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Hypothesis: The adsorption of heavy metal ions such as Pb(II) onto negatively charged minerals such as silica is expected to alter the structure and the interactions at the silica/aqueous interfaces. Besides the solution pH, the inner-sphere sorption of Pb(II) is expected to regulate the surface charge/potential, hypothesized to control the actions of monovalent anions in the aqueous environment. These complex pictures can be probed directly using surface-sensitive sum-frequency generation (SFG) spectroscopy., Experiments: The pH-dependent water structure within the double layer at silica/aqueous interfaces under the influence of different ions was examined using SFG. The recorded SFG spectra were deconvoluted into the Stern layer (SL) and diffuse layer (DL) using the maximum entropy method in conjunction with the electrical double-layer theory., Findings: Standalone monovalent sodium salts do not exhibit ion-specific effects on the silica/aqueous interfaces. However, the mixture of Pb(II) species and each of these salts display profound ion-specific effects on the structure of silica/aqueous interfaces, indicating the role of Pb(II) as an enabler of the ion-specificity of the investigated monovalent anions. The interesting effect arises from a complex interplay between the physical processes (i.e., electrostatic interactions, screening effects, etc.) and chemical processes such as the hydrolysis of Pb(II) ions, ion complexation, protonation and deprotonation of the surface silanol group., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2024 The Author(s). Published by Elsevier Inc. All rights reserved.)

Mineral tailings dams pose high pollution risks to the environment and catastrophic failures. Dry stacking has been identified as a promising alternative to mitigate these risks and offers various benefits to the mining industry but lacks systematic research outcomes. To facilitate dry stacking, coal tailings slurries were dewatered using either filtration or centrifugation methods, resulting in a semi-solid form (cake) that can be safely disposed of. The handleability and disposability of these cakes are greatly influenced by the selection of chemical aids (such as polymer flocculants) and the mechanical dewatering technique employed. The effects of polyacrylamide (PAM) flocculants with a range of molecular weight, charge, and charge density are presented. Coal tailings samples with differences in clay mineralogy were dewatered using press filtration, solid bowl centrifugation, and natural air drying. Handleability and disposability of the tailings were assessed by their rheological properties, including yield stress, adhesive and cohesive stresses, and stickiness. Residue moisture, type of polymer flocculants, and clay mineralogy were found to be crucial factors affecting the handleability and disposability of the dewatered cakes. The tailing yield stress (shear strength) increased as the solid concentration increased. In the semi-solid regime (above 60 wt% solids), the tailings displayed stiff exponential growth. Similar trends were observed for stickiness and adhesive/cohesive energy of the tailings with a steel (truck) surface. Adding polymer flocculants increased the shear strength of the dewatered tailings by 10-15%, thus favouring disposability. However, the polymer selection for coal tailing handling and processing is a trade-off between its disposability and handleability, which requires a multi-criteria decision-making process. The current results also suggested that cationic PAM could be most suitable for dewatering by press filtration, while anionic PAM should be selected for dewatering by solid bowl centrifugation., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier Ltd. All rights reserved.)

Hypothesis: Adsorption of divalent heavy metal ions (DHMIs) at the mineral-water interfaces changes interfacial chemical species and charges, interfacial water structure, Stern (SL), and diffuse (DL) layers. These molecular changes can be detected by probing changing orientation and hydrogen-bond network of interfacial water molecules in response to changing local charges and hydrophobicity., Experiments: Sum-frequency generation (SFG) spectroscopy was used to probe changes in vibrational resonances of interfacial OH vs. DHMI concentration and pH. SFG spectra were deconvoluted using the measured surface potential and maximum entropy method in conjunction with the electrical double-layer theory for the SL and DL structures and correlated by hydrophobicity., Findings: Three surface charge reversals (CRs) were detected at low (CR1), medium (CR2), and high (CR3) pHs. Unlike CR1, SFG signals were minimized at CR2 and CR3 for DHMIs-silica systems highlighting considerable alterations in the structure of interfacial waters due to the inner-sphere sorption of metal hydroxo complexes. SFG results showed "hydrophobic-like" stretching modes at > 3600 cm -1 for Pb-, Cu-, and Zn-treated silica. However, contact angle measurements revealed the hydrophobization of silica only in the presence of Pb(II), as confirmed by an in-depth SFG analysis of the hydrogen-bond network of the interfacial water molecules in the SL., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier Inc. All rights reserved.)

