h vs t at 10-6 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

(a) 9


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(b) Fig. 5. tin vs K132 at 10-6 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces. The measured induction times are used to determine K132 (Fig. 5). For example, when the DAH concentration was 10-6 mol/L and the bubble approach velocity was 50 mm/s for -300+212 µm quartz particles, the measured induction time was 22.5±2.5 ms, and thus the calculated K132 was 6.37×10-18 J and 4.15×10-18 J for immobile bubble surfaces and mobile bubble surfaces, respectively. Considering that K132 values were determined, the relationships between E and h are given in Fig. 6. As seen in Fig. 6, the repulsive forces between particles and bubbles have to be overcome for successful bubble-particle attachment interactions, and thus the flotation recovery of quartz particles at low DAH concentration, i.e. 10-6 mol/L was around 30% [22].

(a) 10


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(b) Fig. 6. E vs h at 10-6 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

Computation of K132 in the presence of high concentration DAH solutions. The Yoon’s theory was also used in the case of a high concentration of DAH solution to obtain the total energy of bubble-particle interactions at different K132 estimates, shown in Fig. 7. When K132 is larger than 1.2×10-19 J, the total energy was always negative, indicating that the energy barriers did not exist i.e. the surface forces between bubbles and particles were always attractive. However, when K132 was less than 1.2×10-19 J, there were both attractive and repulsive forces between a bubble and a particle, and thus the energy barriers can be determined. The relationship between critical film thickness and K132 values is also shown in Fig. 7.

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(a)

(b)

Fig. 7. E vs h and K123 at 10-3 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

The Stefan–Reynolds theory was also used in the case of high DAH concentration to determine the relationship between the liquid film thickness and time as seen in Fig. 8 as well as the relationship between the calculated induction time and K132 values as seen in Fig. 9. However, it is important to highlight that K132 values cannot be determined using the methodology for 10-6 mol/L. The reason is that the calculated induction times were between 1869 and 1876 ms (see Fig. 8) while the measured induction times were between 10 to 80 ms. Therefore, K132 values cannot be determined using the calculated induction time results. In order to address this issue, the fictive hydrophobic force constant, K*132, was obtained using a linear extrapolation as seen in Fig. 10; the red solid line shows the original line based on the calculated results while the red dashed line shows the extended line using the linear extrapolation method. For example, when the DAH concentration was 10-3 mol/L and the bubble approach velocity was 2 mm/s for -300+212 µm quartz particles, the measured induction time was 80 ms, and the calculated K*132 was 2.25×10-17 J and 2.22×10-17 J for immobile bubble surfaces and mobile bubble surfaces. 12


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(a)

(b) Fig. 8. h vs t at 10-3 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

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(b) Fig. 9. tin vs K132 at 10-3 mol/L of DAH for a) immobile surfaces and b) mobile surfaces. Given that K*132 values were calculated, the changes in the total energies with time are given in Fig. 10. As seen in Fig. 10, there were attractive forces between bubbles and particles because the total energies were always negative. Therefore, the flotation recovery of quartz particles at high DAH concentration can reach 80% [22].

(a)

14


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(b) Fig. 10. E vs h at 10-3 mol/L of DAH for a) immobile bubble surfaces and b) mobile bubble surfaces.

Are bubble surfaces mobile or immobile? The results of calculated hydrophobic force constants and critical film thicknesses, obtained for both mobile and immobile bubble surfaces, are shown in Table 2. It should be noted that in the case of mobile bubble surfaces at the DAH concentration and the particle size between -0.105 and 0.0053 µm, it was not possible to determine the hydrophobic constant and the critical film thickness due to the convergence problem of the model. It is known that a liquid film drainage process was four times faster in the case of mobile bubble surfaces than that in the case of immobile bubble surfaces. Therefore, both the hydrophobic constant and the critical film thickness were lower in the case of mobile bubble surfaces than those of immobile bubble surfaces. Yoon and Yordan [25] determined the critical film thickness between air bubble and silica flat plate and found that the critical film thickness in the presence of 10-6 mol/L of DAH was 60 nm while that in the presence of 10-3 mol/L was 80 nm. It means that in this work the air bubble surfaces were mobile considering that the calculated critical film thickness in the case of mobile bubble surface (see Table 2b) was close to the measured critical film thickness. As regards hydrophobic constants, Yoon [16] found that the hydrophobic constant at 10-3 mol/L of DAH was 2.5×10-19 J which was also close to the calculated hydrophobic constant in the case of mobile bubble surfaces (see Table 2b). It appears that the bubble surfaces are mobile rather than immobile. Precisely, when a bubble moves downward through a surfactant solution during induction time measurements, adsorbed surfactant molecules are moved to the back of the bubble causing the surface tension gradient and thus the front bubble surface is mobile [4]. It should be noted that liquid film drainage processes take place between a front bubble surface and a particle during induction time measurements. 15


