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

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A model of the interaction between a charged particle and a pore in a charged membrane surface
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   Advances in Colloid and Interface Science Ž . 81 1999 35  72  A model of the interaction between acharged particle and a pore in a chargedmembrane surface W. Richard Bowen a , Anatoly N. Filippov b,  , Adel O. Sharif  c , Victor M. Starov b a Centre for Complex Fluids Processing, Department of Chemical and Biological Process Engineering, Uni    ersity of Wales Swansea, Swansea SA2 8PP, UK  b  Department of Pure and Applied Mathematics, Moscow State Uni    ersity of Food Industry,Volocolamskoye shosse 11, Moscow 125 080, Russia c  Department of Chemical and Process Engineering, School of Engineering in the En   ironment,Uni    ersity of Surrey, Guildford GU2 5XH, Surrey, UK   Abstract  A model of the electrostatic and molecular interactions of a charged colloid particle with acharged membrane surface in an electrolyte solution has been developed. In Derjaguin’sapproximation, the force between a spherical colloid particle and a cylindrical membrane Ž . pore with a rounded inlet is calculated taking into account both electrostatic and van derWaals interactions. The force and energy are strongly dependent on the zeta-potential of both the particle and the membrane pore, the electrolyte concentration, and geometricalparameters. 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 equilibriumposition of the particle above the membrane surface. Therefore, there is a possibility in thiscase of removing the particle by a tangential flow, preventing pore blocking. The model was Ž .  verified using a  Finite Element Method  FEM analysis developed earlier for colloidalinteractions by two co-authors. It has been found that the accuracies of analytical formulaeobtained for the interaction energy and force are within 10 and 20%, respectively, forpractical application ranges of physico  chemical and geometrical parameters. Two major  Corresponding author. Tel.:   7-95-1586849; fax:   7-95-1580371; e-mail: filippov@inmech.msu.su0001-8686  99  $ - see front matter    1999 Elsevier Science B.V. All rights reserved. Ž . PII: S 0 0 0 1 - 8 6 8 6 9 9 0 0 0 0 4 - 4  ( )W.R. Bowen et al.   Ad    . Colloid Interface Sci. 81 1999 35  72 36 Ž . advantages of the model proposed compared to FEM calculations are: 1 the possibility of  Ž non-centerline calculations when a particle is not moving along the axis of a membrane . Ž . pore without a three-dimensional solution; and 2 speed of calculations using the analyticalformulae is much higher. Using a simplified expression for hydrodynamic force, critical values of pressure gradients across the membrane pore have been calculated analytically.   1999 Elsevier Science B.V. All rights reserved.  Keywords:  Charged particle; Membrane pore; Electrostatic and molecular interactions; Electricaldouble layers; DLVO; Finite element method; Critical pressure gradient Contents 1. Introduction  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362. Formulation of the problem  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383. Analytical and numerical analysis of interactions inside the main regions  . . . . . . . . .  413.1. Interaction between charged particle and charged plane  . . . . . . . . . . . . . . . . . 41 Ž . 3.2. Interaction between charged particle and inner charged cylindrical membranesurface  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3. Interaction between charged particles and charged surfaces of the curved poreentrance  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484. Influence of hydrodynamic force on equilibrium position of the particle entering themembrane pore  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595. Conclusions  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656. Nomenclature  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Greek  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Subscripts  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Superscripts  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Acknowledgements  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Appendix A   . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Appendix B  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Appendix C  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71References  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 1. Introduction The interaction of colloidal particles with membranes determines to a greatextent fouling processes and, hence, membrane capacity. That is why it is importantto understand the nature of physicochemical interactions between colloid particlesand membranes. In turn, this allows the possibility of calculating forces betweenparticles and membranes that are important in controlling fouling processes.   Previous works 1,2 have studied the electrostatic interaction of a charged spheri-cal particle with an inner part of a charged cylindrical pore of infinite lengthconsidered in terms of equilibrium partition coefficients using the approximate   linear form of the Poisson  Boltzmann equation. Adler 3 evaluated the interac-tion of van der Waals and EDL energy for a closely fitting sphere in an infinite  ( )W.R. Bowen et al.   Ad    . Colloid Interface Sci. 81 1999 35  72  37 tube in the case when the surface potentials are small. The influence of theelectrochemical conditions on the equilibrium position of the sphere center werealso investigated. The additional pressure drop due to the presence of the sphereand surface forces were calculated.The collection efficiency of pores has also been considered based on a stream-line approach with consideration of hydrodynamic and long-range attractive forces   on the entry to a pore and deposition of an uncharged particle 4 . A furtherdevelopment has been the calculation of the energy of electrostatic and van der   Waals interactions between a particle and a rounded pore entrance 5 , on the Ž .    