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An electrokinetic (zeta) potential of charged permeable porous films on solid supports generally exceeds their surface potential, which often builds up to a quite high value itself. Recent work provided a quantitative understanding of zeta potentials of thick, compared to the extension of an inner electrostatic diffuse layer, porous films. Here, we consider porous coatings of a thickness comparable or smaller than that of the inner diffuse layer. Our theory, which is valid even when electrostatic potentials become quite high and accounts for a finite hydrodynamic permeability of the porous materials, provides a framework for interpreting the difference between values of surface and zeta potentials in various situations. Analytic approximations for the zeta potential in the experimentally relevant limits provide a simple explanation of transitions between different regimes of electro-osmotic flows, and also suggest strategies for its tuning in microfluidic applications.

We discuss an electro-osmotic flow near charged porous coatings of a finite hydrodynamic permeability, impregnated with an outer electrolyte solution. It is shown that their electrokinetic (zeta) potential is generally augmented compared to the surface electrostatic potential, thanks to a large liquid slip at their surface emerging due to an electro-osmotic flow in the enriched by counter-ion porous films. The inner flow shows a very rich behavior controlled by the volume charge density of the coating, its Brinkman length, and the concentration of added salt. Interestingly, even for a relatively small Brinkman length, the zeta-potential can, in some cases, become huge, providing a very fast outer flow in the bulk electrolyte. When the Brinkman length is large enough, the zeta-potential could be extremely high, even at practically vanishing surface potential. To describe the slip velocity in a simple manner, we introduce a concept of an electro-osmotic slip length and demonstrate that the latter is always defined by the hydrodynamic permeability of the porous film and also, depending on the regime, either by its volume charge density or by the salt concentration. These results provide a framework for the rational design of porous coatings to enhance electrokinetic phenomena, and for tuning their properties by adjusting bulk electrolyte concentrations, with direct applications in microfluidics.

Permeable porous coatings on a flat solid support significantly impact its electrostatic and electrokinetic properties. Existing work has focused on simplified cases, such as weakly charged and/or thick porous films, with limited theoretical guidance. Here, we consider the general case of coatings of any given volume charge density and obtain analytic formulas for electrostatic potential profiles, valid for any film thickness and salt concentration. They allow us to calculate analytically the difference between potentials at solid support and at interface with an outer electrolyte, that is the key parameter ascertaining the functionality of permeable coatings. Our analysis provides a framework for interpreting and predicting specific for porous films super-properties, from an enhanced ion absorption to a giant amplification of electro-osmotic flows. The results are relevant for hydrogel and zeolite coatings, porous carbon and ion-exchange resins, polyelectrolyte brushes, and more.

Based on the Nernst–Planck model for ion transport, the electroosmotic flow of a non-Newtonian fluid near a surface potential heterogeneity is studied numerically. The objectives of this study are to highlight the limitations of the linear slip-model and the nonlinear Poisson–Boltzmann model at various flow conditions as well as to develop vortical flow to promote mixing of neutral solutes within the micro-channel. A power-law fluid, both shear-thinning and shear-thickening, for the pseudoplastic behavior of the non-Newtonian fluid or viscoplastic fluid with yield stress is adopted to describe the transport of electrolyte, which is coupled with the ion transport equations governed by the Nernst–Planck equations and the Poisson equation for electric field. The viscoplastic fluid is modeled as either Casson, Bingham or Hershel–Buckley fluid. A pressure-correction based control volume approach has been adopted for the numerical computations of the governing equations. The nonlinear effects are found to be pronounced for a shear thinning liquid, whereas, the electroosmotic flow is dominated by the diffusion mechanisms for the shear thickening liquid. A maximum difference of 39% between the existing analytic solution based on the Debye–H u ¨ ckel approximation and the present numerical model is found for a shear thinning power-law fluid. A vortex, which resembles a Lamb vortex, develops over the potential patch when the patch potential is of opposite sign to that of the homogeneous surface potential. Enhanced mixing of a neutral solute is also analyzed in the present analysis. The yield stress reduces the electroosmotic flow however, promotes solute mixing.

