Capillary force required to detach micron-sized particles from solid surfaces—Validation with bubbles circulating in water and 2 μm-diameter latex spheres

The adhesion forces holding micron-sized particles to solid surfaces can be studied through the detachment forces developed by the transit of an air–liquid interface in a capillary. Two key variables affect the direction and magnitude of the capillary detachment force: (i) the thickness of the liquid film between the bubble and the capillary walls, and (ii) the effective angle of the triple phase contact between the particles and the interface. Variations in film thickness were calculated using a two-phase flow model. Film thickness was used to determine the time-variation of the capillary force during transit of the bubble. The curve for particle detachment was predicted from the calculated force. This curve proved to be non-linear and gave in situ information on the effective contact angle developing at the particle–bubble interface during detachment. This approach allowed an accurate determination of the detachment force. This theoretical approach was validated using latex particles 2 μm in diameter.     

Microflotation Suppression and Enhancement Caused by Particle/Bubble Electrostatic Interaction

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

两个液滴之间的聚并

对于系统中不含杂质时两个液滴在不互溶液体中的聚并过程进行理论分析，得到聚并所需时间与两相物理性质一范德华力的关系，该结果也适用于气泡在液体中的聚并，只要知道系统的物性数据和液滴半径，就可以计算聚并时间，理论预测与实验结果符合较好。     

Drag of a Solid Particle Trapped in a Thin Film or at an Interface: Influence of Surface Viscosity and Elasticity

We propose a theoretical model for the motion of a spherical particle entrapped in a thin liquid film or in a monolayer of insoluble surfactant at the air/water interface. Both surface shear and dilational viscosity, surface diffusion, and elasticity of the film are taken into consideration. The drag force acting on the particle is analytically calculated and asymptotic expressions of the problem are provided. The relevance of the model is discussed by comparing the calculated "viscoelastic" drag, gamma(vel), to the one predicted by Saffman's theory, gamma(S), for cylindrical inclusions in membranes. Numerical analyses are performed to evaluate the contributions of the surface viscosity and the diffusion coefficient of the layer on the hydrodynamical resistance experienced by the particle. Copyright 2000 Academic Press.     

Bubble-solid interactions in water and electrolyte solutions

Surface forces between an air bubble and a flat mica surface immersed in aqueous electrolyte solutions have been investigated using a modified surface force apparatus. An analysis of the deformation of the air bubble with respect to the mutual position of the bubble and the mica surface, the capillary pressure, and the disjoining pressure allows the air-liquid surface electrical potential to be determined. The experiments show that a long-range, double-layer repulsion acts between the mica (which is negatively charged) and an air bubble in water and in various electrolyte solutions at low concentration, thereby indicating that the air bubble surface is negatively charged. However, there is clear evidence that charge regulation occurs at the air-water interface to maintain a constant surface potential, and as a result of this, the charge at this interface changes from negative to positive as the bubble approaches the mica surface. Because of the attraction that arises as a result of the charge reversal, a finite force is required to separate the bubble from the mica, though the mica remains wetted by the aqueous phase. At the low concentrations investigated, the potential on the gas-liquid interface is independent of the electrolyte type within experimental uncertainty.     

Analysis of the gas states at a liquid/solid interface based on interactions at the microscopic level

The states of gas accumulated at the liquid/solid interface are analyzed on the basis of the continuum theory, in which the Hamaker constant is used to describe the long-range interaction at the microscopic scale. The Hamaker constant is always negative, whereas the "gas" spreading coefficient can be either negative or positive. Despite the complexity of gas, including that the density profile may not be uniform due to absorption on both solid and liquid surfaces, we predict three possible gas states at the liquid/solid interface, that is, complete "wetting", partial "wetting", and pseudopartial "wetting". These possible gas states correspond, respectively, to a gas pancake (or film) surrounded by a wet solid, a gas bubble with a finite contact angle, and a gas bubble(s) coexisting with a gas pancake. The typical thickness of the gas pancakes is at the nano scale within the force range of the long-range interaction, whereas the radius of the gas bubbles can be large. The state of a gas bubble(s) coexisting with a gas film is predicted theoretically for the first time. Our theoretical results can contribute to the development of a unified picture of gas nucleation at the liquid/solid interface.     