basis of the Derjaguin  Landau  Verwey  Overbeek DLVO theory 6 , subse-   quently taking into account a simplified hydrodynamic force 7 . The surface andhydrodynamic forces on a particle approaching a conical constriction in a pore   have also been calculated 8 . The purely hydrodynamic interactions in pores areusually expressed in terms of enhanced drag coefficients. An approach using the   finite element technique for such calculations has been developed 9 for the case   of a centerline formulation and has been extended afterwards 10 for a stationaryspherical particle at various distances in the approach and entry to a cylindricalpore in the planar surface. The electrostatic repulsive force due to partial overlap-ping of electrical double layers has also been taken into account. The numerical Ž . procedure for calculation of this force is based on the  finite element method  FEM   and was improved later 11 .   Carnie et al. 12 developed a numerical scheme to calculate the electricaldoublelayer force between two spherical colloidal particles based on the non-linear   Poisson  Boltzmann theory. Carnie et al. 12 used their method for delineating theaccuracy of approximate methods such as the Derjaguin approximation, the super-position approximation, as well as numerical solutions of the problem based on the Ž . linearized Poisson  Boltzmann Debye  Huckel theory. They found the domains of 10% errors for all approximated methods. The Derjaguin approximation in thecase of constant surface potentials gives good results for thin double layers Ž . Ž .    a  2 and large surface separation     h   1 . This suggests that such an 0 approach could be useful in the case of interaction between a colloidal particle anda membrane pore.     Recent work 13 developed the general approach of Smith and Deen 1,2 forthe case of the linearized charge regulation condition for a surface with multiple Ž .    charge residues and  or binding sites that was proposed by Carnie and Chan 14 . Ž .     A new  Surface Element Integration  SEI method was suggested recently 15 todetermine the interaction energy of a spherical particle in a cylindrical pore. Themain idea is to use Derjaguin’s approach locally for small, almost cylindrical piecesof the particle surface and the inner part of the cylindrical cavity and thenintegrate appropriately to obtain the interaction energy. Bhattacharjee and Sharma   15 believe that their technique is accurate for all realistic combinations of particleand pore radii, radial position of the particle as well as decay length of interaction.Later we compare our results for interaction energy with values which can beobtained by SEI.   It should be noted that recently an alternative approximate method 16 has been  ( )W.R. Bowen et al.   Ad    . Colloid Interface Sci. 81 1999 35  72 38 developed for calculation of the interaction energy, which also seeks to removegeometrical restrictions of Derjaguin’s approach by introducing an  Interfacial Stress Ž . Tensor   IST .In the present paper, we present explicit analytical expressions for both theenergy and the force between two interacting charged surfaces: a spherical particleand a membrane pore with a curved entrance. These expressions take into accountthe electrostatic and van der Waals interactions as well as a simplified hydrody-   namic force and are based on our previous studies 5,7 . We then assess theexpressions for energy and force obtained in Derjaguin’s approximation using our   previously developed 9  11 numerical FEM in the case of the centerline approach.The main aims are:1. to search for the limits of reasonable application of the approximated solution;2. to investigate of the influence of non-centerline movement of a particle on theenergy and force of interaction; and Ž . 3. the determination of critical pressure gradients CPG as a function of geomet-rical parameters of the membrane  particle system, ionic strength of the electro-lyte solution and magnitude of zeta potentials of the interacting surfaces. 2. Formulation of the problem  A spherical particle of radius  a  in the vicinity of a membrane pore mouth having Ž . a rounded edge with radius  b  Fig. 1 is considered. We suppose that theinteracting space is filled by an electrolyte solution, and electric potentials of the Ž . Ž . particle     and the membrane surfaces     are constant and known. According  p m to Derjaguin’s approximation, principal radii of curvature of interacting bodies areassumed to be large compared with the radius of action of surface forces. The totalinteraction energy,  U   in this case is a sum of all interaction energies between Ž . Ž . opposite ‘small segments’ of surfaces  S  and  S  Fig. 2 . If   f h  is an interaction 1 2 Ž . energy per unit area and  h x ,  y  is the local distance between surfaces then: Ž Ž .. Ž . U    f h x ,  y  d  x  d  y , 1 HH Ž .  where  x ,  y  are local Cartesian coordinates in the vicinity of the closest distancebetween the particle and the membrane pore surface. The integration limits in Eq. Ž . Ž . 1 can be chosen to be infinite because of the rapid decay of   f h  dependency. Ž . Both surfaces the particle and the pore can be expressed in local coordinates asfollows: 1 1 2 2 Ž . Ž .  z      x       y  ,  i   1,2 2 i i i i i 2 2  ( )W.R. Bowen et al.   Ad    . Colloid Interface Sci. 81 1999 35  72  39 Fig. 1. Scheme of interaction of a spherical particle with a membrane pore:  a , radius of the particle;  b ,radius of the rounding of the pore entrance;  r   , radius of the membrane pore;  R , radius of ‘influence’ of  0 the pore; I, II and III, main regions of interaction.  where    ,     are the principal curvatures of interacting surfaces. After some i i Ž . Ž . transformation Eq. 1 can be rewritten for derivation see Appendix A ,  2   Ž . Ž . U    f h  d  h . 3 H  g   h 0 Surface interaction between charged particles and charged membranes accordingto the DLVO theory includes two different kinds of interaction:1. an electrostatic repulsion caused by the overlapping of electrical double layers Ž . Ž EDL of the particle and the membrane pore it is assumed below that theparticle and the membrane pore surfaces possess a charge of the same sign, . usually negative ; and   2. van der Waals molecular attraction 6 .Let us suppose that the dimensionless surface electric potentials are small,  e     kT   1, so that the linearised version of the planar Poisson  Boltzmann i Ž . Ž . Ž . equation can be used for calculating the electrostatic part of   f h    f h    f h  e m   6 :
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