Hypothesis: Flotation of water-soluble KCl and NaCl minerals in brines is significant for K-fertilizer production, but its mechanism is controversial. Dissolved salt ions are expected to change the physicochemical properties of solvents, interfaces, and collector colloids, thereby affecting flotation significantly., Experiments: Flotation experiments of KCl and NaCl crystals in brines were conducted using potassium and sodium laurates as collectors. Contact angle (CA) and surface tension measurements, X-ray photoelectron spectroscopy (XPS) analysis, and molecular dynamics simulations (MD) were applied to gain a molecular understanding of changing interfacial properties and crystal-collector colloid interactions in the presence of dissolved ions in terms of salt flotation., Findings: While K + ions activate the NaCl crystal flotation, Na + ions depress the KCl crystal flotation, in agreement with the studies of CA, XPS, and MD results with these crystals. XPS results showed no collector adsorption at crystal surfaces which is a requirement of conventional flotation and presents a new theoretical challenge. We argue the crucial role of ion specificity: Na-laurate colloids adsorb at the bubble surface as a monolayer but solvent-separated from KCl crystals, inhibiting their flotation, or in interactive contact with NaCl crystals, enhancing their flotation. Increasing K + concentration weakens NaCl crystal hydration, increasing Na-laurate colloid attraction with crystals for better flotation. The Contact Interactive Collector Colloid (CICC) and Solvent-separated Interactive Collector Colloid (SICC) hydration states are critical to salt crystal flotation via collector colloid-crystal attraction by dispersion forces., Competing Interests: Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Copyright © 2023 Elsevier Inc. All rights reserved.)

Hypothesis: Bubble attachment to hydrophobic solid surfaces is influenced by the liquid film instability. Inclusion of transiently formed holes within the film rather than the so-called hydrophobic force in the theory is expected to better describe and explain film rupture and triple contact line formation in the bubble-surface attachment process. The significance of surface hydrophobicity and hole formation renders the stochastic nature of the induction time of attachment., Experiments: A combination of high-speed video microscopy and theoretical analysis was applied to investigate the induction time of attachment and critical film thickness of air bubbles rising freely perpendicularly to silica surfaces of different hydrophobicities., Findings: Film rupture occurred statistically for shorter induction times and thicker films on the more hydrophobic surface, rejecting the conjecture of hydrophobic force. Computed results of the critical base radius of the transient holes causing film rupture were merged together nicely, independently of surface hydrophobicity. The paper sheds light on the significance of hydrophobicity on the attachment process by means of a novel and easily implemented methodology, without relying on the debatable hydrophobic force., Competing Interests: Declaration of Competing Interest On behalf of all authors, I write to declare that we have no conflict of interest in whatsoever forms., (Copyright © 2020 Elsevier Inc. All rights reserved.)

Carbonation using CO 2 appears as an attractive solution for disposing of red mud suspensions, an aluminum industry hazardous waste since it also offers an option for CO 2 sequestration. Here we report the novel findings that CO 3 2- together with Ca 2+ can significantly affect the surface properties and settling of goethite, a major component of red mud. Specifically, their effects on the goethite surface chemistry, colloidal interaction forces and settling in alkaline solutions are investigated. The surface potential becomes more negative by the formation of carbonate inner-sphere complexes on goethite surface. It is consistent with the strong repulsion, decreased particle size and settling velocity with increased carbonate concentrations as measured by atomic force microscopy, particle size analysis, and particle settling. Adding Ca 2+ that forms outer-sphere complexes with pre-adsorbed carbonate changes goethite surface charge negligibly. Changing repulsion to the attraction between goethite surfaces by increasing calcium dosage indicates the surface bridging, in accordance with the increased settling velocity. The adverse effect of carbonate on goethite flocculation is probably due to its specific chemisorption and competition with flocculants. By forming outer-sphere complexes together with the flocculant-calcium bridging effect, calcium ions can eliminate the negative influence of carbonate and improve the flocculation of goethite particles. These findings contribute to a better understanding of goethite particle interaction with salt ions and flocculants in controlling the particle behavior in the handling processes, including the red mud carbonation., (Copyright © 2018 Elsevier Inc. All rights reserved.)