Exact and global rational approximate expressions for resistance coefficients for a colloidal solid sphere moving in a quiescent liquid parallel to a slip gas-liquid interface

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.     

Direct measurement of hydrophobic particle–bubble interactions in aqueous solutions by atomic force microscopy: Effect of particle hydrophobicity

Direct measurements of the interaction forces between a spherical silica particle and a small air bubble have been conducted in aqueous electrolyte solutions by using an atomic force microscope (AFM). The silica particle was hydrophobized with a silanating reagent, and the interaction forces were measured by using several particles with different surface hydrophobicities. In the measured force curves, a repulsive force was observed at large separation distances as the particle moved towards the bubble. The origin of the repulsive force was attributed to an electrostatic double-layer force because both the particle and bubble were negatively charged. After the repulsive force, an extremely long-range attractive force acted between the surfaces. These results indicate that the intervening thin water film between the particle and bubble rapidly collapsed, resulting in the particle penetrating the bubble.

The instability of the thin water film between the surfaces suggests the existence of an additional attractive force. By comparing the repulsive forces of the obtained force curves with the DLVO theory, the rupture thickness was estimated. The hydrophobicity of the particle did not significantly change the rupture thickness, whereas the pH of the solution is considered to be a critical factor.     

Simultaneous thermal and surfactant-induced Marangoni effects in thin liquid films

The deformation of a thin liquid film in the presence of a surfactant monolayer, varying temperature distributions, and limited mass flux is considered. Use of lubrication theory yields a coupled pair of partial differential equations for the film height and surfactant surface monolayer concentration. The long-wave stability of the isothermal film is examined over a wide range of parameter values. It is shown that droplet patterns are obtained under certain thermal conditions for both an isothermal and nonisothermal underlying substrate. For the case of a localized thermal gradient initially imposed at the air-liquid interface, severe film thinning beneath the heat source was observed, which was not accompanied by droplet formation; pseudo steady states are observed in this case. In all situations the surfactant is found to rigidify the air-liquid interface, retarding thermally driven flow, while evaporation (condensation) acts to destabilize (stabilize) the film.     

Optical and photocatalytic properties of single crystalline ZnO at the air-liquid interface by an aminolytic reaction

Crystalline flowerlike ZnO was synthesized by an aminolytic reaction at the air-liquid interface in an aqueous media at an alkaline pH. A thin visible film was formed at the air-liquid interface by self-assembly of flowerlike ZnO. Diffraction studies show rearrangement of the single crystalline units at the air-liquid interface leading to the formation of nanobelts. These nanobelts overlap systematically to form petals of the flowerlike structure; individual petals get curved with time. Each nanobelt is found to be single crystalline and can be indexed as the hexagonal ZnO phase. The organic product formed in the aminolytic reaction and dissolution-reprecipitation mechanism is the driving force for the formation of flowerlike ZnO at the air-liquid interface. A clear relationship between the surface, photocatalytic, and photoluminescent properties of ZnO is observed. The flowerlike structure exhibits a blue shift (3.56 eV) in the band emission as compared to bulk ZnO (3.37 eV). The photodegradation of methylene blue over the flowerlike ZnO catalyst formed at the air-liquid interface and in the sediments shows enhanced photocatalytic activity. The sub-bands formed due to surface defects facilitate separation of charge carriers increasing their lifetime, leading to enhanced photocatalytic activity of flowerlike ZnO.     