This article deals with the modulation of electroosmotic flow (EOF) and transport of ionic species through the parallel plate soft nanochannel. The charged rigid walls of the channel are covered by diffuse polyelectrolyte layer (PEL) which entraps immobile charges. A diffuse distribution of the polymer segment density and charge density is assumed. A nonlinear model based on the Poisson-Nernst-Planck equations coupled with the Darcy-Brinkman equations is adopted. Going beyond the widely employed Debye-H $\ddot {u}$ ckel linearization, we adopt a sophisticated numerical tool to study the effect of pertinent parameters on the modulation of EOF through the soft nanochannel. Several interesting key features including the flow reversal, occurrence of zero flow rate, and perm selectivity are studied by regulating the charges entrapped within the diffuse PEL and the surface charge distributed along the channel wall. The results indicate that the channel can be cation-selective, anion-selective, and non-selective based on the nature of the charges within the PEL and wall charge. We have also identified the parameter range for validity of the linearized model for the case of step-like PEL.

An enhanced electroosmotic flow through a surface-modulated microchannel is considered. The microchannel is modulated by periodically arranged rectangular grooves filled with polyelectrolyte materials. The flat surface of the walls is maintained at a constant charge density. A nonlinear model based on the Poisson–Nernst–Planck equations coupled with the Darcy–Brinkman–Forchheimer equation in the polyelectrolyte region and Navier–Stokes equations in the clear fluid region is adopted. Going beyond the widely employed Debye–Huckel linearization, we adopt numerical methods to elucidate the effect of pertinent parameters on electroosmosis in the patterned channel. The patterned microchannel results in an enhancement in the average EOF by creating an intrinsic velocity slip at the polyelectrolyte–liquid interface. An analytical solution of the EOF for a limiting case in which the groove width is much higher than the channel height is obtained based on the Debye–Huckel approximation. This analytical solution is in good agreement with the present numerical model when a low charge density and a thin Debye layer are considered. We have also established an analogy between the EOF in a polyelectrolyte-filled grooved-channel with the EOF in which the grooves are replaced by the charged slipping planes.

In this paper, we have analyzed an enhanced electroosmotic flow (EOF) by geometric modulation of the surface of a charged nanochannel. Otherwise, flat walls of the channel are modulated by embedding rectangular grooves placed perpendicular to the direction of the applied electric field in a periodic manner. The modulated channel is filled with a single electrolyte. The EOF within the modulated channel is determined by computing the Navier–Stokes–Nernst–Planck–Poisson equations for a wide range of Debye length. The objective of the present study is to achieve an enhanced EOF in the surface modulated channel. A significant enhancement in average EOF is found for a particular arrangement of grooves with the width of the grooves much higher than its depth and the Debye length is in the order of the channel height. However, the formation of vortex inside the narrow grooves can reduce the EOF when the groove depth is in the order of its width. Results are compared with the cases in which the grooves are replaced by superhydrophobic patches along which a zero shear stress condition is imposed.