The influence of droplet orientation on the flow directed organization of nanoparticles in evaporating nanofluid droplets is important for the efficiency of foliar applied fertilizers and contamination adhesion to the exterior of buildings. The so called "coffee ring" deposit resulting from the evaporation of a sessile nanofluid drop on a hydrophilic surface has received much attention in the literature. Deposits forming on hydrophobic surfaces in the pendant drop position (i.e. hanging drop), which are of importance in foliar fertilizer and exterior building contamination, have received much less attention. In this study, the deposit patterns resulting from the evaporation of water droplets containing silica nanoparticles on hydrophobic surfaces orientated in the sessile or pendant configuration are compared. In the case of a sessile drop the well known coffee ring pattern surrounding a thin nanoparticle layer was formed. A deposit consisting of a thin coffee ring surrounding a bump was formed in the pendant position. A mechanism involving flow induced aggregation at the droplet waist, settling, orientation dependant accumulation within the drop and pinning of the contact line is suggested to explain the findings. Differences in the contact area and adhesion of deposits with surface orientation will affect the efficiency and rainfastness of foliar fertilizers and the cleanliness of building exteriors., (Copyright © 2012 Elsevier Inc. All rights reserved.)

The surface charge densities of the silica face surface and the alumina face surface of kaolinite particles, recently determined from surface force measurements using atomic force microscopy, show a distinct dependence on the pH of the system. The silica face was found to be negatively charged at pH>4, whereas the alumina face surface was found to be positively charged at pH<6, and negatively charged at pH>8. The surface charge densities of the silica face and the alumina face were utilized in this study to determine the interaction energies between different surfaces of kaolinite particles. Results indicate that the silica face-alumina face interaction is dominant for kaolinite particle aggregation at low pH. This face-face association increases the stacking of kaolinite layers, and thereby promotes the edge-face (edge-silica face and edge-alumina face) and face-face (silica face-alumina face) associations with increasing pH, and hence the maximum shear-yield stress at pH 5-5.5. With further increase in pH, the face-face and edge-face association decreases due to increasing surface charge density on the silica face and the edge surfaces, and decreasing surface charge density on the alumina face. At high pH, all kaolinite surfaces become negatively charged, kaolinite particles are dispersed, and the suspension is stabilized. The face-face association at low pH has been confirmed from cryo-SEM images of kaolinite aggregates taken from suspension which show that the particles are mostly organized in a face-face and edge-face manner. At higher pH conditions, the cryo-SEM images of the kaolinite aggregates reveal a lower degree of consolidation and the edge-edge association is evident., (Copyright © 2011 Elsevier Inc. All rights reserved.)

The crystal lattice images of the two faces of kaolinite (the silica face and the alumina face) have been obtained using contact-mode atomic force microscopy (AFM) under ambient conditions. Lattice resolution images reveal the hexagonal surface lattice of these two faces of kaolinite. Analysis of the silica face of kaolinite showed that the hexagonal surface lattice ring of oxygen atoms had a periodicity of 0.50±0.04nm between neighboring oxygen atoms, which is in good agreement with the surface lattice structure of the mica basal plane. The center of the hexagonal ring of oxygen atoms is vacant. Analysis of the alumina face of kaolinite showed that the hexagonal surface lattice ring of hydroxyls surround a hydroxyl in the center of the ring. The atomic spacing between neighboring hydroxyls was determined as 0.36±0.04nm. Ordering of the kaolinite particles for examination of the silica and alumina surfaces was accomplished using different substrates, a procedure previously established. Crystal lattice imaging supports previous results and independently confirms that the two faces of kaolinite have been properly identified., (Copyright © 2010 Elsevier Inc. All rights reserved.)