The detachment of particles from coalescing bubble pairs

This paper is concerned with the detachment of particles from coalescing bubble pairs. Two bubbles were generated at adjacent capillaries and coated with hydrophobic glass particles of mean diameter 66 μm. The bubbles were then positioned next to each other until the thin liquid film between them ruptured. The particles that dropped from the bubble surface during the coalescence process were collected and measured. The coalescence process was very vigorous and observations showed that particles detached from the bubble surfaces as a result of the oscillations caused by coalescence. The attached particles themselves and, to some extent the presence of the surfactant had a damping affect on the bubble oscillation, which played a decisive role on the particle detachment phenomena. The behaviour of particles on the surfaces of the bubbles during coalescence was described, and implications of results for the flotation process were discussed.     

A Note on the Formation and Rise of a Bubble from a Submerged Nozzle, Including Effects of Bulk- and Surface-Dilatational Viscosity

A model of the formation, detachment, and rise of a bubble from a submerged orifice is derived, based upon a study using a modified form of the Rayleigh–Plesset equation. Similar models have previously been proposed by Oguz and Prosperetti (1), Avramidis and Jiang (2), and also Chakraborty and Tuteja (3). We seek to re-examine these models and implement a number of additional physical features. In particular, we demonstrate the relative importance of the surface dilatational viscosity of surfactant added to the liquid in the growth and detachment of the bubble from the orifice. It is found that “large” surface dilatational viscosities significantly increase the time to detachment of the bubble. In addition, through a drastic reduction in the rate of radial expansion of the bubble in the early stages of development (given an initial condition on the radial velocity for “fast” bubble growth), the rise velocity of the bubble centroid at this time is greater with a large surface dilatational viscosity than when this property is neglected.     

Measurement and modeling on hydrodynamic forces and deformation of an air bubble approaching a solid sphere in liquids

The interaction between bubbles and solid surfaces is central to a broad range of industrial and biological processes. Various experimental techniques have been developed to measure the interactions of bubbles approaching solids in a liquid. A main challenge is to accurately and reliably control the relative motion over a wide range of hydrodynamic conditions and at the same time to determine the interaction forces, bubble–solid separation and bubble deformation. Existing experimental methods are able to focus only on one of the aspects of this problem, mostly for bubbles and particles with characteristic dimensions either below 100 μm or above 1 cm. As a result, either the interfacial deformations are measured directly with the forces being inferred from a model, or the forces are measured directly with the deformations to be deduced from the theory. The recently developed integrated thin film drainage apparatus (ITFDA) filled the gap of intermediate bubble/particle size ranges that are commonly encountered in mineral and oil recovery applications. Equipped with side-view digital cameras along with a bimorph cantilever as force sensor and speaker diaphragm as the driver for bubble to approach a solid sphere, the ITFDA has the capacity to measure simultaneously and independently the forces and interfacial deformations as a bubble approaches a solid sphere in a liquid. Coupled with the thin liquid film drainage modeling, the ITFDA measurement allows the critical role of surface tension, fluid viscosity and bubble approach speed in determining bubble deformation (profile) and hydrodynamic forces to be elucidated. Here we compare the available methods of studying bubble–solid interactions and demonstrate unique features and advantages of the ITFDA for measuring both forces and bubble deformations in systems of Reynolds numbers as high as 10. The consistency and accuracy of such measurement are tested against the well established Stokes–Reynolds–Young–Laplace model. The potential to use the design principles of the ITFDA for fundamental and developmental research is demonstrated.     

Direct measurement of the viscous force between two spherical particles trapped in a thin wetting film

Here we present the first direct measurement of the viscous drag force between two spherical particles of millimeter size trapped in a thin wetting film. Each particle is constrained by the liquid/air interface and the solid substrate. The viscous force is counterbalanced by another known force, the attractive capillary immersion force between identical particles protruding from the film surface. The results of the measurements provide evidence for an increased hydrodynamic force due to a non-Stokesian resistance to the particle motion. Our findings can be applied to the self-assembly of colloidal particles in a two-dimensional array for coating and to the friction between small species and a solid. Received: 19 March 1999 Accepted in revised form: 11 May 1999     

Effective spring description of a bubble or a droplet interacting with a particle

It is shown that the interaction of a particle with a liquid drop or a gas bubble may be quantitatively described over the whole distance regime by treating the fluid interface as a Hookean spring. An algorithm suitable for analyzing atomic force microscopy data suitable for a calculator or a spread-sheet is given and applied to data for oil drops.     

