A. N. Filippov

A model of the interaction between a charged particle and a pore in a charged membrane surface

Abstract A model of the electrostatic and molecular interactions of a charged colloid particle with a charged membrane surface in an electrolyte solution has been developed. In Derjaguins approximation, the force between a spherical colloid particle and a cylindrical membrane pore (with a rounded inlet) is calculated taking into account both electrostatic and van der Waals interactions. The force and energy are strongly dependent on the zeta-potential of both the particle and the membrane pore, the electrolyte concentration, and geometrical parameters. Conditions are found for which a potential barrier exists at the pore entrance. This barrier prevents a particle from entering the pore and, hence, gives an equilibrium position of the particle above the membrane surface. Therefore, there is a possibility in this case of removing the particle by a tangential flow, preventing pore blocking. The model was verified using a Finite Element Method (FEM) analysis developed earlier for colloidal interactions by two co-authors. It has been found that the accuracies of analytical formulae obtained for the interaction energy and force are within 10 and 20%, respectively, for practical application ranges of physico–chemical and geometrical parameters. Two major advantages of the model proposed compared to FEM calculations are: (1) the possibility of non-centerline calculations (when a particle is not moving along the axis of a membrane pore) without a three-dimensional solution; and (2) speed of calculations using the analytical formulae is much higher. Using a simplified expression for hydrodynamic force, critical values of pressure gradients across the membrane pore have been calculated analytically.

Hydrodynamic permeability of membranes built up by particles covered by porous shells: Cell models

A review is presented on an application of a cell method for investigations of hydrodynamic permeability of porous/dispersed media and membranes. Based on the cell method, a hydrodynamic permeability is calculated of a porous layer/membrane built up by solid particles with a porous shell and non-porous impermeable interior. Four known boundary conditions on the outer cell boundary are considered and compared: Happels, Kuvabaras, Kvashnins and Cunninghams (usually referred to as Mehta-Morses condition). For description of a flow inside the porous shell Brinkmans equations are used. A flow around an isolated spherical particle with a porous shell is considered and a number of limiting cases are shown. These are compared with the corresponding results obtained earlier.

Hydrodynamic permeability of aggregates of porous particles with an impermeable core.

A hydrodynamic permeability of membranes built up by porous cylindrical or spherical particles with impermeable core is investigated. Different versions of a cell method are used to calculate the hydrodynamic permeability of the membranes. Four known boundary conditions, namely, Happels, Kuwabaras, Kvashnins and Cunningham/Mehta-Morses, are considered on the outer surface of the cell. Comparison of the resulting hydrodynamic permeability is undertaken. A possible jump of a shear stress at the fluid-membrane interface, its impact on the hydrodynamic permeability is also investigated. New results related to the calculated hydrodynamic permeability and the theoretical values of Kozeny constant are reported. Both transversal and normal flows of liquid with respect to the cylindrical fibers that compose the membrane are studied. The deduced theoretical results can be applied for the investigation of the hydrodynamic permeability of colloidal cake layers on the membrane surface, the hydrodynamic permeability of woven materials.

Cell models for flows in concentrated media composed of rigid impenetrable cylinders covered with a porous layer

AbsractThe hydrodynamic permeability of a membrane simulated by a set of identical impenetrable cylinders covered with a porous layer is calculated by the Happel-Brenner cell method. Both transverse and longitudinal flows of filtering liquid with respect to the cylindrical fibers that compose the membrane are studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham models are considered. Brinkman equations are used to describe the flow of liquid in the porous layer. Results that correspond to previously published data are obtained for the limiting cases. Theoretical values of Kozeny constants are calculated. The models proposed can be used to describe the processes of reverse osmosis, as well as nano- and ultrafiltration.

