Enhanced electroosmotic flow and ion selectivity in a channel patterned with periodically arranged polyelectrolyte-filled grooves

  • S. BhattacharyyaEmail author
  • Naren Bag
Research Paper


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–Hückel 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–Hückel 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.


Electroosmotic flow Polyelectrolyte-infused grooves Hydrophobic surfaces Ion selectivity 


Supplementary material

10404_2019_2213_MOESM1_ESM.pdf (64 kb)
Supplementary material 1 (pdf 63 KB)


  1. Alexander S (1977) Polymer adsorption on small spheres. a scaling approach. J Phys 38(8):977–981CrossRefGoogle Scholar
  2. Asmolov ES, Nizkaya TV, Vinogradova OI (2018) Enhanced slip properties of lubricant-infused grooves. Phys Rev E 98(3):033103CrossRefGoogle Scholar
  3. Bag N, Bhattacharyya S, Gopmandal PP, Ohshima H (2018) Electroosmotic flow reversal and ion selectivity in a soft nanochannel. Colloid Polym Sci 296(5):849–859CrossRefGoogle Scholar
  4. Bahga SS, Vinogradova OI, Bazant MZ (2010) Anisotropic electro-osmotic flow over super-hydrophobic surfaces. J Fluid Mech 644:245–255CrossRefGoogle Scholar
  5. Belyaev AV, Vinogradova OI (2011) Electro-osmosis on anisotropic superhydrophobic surfaces. Phys Rev Lett 107(9):098301CrossRefGoogle Scholar
  6. Bhattacharyya S, Bag N (2017) Enhanced electroosmotic flow through a nanochannel patterned with transverse periodic grooves. J Fluids Eng 139(8):081203CrossRefGoogle Scholar
  7. Bhattacharyya S, Pal S (2018) Enhanced electroosmotic flow in a nano-channel patterned with curved hydrophobic strips. Appl Math Model 54:567–579MathSciNetCrossRefGoogle Scholar
  8. Bhattacharyya S, Dhinakaran S, Khalili A (2006) Fluid motion around and through a porous cylinder. Chem Eng Sci 61(13):4451–4461CrossRefGoogle Scholar
  9. Busse A, Sandham ND, McHale G, Newton MI (2013) Change in drag, apparent slip and optimum air layer thickness for laminar flow over an idealised superhydrophobic surface. J Fluid Mech 727:488–508MathSciNetCrossRefGoogle Scholar
  10. Chanda S, Sinha S, Das S (2014) Streaming potential and electroviscous effects in soft nanochannels: towards designing more efficient nanofluidic electrochemomechanical energy converters. Soft Matter 10(38):7558–7568CrossRefGoogle Scholar
  11. Chen G, Das S (2015) Streaming potential and electroviscous effects in soft nanochannels beyond Debye–Hückel linearization. J Colloid Interface Sci 445:357–363CrossRefGoogle Scholar
  12. Chen G, Das S (2017) Massively enhanced electroosmotic transport in nanochannels grafted with end-charged polyelectrolyte brushes. J Phys Chem B 121(14):3130–3141CrossRefGoogle Scholar
  13. Coster HG (1973) The double fixed charge membrane. Biophys J 13(2):118–132CrossRefGoogle Scholar
  14. De Gennes P (1976) Scaling theory of polymer adsorption. J Phys 37(12):1445–1452CrossRefGoogle Scholar
  15. De Gennes P (1980) Conformations of polymers attached to an interface. Macromolecules 13(5):1069–1075CrossRefGoogle Scholar
  16. DeGroot C, Wang C, Floryan J (2016) Drag reduction due to streamwise grooves in turbulent channel flow. J Fluids Eng 138(12):121201CrossRefGoogle Scholar
  17. Duval JF (2005) Electrokinetics of diffuse soft interfaces. 2. Analysis based on the nonlinear Poisson–Boltzmann equation. Langmuir 21(8):3247–3258CrossRefGoogle Scholar
  18. Duval JF, van Leeuwen HP (2004) Electrokinetics of diffuse soft interfaces. 1. Limit of low donnan potentials. Langmuir 20(23):10324–10336CrossRefGoogle Scholar
  19. Duval JF, Zimmermann R, Cordeiro AL, Rein N, Werner C (2009) Electrokinetics of diffuse soft interfaces. IV. Analysis of streaming current measurements at thermoresponsive thin films. Langmuir 25(18):10691–10703CrossRefGoogle Scholar
  20. Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94Google Scholar
  21. Feuillebois F, Bazant MZ, Vinogradova OI (2010) Transverse flow in thin superhydrophobic channels. Phys Rev E 82(5):055301CrossRefGoogle Scholar