The constant thickness (H) of metastable free films of various non-ionic surfactant solutions was measured at surfactant concentrations less than the critical micelle concentrations or solubility limits with fixed 5x10(-5) M sodium chloride (NaCl) serving as the background electrolyte. The surfactants include n-pentanol, n-octanol, methyl isobutyl carbinol (MIBC), polypropylene glycol (PPG-400), tetraethylene glycol monooctyl ether (C(8)E(4)), and tetraethylene glycol monodecyl ether (C(10)E(4)). H was interferometrically measured. For each surfactant in this study, the H-versus-surfactant-concentration curve finds a peak at a concentration around 5x10(-6)-1x10(-5) M and a valley at a higher concentration. The measured H values were compared to those predicted from the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, which considers solely the contribution from electrostatic double-layer repulsion with van der Waals attraction being neglected in the present work. In determining the double-layer repulsion, the ionic strength was determined from the electrolytic conductivity measurement of the film-forming solutions and the surface potential was estimated from the zeta-potential measurement of air bubbles. It was found that the DLVO theory failed to explain the thickness variance with surfactant concentration, implying that additional non-DLVO attractive forces might be required to explain the experimental results. Finally, the possible origins of these attractive forces were discussed.

Atomic force microscopy (AFM) was used to examine how ethanol/water concentration affects the nanobubble bridging capillary force between a hydrophobic silica colloidal probe and a hydrophobic silica wafer. Nanobubbles were produced on the solid surfaces by a previously utilised method which uses solvent-exchange and surface scanning. In pure water a strong, long range attractive force ( approximately 230 nm) with a single jump in step was measured, typical of an interaction between two nanobubbles attached to the hydrophobic surfaces. An increase in the ethanol concentration had little effect on the range of the force but dramatically reduced its magnitude. At an ethanol concentration of 40% by mass, the force became repulsive after the initial attractive jump in. Above an ethanol concentration of 40% by mass, the capillary force disappeared. The change in the force with ethanol concentration was explained using a capillary force model with constant volume and contact angle. The bridge geometry, contact angle, volume and rupture distance were determined for different ethanol concentrations.

The latest developments in profile analysis tensiometry (PAT) for determining surface tension and interfacial viscoelastic parameters involve the determination of a digital interface profile and its best fit with the Young-Laplace equation. In this short communication, we show that the results for surface tension and other interfacial parameters determined by PAT are extremely sensitive to the (aspect) ratio of length to width of a pixel. Fine calibration (to five decimal digits) for the aspect ratio required to obtain physically consistent results is not always achieved with conventional numerical procedures due to nanometer resolution limit of optical imaging devices but can be manually adjusted using the known surface tension of pure water and/or surfactant solutions at reference temperature.

The interaction between hydrophobic surfaces in aqueous solutions is particularly important because it is encountered in many industrial processes. Even though advances in surface science have been tremendous, the nature of the hydrophobic interaction remains one of the greatest challenges to the field. In this work an atomic force microscope (AFM) was used to measure the normal and lateral interactions between a silica bead and a smooth silica substrate hydrophobized by esterification with 1-octanol. The experiments were performed in water and in water after alcohol-water exchange, a method that has been shown to increase the occurrence and size of nanobubbles at the hydrophobic surface and in turn result in a longer range hydrophobic force due to capillary bridge formation. It was found that the alcohol-water exchange had a significant impact on the friction force due to the increased size of the capillary, which increased adhesion.

Atomic force microscopy (AFM) was used to examine how different alcohols affect the hydrophobic attraction between a hydrophobic silica colloidal probe and a hydrophobic silica wafer. The experiments were performed in water and in water after rinsing alcohol (methanol, ethanol, or 1-propanol) throughout the AFM system. In all three cases the range of the attractive force increased after alcohol-water exchange, with 1-propanol showing the largest increase in range followed by ethanol and methanol. Additionally, experiments were performed before and after scanning the flat substrate with the colloidal probe. The range of the attractive force substantially increased with increasing scanning area. The attraction was explained by nanobubble bridging with a capillary force model with constant bridge volume proposed. The bridge volume (constant during each of the force curve measurements), contact angle and rupture distance were also determined for different scan sizes. The correlation between the rupture distance and bridge volume agreed with the available prediction.

This paper presents new theoretical and experimental results that quantify the role of surfactant adsorption and the related interfacial tension changes and interfacial forces in the emulsion film drainage and equilibrium. The experimental results were obtained with plane-parallel microscopic films from aqueous sodium dodecyl sulphate solutions formed between two toluene droplets using an improved micro-interferometric technique. The comparison between the theory and the experimental data show that the emulsion film drainage and equilibrium are controlled by the DLVO interfacial forces. The effect of interfacial viscosity and interfacial tension gradient (the Marangoni number) on the film drainage is also significant.