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Table 2 shows that the hydrophobic constants were higher at 10-3 mol/L DAH than those at 10-6 mol/L DAH. This is in agreement with our induction time results as well as the contact angle results obtained by Albijanic et al [15]. The increase in DAH concentration results in higher critical film thickness, which is in agreement with the critical film thickness results obtained for the silica plate-air bubble system in various DAH solutions [25]; this can be explained due to the increase in hydrophobic attractions between a bubble and a particle at higher DAH concentration. The increase in particle size significantly reduced the critical film thickness probably because the displacement of the liquid film on small particle surfaces was faster than that on large particle surfaces. When the pressure applied by a bubble on a liquid film is high, the liquid film ruptures at higher critical film thickness [9]. Therefore, the critical film thicknesses at high bubble approach velocities were higher than those at low bubble approach velocities. Table 2. Hydrophobic constants and critical film thicknesses under various conditions for a) immobile bubble surfaces and b) mobile bubble surfaces. a) immobile bubble surfaces DAH

Bubble approach velocity

Particle size

K132/K*132

hcr

(mol/L)

(mm/s)

(mm)

(J)

(nm)

-0.300+0.212

(6.08±0.29)×10-18

263.23±7.14

-0.300+0.212

(2.62±0.18)×10-18

174.81±3.97

-0.105+0.053

(1.64±0.01)×10-18

144.15±0.22

-0.105+0.053

(6.94±0.04)×10-19

108.49±1.17

-0.300+0.212

(2.33±0.01)×10-17

285.40±4.57

2

-0.300+0.212

(2.23±0.02)×10-17

225.10±14.07

50

-0.105+0.053

(2.04±0.01)×10-17

150.50±11.91

2

-0.105+0.053

(2.04±0.01)×10-17

150.50±11.91

50 10-6

2 50 2 50

10-3

K*132 represents the fictive value of hydrophobic force constant for 10-3 mol/L DAH solutions.

b) mobile bubble surfaces DAH

Bubble approach velocity

Particle size

K132/K*132

hcr

(mol/L)

(mm/s)

(mm)

(J)

(nm)

-0.300+0.212

(3.69±0.47)×10-18

201.72±12.72

2

-0.300+0.212

(2.68±1.20)×10-18

100.89±3.69

50

-0.105+0.053

-

-

2

-0.105+0.053

-

-

50

-0.300+0.212

(2.30±0.02)×10-17

253.70±12.34

2

-0.300+0.212

(2.20±0.02)×10-17

155.06±12.02

-0.105+0.053

(1.55±0.01)×10-19

90.54±9.86

-0.105+0.053

(1.55±0.01)×10-19

90.54±9.86

50 10-6

10-3

50 2

K*132 represents the fictive value of hydrophobic force constant for 10-3 mol/L DAH solutions.