Electrotransport properties and morphology of MF-4SK membranes after surface modification with polyaniline

The method of template synthesis is used for the surface modification of MF-4SK membranes with polyaniline. The influence of the time of polyaniline synthesis in the surface layer of a perfluorinated MF-4SK membrane on its morphology and electrotransport properties is investigated. It is established that under the synthesis conditions, a gradient distribution of polyaniline develops across the thickness of the membrane, and as a result of this, an anisotropic composite structure is formed. It is shown that the specific electrical conductivity and the electroosmotic and diffusion permeability exhibit an extremal character as functions of the time of polyaniline synthesis. When the orientation of these composite membranes is changed with respect to electrolyte flow, an asymmetry effect in their diffusion characteristics is found. With the application of the bilayer fine porous membrane model, the modified-layer thickness is estimated, and the determining influence of the difference in absolute values of effective fixed-charge volume densities on the development of the asymmetry effect is found.

Permeability of complex porous media

AbsractHydrodynamic permeability of membrane composed of a set of porous spherical particles with rigid impenetrable cores is calculated. The cell method proposed by Happel and Brenner is used in calculations. All known boundary conditions on the cell surface, such as the Happel, Kuwabara, Kvashnin, and Cunningham (Mehta-Morse) models, are considered. The flow of liquid is described by the Brinkman equations. The problem of flow around a single spherical particle covered with porous layer by the uniform flow of viscous incompressible liquid is solved. Theoretical and empirical results are compared. Different limiting cases for which the derived formulas lead to results known from published literature are considered.

Theoretical and experimental study of asymmetry of diffusion permeability of composite membranes

Theoretical approaches to the description of asymmetry effects of transport characteristics of two-layer membranes are discussed. Based on a uniform finely porous model membrane, a new method is proposed for calculating mass transfer through asymmetric membranes with allowance for the physicochemical properties of their layers. The difference between the effective densities of charges fixed in membrane layers is shown to be the key factor that governs the degree of asymmetry of diffusion permeability. The adequacy of the proposed approach is supported by the comparison between theoretical calculations and experimental data on the diffusion permeability of modified ion-exchange membranes formed on polystyrene and perfluorinated matrices. It is shown that this approach is applicable to the description of the asymmetry effects of diffusion characteristics of MF-4SK perfluorinated membranes, the surface layer of which is modified with polyaniline.

Synthesis and diffusion permeability of MF-4SK/polyaniline composite membranes with controlled thickness of the modified layer

A new method is proposed for the synthesis of anisotropic membranes based on F-4SF films, which ensures their modification with polyaniline in a layer with a fixed thickness. The concentration dependences of conductivity and diffusion permeability of the composite membranes are studied. These parameters decrease with increasing thickness of the polyaniline-modified layer. The quantitative correlation is revealed between the regularities of variations in both transport characteristics and the thickness of the modified layer. The absence of the asymmetry of the diffusion permeability of the examined composite membranes is explained in terms of the model of a charged bilayer membrane.

Asymmetric Transmembrane Transfer Caused by a Difference in Adsorption Characteristics at Interfaces

A surface adsorption mechanism is proposed for the effect of asymmetric diffusion permeability of a modified membrane. A model is developed to describe diffusion transfer through the modified membrane with regard to the finiteness of the rate of adsorption on external membrane surfaces. Expressions are derived for diffusion fluxes as functions of transfer direction, and asymmetry coefficient is calculated for the diffusion permeability of the membrane. It is shown that the difference in the kinetic properties of the external surfaces of the membrane, which results from the modification of one of the surfaces, may induce the effect of asymmetric diffusion permeability.

Effect of the magnetic field on the hydrodynamic permeability of a membrane

The present paper concerns the influence of the magnetic field on the permeability of a membrane of solid cylindrical particles covered with porous layer. Here, we have considered the flow along the axis of cylinder and the alignment of uniform magnetic field is assumed to be perpendicular to the axis. The Brinkman equation is used for flow through porous region and Stokes equation is used for flow through clear fluid region. To model flow through assemblage of particles, cell model technique has been used i.e. the porous cylindrical shell is assumed to be confined within a hypothetical cell of same geometry. The stress jump condition has been employed at the fluid-porous interface and all four alternative conditions Happel, Kuwabara, Kvashnin and Mehta-Morse/Cunningham are used at the hypothetical cell. Effect of the Hartmann number on the hydrodynamic permeability of the membrane is discussed.