Victor Starov

A unifying model for concentration polarization, gel-layer formation and particle deposition in cross-flow membrane filtration of colloidal suspensions

Abstract A model is proposed to describe the cross-flow filtration of colloidal particles and molecules. This two-dimensional model depicts both concentration polarization and gel or cake formation in a tubular filtration device. A description of transport phenomena in a concentrated colloidal suspension is the core of the model. Surface and hydrodynamic interactions are used to predict the variation of the osmotic pressure and diffusion coefficient with the volume fraction of the suspension. The mathematical development leads to an analytical equation used for calculating the stationary permeate flux from integral calculations. The two-dimensional concentration profile along the membrane, together with the corresponding permeate flux is obtained. This paper illustrates how mass transfer equations coupled with a realistic description of the fluid can describe both concentration polarization and gel or cake formation. The paper includes a discussion on the differences between limiting and critical fluxes, and between particles and macromolecular cross-flow filtrations.

Surface forces and wetting phenomena

Conditions for thermodynamic equilibrium of liquid drops on solid substrates are presented. It is shown that if surface force (disjoining/conjoining Derjaguin pressure) action in a vicinity of the three-phase contact line is taken into account the condition of thermodynamic equilibrium is duly satisfied. Then the thermodynamic expressions for equilibrium contact angles of drops on solid substrates and menisci in thin capillaries are expressed in terms of the corresponding Derjaguin isotherm. It is shown that equilibrium contact angles of drops vary significantly depending on the vapour pressure in the ambient atmosphere, while there is a single, unique equilibrium contact angle in thin capillaries. It is also shown that the static advancing contact angle of a drop depends on its volume, in agreement with experimental data. In the case of menisci in capillaries, the expression for the receding contact angle is deduced, with results that are also in agreement with known experimental data.

Spreading of liquid drops over porous substrates.

The spreading of small liquid drops over thin and thick porous layers (dry or saturated with the same liquid) has been investigated in the case of both complete wetting (silicone oils of different viscosities) and partial wetting (aqueous SDS solutions of different concentrations). Nitrocellulose membranes of different porosity and different average pore size have been used as a model of thin porous layers, glass and metal filters have been used as a model of thick porous substrates. The first problem under investigation has been the spreading of small liquid drops over thin porous layers saturated with the same liquid. An evolution equation describing the drop spreading has been deduced, which showed that both an effective lubrication and the liquid exchange between the drop and the porous substrates are equally important. Spreading of silicone oils over different nitrocellulose microfiltration membranes was carried out. The experimental laws of the radius of spreading on time confirmed the theory predictions. The spreading of small liquid drops over thin dry porous layers has also been investigated from both theoretical and experimental points of view. The drop motion over a dry porous layer appears caused by the interplay of two processes: (a). the spreading of the drop over already saturated parts of the porous layer, which results in a growth of the drop base, and (b). the imbibition of the liquid from the drop into the porous substrate, which results in a shrinkage of the drop base and a growth of the wetted region inside the porous layer. As a result of these two competing processes the radius of the drop base goes through a maximum as time proceeds. A system of two differential equations has been derived to describe the time evolution of the radii of both the drop base and the wetted region inside the porous layer. This system includes two parameters, one accounts for the effective lubrication coefficient of the liquid over the wetted porous substrate, and the other is a combination of permeability and effective capillary pressure inside the porous layer. Two additional experiments were used for an independent determination of these two parameters. The system of differential equations does not include any fitting parameter after these two parameters were determined. Experiments were carried out on the spreading of silicone oil drops over various dry nitrocellulose microfiltration membranes (permeable in both normal and tangential directions). The time evolution of the radii of both the drop base and the wetted region inside the porous layer was monitored. In agreement with our theory all experimental data fell on two universal curves if appropriate scales were used with a plot of the dimensionless radii of the drop base and of the wetted region inside the porous layer using a dimensionless time scale. Theory predicts that (a). the dynamic contact angle dependence on the dimensionless time should be a universal function, (b). the dynamic contact angle should change rapidly over an initial short stage of spreading and should remain a constant value over the duration of the rest of the spreading process. The constancy of the contact angle on this stage has nothing to do with hysteresis of the contact angle: there is no hysteresis in our system. These predictions are in the good agreement with our experimental observations. In the case of spreading of liquid drops over thick porous substrates (complete wetting) the spreading process goes in two similar stages as in the case of thin porous substrates. In this case also both the drop base and the radii of the wetted area on the surface of the porous substrates were monitored. Spreading of oil drops (with a wide range of viscosities) on dry porous substrates having similar porosity and average pore size shows universal behavior as in the case of thin porous substrates. However, the spreading behavior on porous substrates having different average pore sizes deviates from the universal behavior. Yet, even in this case the dynamic contact angle remains constant over the duration of the second stage of spreading as in the case of spreading on thin porous substrates. Finally, experimental observations of the spreading of aqueous SDS solution over nitrocellulose membranes were carried out (case of partial wetting). The time evolution of the radii of both the drop base and the wetted area inside the porous substrate was monitored. The total duration of the spreading process was subdivided into three stages: in the first stage the drop base growths until a maximum value is reached. The contact angle rapidly decreases during this stage; in the second stage the radius of the drop base remains constant and the contact angle decreases linearly with time; finally in the third stage the drop base shrinks while the contact angle remains constant. The wetted area inside the porous substrate expands during the whole spreading process. Appropriate scales were used to have a plot of the dimensionless radii of the drop base, of the wetted area inside the porous substrate, and the dynamic contact angle vs. the dimensionless time. Our experimental data show: the overall time of the spreading of drops of SDS solutions over dry thin porous substrates decreases with the increase of surfactant concentration; the difference between advancing and hydrodynamic receding contact angles decreases with the surfactant concentration increase; the constancy of the contact angle during the third stage of spreading has nothing to do with the hysteresis of contact angle, but determined by the hydrodynamics. Using independent spreading experiments of the same drops on a non-porous nitrocellulose substrate we have shown that the static receding contact angle is equal to zero, which supports our conclusion on the hydrodynamic nature of the hydrodynamic receding contact angle on porous substrates.