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CONCLUSIONS This paper investigates the limitations of the Stefan–Reynolds model (immobile bubble surfaces) and the modified Stefan-Reynolds model (mobile bubble surfaces) for studying attachment interactions between air bubbles and particles in the context of flotation. It was found that at low surfactant concentration, the Stefan–Reynolds model and the modified Stefan-Reynolds model can be used to determine the real bubble–particle hydrophobic constants. However, at high surfactant concentration, the fictive bubble– particle hydrophobic constants were determined using the linear extrapolation method because the real bubble–particle hydrophobic constants cannot be determined. A better agreement between the literature and calculated data was achieved using the modified Stefan-Reynolds model, demonstrating that the surface mobility of air bubble play an essential role in bubble–particle attachment interactions. This work demonstrated that the modified Stefan–Reynolds model can be used for better understanding of the bubble–particle attachment mechanism under a wide range of experimental conditions. However, future work is required to determine whether the calculated hydrophobic constants can be used to predict flotation rate constants and thus mineral recovery by flotation. References [1] Lowson, R.T. Aqueous oxidation of pyrite by molecular oxygen. Chem. Rev. 1982, 82, 461-497. [2] Finch, J. A.; Dobby, G. S. Column Flotation., Pergamon Press. Oxford, 1990. [3] Xia, W.; Xie, G.; Peng, Y. Recent advances in beneficiation for low rank coals, Powder Technol. 2015, 277, 206-221. [4] Nguyen, A.; Schulze, H. J. Colloidal science of flotation., CRC Press, 2004. [5] Glembotsky, V. A. The time of attachment of air bubbles to mineral particles in flotation and its measurement. Izv. Akad. Nauk SSSR (OTN). 1953, 11, 1524-1531. [6] Wang, W.; Zhou, Z.; Nandakumar, K.; Masliyah, J. H.; Xu, Z. An induction time model for the attachment of an air bubble to a hydrophobic sphere in aqueous solutions. Int. J Miner. Process. 2005, 75, 69-82. [7] Albijanic, B.; Bradshaw, D. J.; Nguyen, A. V. The relationships between the bubble–particle attachment time, collector dosage and the mineralogy of a copper sulfide ore. Miner Eng. 2012, 36, 309-313. [8] Albijanic, B.; Amini, E.; Wightman, E.; Ozdemir, O.; Nguyen, A. V.; Bradshaw, D. J. A relationship between the bubble– particle attachment time and the mineralogy of a copper–sulphide ore. Miner. Eng. 2011, 24, 1335-1339. [9] Albijanic, B.; Zhou, Y.; Tadesse, B.; Dyer, L.; Xu, G.; Yang, X. Influence of bubble approach velocity on liquid film drainage between a bubble and a spherical particle. Powder Technol. 2018, 338, 140-144. [10] Subasinghe, G. N.; Albijanic, B. Influence of the propagation of three phase contact line on flotation recovery. Miner. Eng. 2014, 57, 43-49. [11] Albijanic, B.; Subasinghe, G. N.; Bradshaw, D. J.; Nguyen, A. V. Influence of liberation on bubble–particle attachment time in flotation. Miner. Eng. 2015, 74, 156-162. [12] Yoon, R.; Yordan, J. L. Induction time measurements for the quartz—amine flotation system, J Colloid Interf Sci. 1991, 141, 374-383. [13] Schulze, H. J.; Stöckelhuber, K. W.; Wenger, A. The influence of acting forces on the rupture mechanism of wetting films—nucleation or capillary waves, Colloids and Surfaces A: Physicochemical and Engineering Aspects. 2001, 192, 61-72. 17


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[14] Yoon, R.; Mao, L. Application of extended DLVO theory, IV: derivation of flotation rate equation from first principles, J Colloid Interf. Sci. 1996, 181, 613-626. [15] Albijanic, B.; Ozdemir, O.; Hampton, M. A.; Nguyen, P. T.; Nguyen, A. V.; Bradshaw, D. Fundamental aspects of bubble–particle attachment mechanism in flotation separation. Miner Eng. 2014, 65, 187-195. [16] Yoon, R.H. The role of hydrodynamic and surface forces in bubble–particle interaction. Int. J. Miner. Process. 2000, 58, 129-143. [17] Yoon, R. H.; Aksoy, B.S. Hydrophobic forces in thin water films stabilized by dodecylammonium chloride. J. Colloids Interface Sci. 1999, 211, 1–10. [18] Attard, P.; Miklavcic, S. J. Effective spring constant of bubbles and droplets, Langmuir. 2001, 17, 8217-8223. [19] Li, D.; Yin, W.; Liu, Q.; Cao, S.; Sun, Q.; Zhao, C.; Yao, J. Interactions between fine and coarse hematite particles in aqueous suspension and their implications for flotation. Miner Eng. 2017, 114, 74-81. [20] Nguyen, K. T.; Nguyen, A. V.; Evans, G. M. Interfacial Water Structure at Surfactant Concentrations below and above the Critical Micelle Concentration as Revealed by Sum Frequency Generation Vibrational Spectroscopy. J. Phys. Chem. C. 2015, 119, 15477-15481. [21] Verrelli, D. I.; Albijanic, B. A comparison of methods for measuring the induction time for bubble–particle attachment, Miner. Eng. 2015, 80, 8-13. [22] Albijanic, B., Hampton M.A., Nguyen P.T., Ozdemir O., Bradshaw D.J. and Nguyen A.V., 2010. An integrated study of bubble-particle attachment mechanisms, In: XXV International Mineral Processing Congress Proceedings Brisbane, Qld, Australia, (1703-1709). 6-10 September, 2010. [23] Gu, G.; Xu, Z.; Nandakumar, K.; Masliyah, J. Effects of physical environment on induction time of air–bitumen attachment. Int J Miner Process. 2003, 69, 235-250. [24] Alexandrova, L.; Rao, K. H.; Forsberg, K.; Grigorov, L.; Pugh, R. J. Three-phase contact parameters measurements for silica-mixed cationic–anionic surfactant systems, Colloids and Surfaces A: Physicochemical and Engineering Aspects. 2009, 348, 228-233. [25] Yoon, R.; Yordan, J. L. The critical rupture thickness of thin water films on hydrophobic surfaces. J Colloid Interf. Sci. 1991, 146, 565-572.

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The StefanâReynolds model and the modified StefanâReynolds

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