The nature of the air/carbonate solution interface is considered with respect to water structure by sum-frequency vibrational spectroscopy (SFVS) and molecular dynamics simulations (MDS). Results from this study provide further understating regarding previous observations that the surface tensions of structure making sodium carbonate solutions have been shown to be significantly greater than the surface tensions of structure breaking bicarbonate solutions at equivalent concentrations. This difference in surface tension and its variation with salt concentration is related to the organization of water and ions at the air/solution interface. Spectral results from SFVS show at equivalent concentrations that, for the carbonate solution, the strong water structure signal of 3200 cm(-1) at the air/carbonate solution interface is increased by a factor of 4 when compared to the same signal for the air/bicarbonate solution interface, which spectrum is weaker than the spectrum for the air/water interface in the absence of salt. These results from SFVS are explained by the results from MDS which show that in the case of carbonate solutions the structure making carbonate ions are excluded from the interfacial water region which region is extended in depth. On the other hand, in the case of bicarbonate solutions, the bicarbonate ions are accommodated in the interfacial water region and there is no evidence of an increase in the extent of water structure. These SFVS experimental and MD simulation results provide further information to understand interfacial phenomena of soluble salts at the molecular level.

This paper examined the dewetting between a small air bubble and a solid surface in deionised water. Hydrodynamics was used in conjunction with surface molecular kinetics to model and predict the velocity of the moving contact line as a function of the dynamic macroscopic contact angle. The dewetting hydrodynamics was modelled following the approach developed specifically for drops and bubbles using the (absolute) coordinate system with the origin located at the centre of the contact area, which does not move with the moving contact line. The model provides accurate corrections unavailable in the generic hydrodynamic theories developed by Voinov and Cox, and removes the need for a macroscopic length scale employed in their generic theories. Molecular kinetics was used to determine the contact angle of the inner region close to the contact line, where the hydrodynamic approach breaks down due to the singularity. Unlike the generic hydrodynamic theories, the inner (microscopic) angle in our combined model is not a constant (a fitting parameter) but is a function of the moving contact line velocity and other molecular properties of the interfaces. The combined model agreed with the experimental data and produced physically consistent values for the slip length, molecular jumping distance and frequency. The dissolved gases accumulated at the non-wetting solid-liquid interface may influence the slip length.

The combined approach of the molecular-kinetic and hydrodynamic theories for description of the motion of three-phase gas-liquid-solid contact lines has been examined using the Wilhelmy plate method. The whole dynamic meniscus has been divided into molecular, hydrodynamic, and static-like regions. The Young-Laplace equation and the molecular-kinetic and hydrodynamic dewetting theories have been applied to describe the meniscus profiles and contact angle. The dissipative forces accompanying the dynamic dewetting have also been investigated. The experiments with a Wilhelmy plate made from an acrylic polymer sheet were carried out using a computerized apparatus for contact angle analysis (OCA 20, DataPhysics, Germany). The extrapolated dynamic contact angle versus velocity of the three-phase contact line for Milli-Q water and 5x10(-4) M SDBS solution was experimentally obtained and compared with the combined MHD models with low and moderate Reynolds numbers. The models predict similar results for the extrapolated contact angle. SDBS decreases the equilibrium contact angle and increases the molecular jumping length but does not affect the molecular frequency significantly. The hydrodynamic deformation of the meniscus, viscous dissipation, and friction were also influenced by the SDBS surfactant.

In this paper mathematical expressions have been developed to describe the hydrodynamic resistance force on a colloidal particle as it slides along a slip surface of a gas bubble held stationary in a quiescent liquid. The particle size was considered to be sufficiently small relative to the bubble size so that the bubble surface could be locally approximated to a planar interface. The modeling incorporated a bispherical coordinate transformation to solve the equations governing the liquid creeping flow disturbed by the particle. Exact numerical solutions for the resistance coefficients of the particle-shearing motion parallel to the slip bubble surface were obtained as a function of the separation distance from the bubble surface. Finally, simplified analytical rational approximations for the whole range of the separation distance were presented, which were in good agreement with the exact numerical result. Importantly, the approximations for the modeling and simulation of the bubble-particle interactions are mathematically tractable.