Concentrated dispersions of charged colloidal particles: Sedimentation, ultrafiltration and diffusion☆

Abstract The formation and structure of sedimentation and/or filtration layers built up by charged colloidal particles were studied. For that purpose a theoretical model based on the force balance on a given particle in the layer was developed. The colloid stability problem and appearance of coagulation were considered. The hydrodynamic permeability of a filtration layer, consisting of charged particles, was calculated for different values of the electrolyte concentration, filtration rate, particle surface potential and radius. A simple expression for estimation of the collective diffusion coefficient of charged colloidal particles at relatively high volume fractions was also proposed.

Particle laden fluid interfaces: Dynamics and interfacial rheology

We review the dynamics of particle laden interfaces, both particle monolayers and particle+surfactant monolayers. We also discuss the use of the Brownian motion of microparticles trapped at fluid interfaces for measuring the shear rheology of surfactant and polymer monolayers. We describe the basic concepts of interfacial rheology and the different experimental methods for measuring both dilational and shear surface complex moduli over a broad range of frequencies, with emphasis in the micro-rheology methods. In the case of particles trapped at interfaces the calculation of the diffusion coefficient from the Brownian trajectories of the particles is calculated as a function of particle surface concentration. We describe in detail the calculation in the case of subdiffusive particle dynamics. A comprehensive review of dilational and shear rheology of particle monolayers and particle+surfactant monolayers is presented. Finally the advantages and current open problems of the use of the Brownian motion of microparticles for calculating the shear complex modulus of monolayers are described in detail.

Kinetics of wetting and spreading by aqueous surfactant solutions.

Interest in wetting dynamics processes has immensely increased during the past 10-15 years. In many industrial and medical applications, some strategies to control drop spreading on solid surfaces are being developed. One possibility is that a surfactant, a surface-active polymer, a polyelectrolyte or their mixture are added to a liquid (usually water). The main idea of the paper is to give an overview on some dynamic wetting and spreading phenomena in the presence of surfactants in the case of smooth or porous substrates, which can be either moderately or highly hydrophobic surfaces based on the literature data and the authors own investigations. Instability problems associated with spreading over dry or pre-wetted hydrophilic surfaces as well as over thin aqueous layers are briefly discussed. Toward a better understanding of the superspreading phenomenon, unusual wetting properties of trisiloxanes on hydrophobic surfaces are also discussed.

Spreading of liquid drops over dry surfaces

Abstract Spreading of thin, axisymmetric, non-volatile, Newtonian liquid drops over a dry, smooth, flat solid surface is considered both theoretically and experimentally in the case of complete wetting. The drop profile is solved analytically by matching the “outer” solution for large film thicknesses, where only the capillary effects are important, with the “inner” solution for small film thicknesses, where the viscous and disjoining pressure effects are comparable to capillary effects. It is shown that the apparent radius of the wetted spot, the apex height of the drop, and the apparent advancing dynamic contact angle follow different power laws in time and the advancing dynamic contact angle follows a power law in capillary number. Both the prefactor and the exponent of each power law are derived theoretically. Good agreement between the theory predictions and experimental measurements is shown for both the prefactor and exponent of each power law. It is necessary to emphasize that the theory suggested does not include any fitting parameters.

Adhesion models: From single to multiple asperity contacts

This review presents a summary of the current adhesion models available to date, between real rough surfaces, starting from single asperity models and expanding to multiple asperity contacts. The focus is made on multi-asperity contact interactions. Both van der Waals and contact mechanics approaches have been considered and relevant adhesion models are reviewed and discussed. The influence of the meniscus forces on adhesion has been considered, along with a summary of the various meniscus models. The effect of surface geometry, its topography and environmental conditions on meniscus action are also discussed along with its integration into multi-asperity adhesion models.

Reverse osmosis of multicomponent electrolyte solutions Part I. Theoretical development

A model is developed to treat reverse osmotic separations of electrolyte solutions. Transport in the model is based on the Extended Nernst-Planck equation, which includes diffusion, convection, and electromigration. Boundary conditions include a distribution coefficient that is due to a specific interaction potential representing repulsion of ions from the membrane material. Boundary conditions also include potential jumps known as Donnan Potentials. The model incorporates a mechanism for varying-membrane-fixed-charge as a function of ion concentrations and pH inside the membrane. In addition to salt ions, hydrogen and hydroxide ions are also considered, as pH changes indicate they take part in the transport. A high Peclet approximation is developed which simplifies calculations.

Fluoro- vs hydrocarbon surfactants: why do they differ in wetting performance?

Fluorosurfactants are the most effective compounds to lower the surface tension of aqueous solutions, but their wetting properties as related to low energy hydrocarbon solids are inferior to hydrocarbon trisiloxane surfactants, although the latter demonstrate higher surface tension in aqueous solutions. To explain this inconsistency available data on the adsorption of fluorosurfactants on liquid/vapour, solid/liquid and solid/vapour interfaces are discussed in comparison to those of hydrocarbon surfactants. The low free energy of adsorption of fluorosurfactants on hydrocarbon solid/water interface should be of a substantial importance for their wetting properties.