In this study, we have analyzed the transport of analytes under a two dimensional steady incompressible flow of power-law fluids through rectangular nanochannel. A mathematical model based on the Cauchy momentum-Nernst-Planck-Poisson equations is considered to study the combined effect of mixed electroosmotic (EO) and pressure driven (PD) flow. The coupled governing equations are solved numerically by finite volume method. We have studied extensively the effect of key parameters, e.g., flow behavior index, concentration of the electrolyte, surface potential, imposed pressure gradient and imposed electric field strength on the net average flow across the channel. In addition to study the effect of mixed EOF and PD on the analyte distribution across the channel, we consider a nonlinear model based on general convective-diffusion-electromigration equation. We have also presented the retention factor for various values of electrolyte concentration and flow behavior index., {"references":["R. J. Hunter, Foundations of colloid science, Oxford University Press,\n2001.","R. F. Probstein, Physicochemical hydrodynamics, Wiley, 1994.","J. H. Masliyah, S. Bhattacharjee, Electrokinetic and colloid transport\nphenomena, John Wiley & Sons, 2006.","H. A. Stone, A. D. Stroock, A. Ajdari, Engineering flows in small\ndevices: microfluidics toward a lab-on-a-chip, Annu. Rev. Fluid Mech.\n36 (2004) 381–411.","X. Wang, C. Cheng, S. Wang, S. Liu, Electroosmotic pumps and their\napplications in microfluidic systems, Microfluidics and Nanofluidics\n6 (2) (2009) 145–162.","F. Kamis¸li, Flow analysis of a power-law fluid confined in an\nextrusion die, International journal of engineering science 41 (10) (2003)\n1059–1083.","W. Zimmerman, J. Rees, T. Craven, Rheometry of non-newtonian\nelectrokinetic flow in a microchannel t-junction, Microfluidics and\nNanofluidics 2 (6) (2006) 481–492.","M. Das, V. Jain, P. Ghoshdastidar, Fluid flow analysis of\nmagnetorheological abrasive flow finishing (mraff) process, International\nJournal of Machine Tools and Manufacture 48 (3) (2008) 415–426.","Y. Koh, N. Ong, X. Chen, Y. Lam, J. Chai, Effect of temperature\nand inlet velocity on the flow of a nonnewtonian fluid, International\ncommunications in heat and mass transfer 31 (7) (2004) 1005–1013.\n[10] A. Y. Malkin, Rheology Fundamentals, ChemTec, 1994.\n[11] S. Das, S. Chakraborty, Analytical solutions for velocity, temperature\nand concentration distribution in electroosmotic microchannel flows of a\nnon-Newtonian bio-fluid, Analytica Chimica Acta 559 (1) (2006) 15–24.\n[12] C. Zhao, E. Zholkovskij, J. H. Masliyah, C. Yang, Analysis of\nelectroosmotic flow of power-law fluids in a slit microchannel, Journal\nof colloid and interface science 326 (2) (2008) 503–510.\n[13] C. Rice, R. Whitehead, Electrokinetic flow in a narrow cylindrical\ncapillary, The Journal of Physical Chemistry 69 (11) (1965) 4017–4024.\n[14] G. Tang, X. Li, Y. He, W. Tao, Electroosmotic flow of non-Newtonian\nfluid in microchannels, Journal of Non-Newtonian Fluid Mechanics\n157 (1) (2009) 133–137.\n[15] N. Vasu, S. De, Electroosmotic flow of power-law fluids at high zeta\npotentials, Colloids and Surfaces A: Physicochemical and Engineering\nAspects 368 (1) (2010) 44–52.\n[16] A. Babaie, A. Sadeghi, M. H. Saidi, Combined electroosmotically and\npressure driven flow of power-law fluids in a slit microchannel, Journal\nof Non-Newtonian Fluid Mechanics 166 (14) (2011) 792–798.\n[17] S. Pennathur, J. G. Santiago, Electrokinetic transport in nanochannels.\n1. theory, Analytical chemistry 77 (21) (2005) 6772–6781.\n[18] S. K. Griffiths, R. H. Nilson, Electroosmotic fluid motion and late-time\nsolute transport for large zeta potentials, Analytical chemistry 72 (20)\n(2000) 4767–4777.\n[19] X. Xuan, D. Li, Solute separation in nanofluidic channels:\nPressure-driven or electric field-driven?, Electrophoresis 28 (4) (2007)\n627–634.\n[20] C. A. Fletcher, Computational techniques for fluid dynamics vol 2, 2nd\nedn, Springer, Berlin, 1991.\n[21] S. Patankar, Numerical heat transfer and fluid flow, CRC Press, 1980.\n[22] D. Gillespie, S. Pennathur, Separation of ions in nanofluidic channels\nwith combined pressure-driven and electro-osmotic flow, Analytical\nchemistry 85 (5) (2013) 2991–2998."]}

In this paper, we consider the electroosmotic flow (EOF) of a viscoplastic fluid within a slit nanochannel modulated by periodically arranged uncharged slipping surfaces and no-slip charged surfaces embedded on the channel walls. The objective of the present study is to achieve an enhanced EOF of a non-Newtonian yield stress fluid. The Herschel-Bulkley model is adopted to describe the transport of the non-Newtonian electrolyte, which is coupled with the ion transport equations governed by the Nernst-Planck equations and the Poisson equation for electric field. A pressure-correction-based control volume approach is adopted for the numerical computation of the governing nonlinear equations. We have derived an analytic solution for the power-law fluid when the periodic length is much higher than channel height with uncharged free-slip patches. An agreement of our numerical results under limiting conditions with this analytic model is encouraging. A significant EOF enhancement and current density in this modulated channel are achieved when the Debye length is in the order of the nanochannel height. Flow enhancement in the modulated channel is higher for the yield stress fluid compared with the power-law fluid. Unyielded region develops adjacent to the uncharged slipping patches, and this region expands as slip length is increased. The impact of the boundary slip is significant for the shear thinning fluid. The results indicate that the channel can be cation selective and nonselective based on the Debye layer thickness, flow behavior index, yield stress, and planform length of the slip stripes.