A CCD high-speed video microscopy system operating at 1000 frames per second was used to obtain direct quantitative measurements of the trajectories of fine glass spheres on the surface of air bubbles. The glass spheres were rendered hydrophobic by a methylation process. Rupture of the intervening water film between a hydrophobic particle and an air bubble with the consequent formation of a three-phase contact was observed. The bubble-particle sliding attachment interaction is not satisfactorily described by the available theories. Surface forces had little effect on the particle sliding with a water film, which ruptured probably due to the submicrometer-sized gas bubbles existing at the hydrophobic particle-water interface.

In the early 1930s it was first reported that inorganic electrolytes enhance the floatability of coal and naturally hydrophobic minerals. To date, explanations of coal flotation in electrolytes have not been entirely clear. This research investigated the floatability of coal in NaCl and MgCl2 solutions using a modified Hallimond tube to examine the role of the electrical double-layer interaction between bubbles and particles. Flotation of coal was highly dependent on changes in solution pH, type of electrolyte, and electrolyte concentration. Floatability of coal in electrolyte solutions was seen not to be entirely controlled by the electrical double-layer interaction. Coal flotation in low electrolyte concentration solutions decreases with increase in concentration, not expected from the theory since the electrical double layer is compressed, resulting in diminishing the (electrical double layer) repulsion between the bubble and the coal particles. Unlike in low electrolyte concentration solutions, coal flotation in high electrolyte concentration solutions increases with increase in electrolyte concentration. Again, this behavior of coal flotation in high electrolyte concentration solutions cannot be quantitatively explained using the electrical double-layer interaction. Possible mechanisms are discussed in terms of the bubston (i.e., bubble stabilized by ions) phenomenon, which explains the existence of the submicron gas bubbles on the hydrophobic coal surface.

In this paper the problem of calculating the depression of the gas-liquid meniscus by the particle attachment was solved. The analytical approximate equations obtained for small and large radii, r(tpc), of the three-phase contact were analyzed and compared to the available numerical results. The Derjaguin equation for small r(tpc) and the analytical results for large r(tpc) are accurate for r(tpc)/L< or =0.2 and r(tpc)/L> or =2, respectively, where L is the capillary length. For the meniscus depression with r(tpc)/L from 0.2 to 2, the empirical equations were obtained based on the asymptotic analysis of the analytical approximate solutions. The empirical numerical constants were obtained by fitting to the exact numerical results. The empirical equations together with the analytical approximate equations provide the accurate predictions for the meniscus depression for the whole range of the radius of the three-phase contact and are expected to be useful for modeling the detachment interaction in the flotation separation processes.

This paper reports on an investigation of the influence of the interfacial shear viscosity on the liquid drainage in single Plateau borders of foam. The simplified Navier-Stokes equation governing the liquid flow is solved for the liquid velocity by the numerical computational method. The numerical results show significant influence of the interfacial shear viscosity on the liquid velocity in the Plateau border. Comparison of the numerical results for the average velocity over the cross-section area of the Plateau border to the available analytical solution shows that the available analytical solution underestimates the average velocity. New, simple yet accurate correlations for the dependence of the average velocity on the radius of the cross section of the Plateau border, the pressure gradient, and the interfacial shear viscosity are obtained using the asymptotic analysis and the numerical data.

In this paper the problem of calculating the liquid flow force on a particle in interaction with an air bubble with a mobile surface in flotation as a function of the separation distance was solved. The force equation was obtained by first deriving the disturbed flow confined between the surfaces. The model for the force includes the separation distance between the bubble and the particle, the particle size, the bubble's Reynolds number, the bubble rise velocity, and the polar position of the particle on the bubble surface. The proposed equations provide an exact solution to the situation where the particle and the bubble are very close together. The attractive flow force and the surface forces are of similar orders of magnitude. Consequently, the models presented in this paper should provide a better estimate for calculating the forces on particles interacting with air bubbles in mineral flotation and other separation operations involving colloidal interactions.