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Theory of Electrokinetic Phenomena in Fibrous Porous Media Caused By Colloids and Surfaces A: Physicochem. Eng. Aspects 222 (2003) 301Á/310 www.elsevier.com/locate/colsurfa Theory of electrokinetic phenomena in fibrous porous media caused by gradients of electrolyte concentration Yeu K. Wei, Huan J. Keh * Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, ROC Abstract The steady diffusioosmotic flow of an electrolyte solution in the fibrous medium constructed by an ordered array of parallel charged circular cylinders is analytically studied. The prescribed electrolyte concentration gradient is constant but can be oriented arbitrarily with respect to the axes of the cylinders. The electric double layer surrounding each cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model, which allows for the overlap of the double layers of adjacent cylinders, is employed to account for the effect of fibers on each other. The electrostatic potential distribution in the fluid phase of a cell is obtained by solving the linearized PoissonÁ/Boltzmann equation, which applies to the case of low surface potential or low surface charge density of the cylinders. An analytical formula for the diffusioosmotic/electroosmotic velocity of the electrolyte solution as a function of the porosity of the array of cylinders correct to the second order of their surface charge density or zeta potential is derived as the solution of a modified Stokes equation. In the absence of a macroscopic electric field induced by the electrolyte gradient (or externally imposed), the fluid flows (due to the chemiosmotic contribution) toward lower electrolyte concentration. With an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the fluid flow over a small range of the fiber surface potential. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic and electrophoretic velocities of an isolated circular cylinder caused by the applied electrolyte gradient or electric field. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Diffusioosmosis; Electroosmosis; Diffusiophoresis; Electrophoresis; Charged cylinder; Unit cell model 1. Introduction electrokinetic flow include dynamic pressure dif- ferences between the two ends of the pore (a The flow of electrolyte solutions in a small pore streaming potential is developed as a result of zero with a charged wall is of much fundamental and net electric current) and tangential electric fields practical interest in various areas of science and that interact with the electric double layer adjacent engineering. In general, driving forces for this to the pore wall (electroosmosis). Problems of fluid flow in porous media caused by these well-known driving forces were studied extensively in the past century [1Á/6]. * Corresponding author. Tel.: /886-2-2363-5462; fax: / 886-2-2362-3040. Another driving force for the electrokinetic flow E-mail address: [email protected] (H.J. Keh). in a micropore, which has commanded less atten- 0927-7757/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0927-7757(03)00246-2 302 Y.K. Wei, H.J. Keh / Colloids and Surfaces A: Physicochem. Eng. Aspects 222 (2003) 301Á/310 tion, involves tangential concentration gradients and the bulk fluid. This model involves the of an ionic solute that interacts with the charged concept that a bed of identical particles can be pore wall. This solute-wall interaction is electro- divided into a number of identical cells, one static in nature and its range is the Debye screen- particle occupying each cell at its center. The ing length k1. The fluid motion associated with boundary value problem for multiple particles is this mechanism, known as ‘‘diffusioosmosis’’ thus reduced to the consideration of the behavior (which is the complementary transport phenom- of a single particle and its bounding envelope. In enon to the diffusiophoresis of colloidal particles the past three decades, the unit cell model was used in prescribed solute concentration gradients), has by many researchers to predict various electro- been discussed mainly for solutions near a plane kinetic properties such as the mean electrophoretic wall [4,7Á/9]. Electrolyte solutions with a concen- mobility [14Á/22], effective electric conductivity 1 tration gradient of order 1 M cm along rigid [17,20Á/23], mean sedimentation rate [24Á/27], and surfaces with a zeta potential of order kT/e (Â/25 sedimentation potential [24Á/28] of a suspension of mV; e is the charge of a proton, k is the charged spherical particles as well as the transverse Boltzmann constant, and T is the absolute tem- electroosmotic mobility of electrolyte solutions in perature) can flow by diffusioosmosis at rates of a fibrous porous medium [29Á/32]. Recently, using several microns per second. the cell model, the present authors derived analy- The analytical expression for the diffusioosmo- tical expressions for the mean diffusioosmotic/ tic velocity of electrolyte solutions parallel to a diffusiophoretic velocity in a bed of dielectric