Anomalous Capillary Pressure, Stress, and Stability of Solids-Coated Bubbles

A two-dimensional theoretical model for solids-coated, or "armored," bubbles shows how the armor can support a liquid-vapor interface of reduced or reversed curvature between the particles, giving the bubble zero or even negative capillary pressure. The inward capillary force pulling the particles into the center of the bubble are balanced by large contact forces between the particles in the armor. Thus the bubble is stabilized against dissolution of gas into surrounding liquid, which otherwise would rapidly collapse the bubble. The stresses between particles in such cases are large and could drive sintering of the particles into a rigid framework. Earlier work on solids-coated bubbles assumed that solids can freely enter or leave the bubble surface as the bubble shrinks or expands. In such a case, armored bubbles would not be stable to gas dissolution into surrounding liquid. A new free-energy analysis, however, suggests that a shrunken bubble would not spontaneously expel a solid particle from its armor to relieve stress and allow the bubble to shrink further. Implications and limitations of the theory are discussed. Copyright 1999 Academic Press.     

Three-dimensional opal-like silica foams

The synthesis of novel meso-/macroporous SiO2 monoliths by combining a nano-building-blocks-based approach with the confined geometry of a tailored air-liquid foam structure is described. The resulting macrostructure in which ordered close-packed colloidal silica nanoparticles constitute the monolith's scaffolds very closely resembles the tailored periodic air-liquid foam template. The void spaces between adjacent particles create textural mesoporosity; therefore, the as-prepared silica networks are characterized by hierarchical porosity at the macroscopic and mesoscopic length scales. The fine-tuning of both the liquid foam's fraction and the bubble size allows a rational design over the macroscopic cell morphologies (shape, Plateau border's length, and width). Striking results of this approach are the weak shrinkage of the as-synthesized opal-like scaffolds during the thermally induced sintering process and, in contrast with previous studies, the formation of closed-cell structures. Particle organization and the foam film surface roughness are investigated by atomic force microscopy (AFM), showing the influence of the liquid flow, within the foams' Plateau borders and films, on the final assemblies.     

Dynamics of positronium bubble growth in liquid media and the energy dissipation problem

The dynamics of positronium bubble growth and dissipative energy loss accompanying this process were considered in the hydrodynamic approximation for liquid media with strongly differing values of the surface tension and viscosity. The driving force of the growth of the positronium bubble is repulsive exchange interaction between the electron inherent in the positronium atom and electrons of surrounding molecules. Intermolecular attraction (Laplace pressure) and viscous media retard bubble growth. In liquid helium and glycerol, the Ps formation time is comparable to or even longer than the parapositronium lifetime. Dissipative energy losses accompanying the bubble growth are substantially higher than the surface energy of the equilibrium bubble.Translated from Khimiya Vysokikh Energii, Vol. 39, No. 1, 2005, pp. 44–51.Original Russian Text Copyright © 2005 by Mikhin, Stepanov, Byakov.     

Wetting of a particle in a thin film

When a particle is placed in a thin liquid film on a planar substrate, the liquid either climbs or descends the particle surface to satisfy its wetting boundary condition. Analytical solutions for the film shape, the degree of particle immersion, and the downward force exerted by the wetting meniscus on the particle are presented in the limit of small Bond number. When line tension is significant, multiple solutions for the equilibrium meniscus position emerge. When the substrate is unyielding, a dewetting transition is predicted; that is, it is energetically favorable for the particle to rest on top of the film rather than remain immersed in it. If the substrate can bend, the energy to drive this bending is found in the limits of slow or rapid solid deflection. These results are significant in a wide array of disciplines, including controlled delivery of drugs to pulmonary airways, the probing of liquid film/particle interface properties using particles affixed to AFM tips and the positioning of small particles in thin films to create patterned media.