charged plane wall [7] can be applied to the spheres [33] and within an array of parallel corresponding flow in capillary tubes and slits charged circular cylinders caused by a transversely when the thickness of the double layer adjacent to imposed electrolyte concentration gradient [34]. the capillary wall is small compared with the For a general case of diffusioosmosis/electro- capillary radius. However, in some practical ap- osmosis of a fluid solution in a fibrous system plications involving dilute electrolyte solutions in constructed by an array of parallel cylinders, the very fine pores, this condition is no longer satisfied applied electrolyte concentration gradient can be and the dependence of the fluid flow on the taken as a combination of its transversal and electrokinetic radius kR, where R is the radius longitudinal components with respect to the or- of a capillary tube or the half thickness of a ientation of the cylinders. Hence, the problem can capillary slit, must be taken into account. Re- be divided into two, if it is linearized, and they cently, the diffusioosmosis of electrolyte solutions might be separately solved. The overall diffusioos- in a capillary tube or slit with an arbitrary value of motic/electroosmotic velocity of the bulk fluid can kR was analyzed [10] for the case of small surface be obtained by the vectorial addition of the two- potential or surface charge density at the capillary component results. Since the problem of the wall. Closed-form formulas for the fluid velocity transverse diffusioosmotic flow in a homogeneous profile and average fluid velocity on the cross array of parallel charged circular cylinders has section of the capillary tube or slit were derived. been solved in a previous paper [34], in this work The capillary model of porous media is not we only need to treat the diffusioosmosis in the realistic for either granular or fibrous systems, for array generated by a longitudinal electrolyte it does not allow for the convergence and diver- gradient. The unit cell model is still used in the gence of flow channels. For electrokinetic flow analysis. The electric potential at the cylinder within beds of particles or synthetic microporous surfaces (or the surface charge density) is assumed membranes, it is usually necessary to account for to be uniform and low, but no assumption is made the effects of pore geometry, tortuosity, etc. To as to the thickness of the electric double layers avoid the difficulty of the complex geometry relative to the radius of the cylinders. A closed- appearing in beds of particles, a unit cell model form expression for the diffusioosmotic/electroos- [11Á/13] was often employed to predict these effects motic velocity profile in a cell is obtained in Eqs. on the relative motions between a granular bed (16), (17a) and (17b). The information provided by Y.K. Wei, H.J. Keh / Colloids and Surfaces A: Physicochem. Eng. Aspects 222 (2003) 301Á/310 303 this work may prove relevant in understanding the concentrations in the electric double layer adjacent chemotactic flow of fluids and transport of parti- to the cylinder surface with the axial position can cles in physiological media. be ignored. 2.1. Electrostatic potential distribution 2. Analysis We first deal with the electric potential distribu- We consider the diffusioosmotic flow of a fluid tion in the fluid phase on a plane normal to the solution of a symmetrically charged binary elec- axis of the cylinder in a unit cell. If c(r) represents Z Z trolyte of valence (where is a positive integer) the electrostatic potential at a point distance r through a uniform array of parallel, identical, from the axis of the cylinder relative to that in the circular cylinders at the steady state. The discrete bulk solution and n(r, z) and n(r, z) denote nature of the surface charges, which are uniformly the local concentrations of the cations and anions, distributed on each cylinder, is neglected. The respectively, then Poisson’s equation gives: applied electrolyte concentration gradient 9n is a constant along the axial (z) direction of the 1 d dc r cylinders, where n (z) is the linear concentration r dr dr distribution of the electrolyte in the bulk solution phase in equilibrium with the fluid inside the 4pZe [n(r; 0)n(r; 0)]: (1) array. The electrolyte ions can diffuse freely in o the fibrous porous medium, so there exists no In this equation, o/4po o , where o is the relative regular osmotic flow of the solvent. As shown in 0 r r permittivity of the electrolyte solution and o0 is the Fig. 1, we employ a unit cell model in which each permittivity of a vacuum. The local ionic concen- a dielectric cylinder of radius is surrounded by a trations can also be related to the electrostatic coaxial circular cylindrical shell of the fluid solu- potential by the Boltzmann equation, tion having an outer radius of b such that the fluid/cell volume ratio is equal to the porosity 1/ Zec 2 n9 n exp : (2) 8 of the fiber matrix; viz.
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