  22. Fletcher CA (1991) Computational techniques for fluid dynamics, vol 2, 2nd edn. Springer, BerlinzbMATHGoogle Scholar
  23. Hill RJ, Saville D (2005) Exact solutions of the full electrokinetic model for soft spherical colloids: electrophoretic mobility. Colloids Surf A 267(1–3):31–49CrossRefGoogle Scholar
  24. Hill RJ, Saville D, Russel W (2003) Electrophoresis of spherical polymer-coated colloidal particles. J Colloid Interface Sci 258(1):56–74CrossRefGoogle Scholar
  25. Hsu C, Cheng P (1990) Thermal dispersion in a porous medium. Int J Heat Mass Transf 33(8):1587–1597CrossRefGoogle Scholar
  26. Israelachvili JN (2011) Intermolecular and surface forces. Academic Press, LondonGoogle Scholar
  27. Kim SJ, Ko SH, Kang KH, Han J (2010) Direct seawater desalination by ion concentration polarization. Nat Nanotechnol 5(4):297CrossRefGoogle Scholar
  28. Lauga E, Stone HA (2003) Effective slip in pressure-driven Stokes flow. J Fluid Mech 489(6):55–77MathSciNetCrossRefGoogle Scholar
  29. Lauga E, Brenner MP, Stone HA (2007) Microfluidics: the no-slip boundary condition. In: Foss J, Tropea C, Yarin A (eds) Handbook of experimental fluid mechanics. Springer, BerlinGoogle Scholar
  30. Lee C, Choi C-H et al (2008) Structured surfaces for a giant liquid slip. Phys Rev Lett 101(6):064501CrossRefGoogle Scholar
  31. Lee C, Choi C-H, Kim C-J (2016) Superhydrophobic drag reduction in laminar flows: a critical review. Exp Fluids 57(12):176CrossRefGoogle Scholar
  32. Leonard BP (1979) A stable and accurate convective modelling procedure based on quadratic upstream interpolation. Comput Methods Appl Mech Eng 19(1):59–98CrossRefGoogle Scholar
  33. López-Garcıa JJ, Horno J, Grosse C (2003) Suspended particles surrounded by an inhomogeneously charged permeable membrane. Solution of the Poisson—Boltzmann equation by means of the network method. J Colloid Interface Sci 268(2):371–379CrossRefGoogle Scholar
  34. Maduar S, Belyaev A, Lobaskin V, Vinogradova O (2015) Electrohydrodynamics near hydrophobic surfaces. Phys Rev Lett 114(11):118301CrossRefGoogle Scholar
  35. Matin MH, Ohshima H (2015) Combined electroosmotically and pressure driven flow in soft nanofluidics. J Colloid Interface Sci 460:361–369CrossRefGoogle Scholar
  36. Matin MH, Ohshima H (2016) Thermal transport characteristics of combined electroosmotic and pressure driven flow in soft nanofluidics. J Colloid Interface Sci 476:167–176CrossRefGoogle Scholar
  37. Mohammadi A, Floryan J (2015) Numerical analysis of laminar-drag-reducing grooves. J Fluids Eng 137(4):041201CrossRefGoogle Scholar
  38. Ng C-O, Chu HC (2011) Electrokinetic flows through a parallel-plate channel with slipping stripes on walls. Phys Fluids 23(10):102002CrossRefGoogle Scholar
  39. Papadopoulos P, Deng X, Vollmer D, Butt H-J (2012) Electrokinetics on superhydrophobic surfaces. J Phys Condens Matter 24(46):464110CrossRefGoogle Scholar
  40. Patankar S (1980) Numerical heat transfer and fluid flow. CRC Press, Boca RatonzbMATHGoogle Scholar
  41. Poddar A, Maity D, Bandopadhyay A, Chakraborty S (2016) Electrokinetics in polyelectrolyte grafted nanofluidic channels modulated by the ion partitioning effect. Soft Matter 12(27):5968–5978CrossRefGoogle Scholar
  42. Schäffel D, Koynov K, Vollmer D, Butt H-J, Schönecker C (2016) Local flow field and slip length of superhydrophobic surfaces. Phys Rev Lett 116(13):134501CrossRefGoogle Scholar
  43. Schönecker C, Hardt S (2013) Longitudinal and transverse flow over a cavity containing a second immiscible fluid. J Fluid Mech 717:376–394MathSciNetCrossRefGoogle Scholar
  44. Schönecker C, Baier T, Hardt S (2014) Influence of the enclosed fluid on the flow over a microstructured surface in the cassie state. J Fluid Mech 740:168–195MathSciNetCrossRefGoogle Scholar
  45. Squires TM (2008) Electrokinetic flows over inhomogeneously slipping surfaces. Phys Fluids (1994–Present) 20(9):092105CrossRefGoogle Scholar
  46. Steffes C, Baier T, Hardt S (2011) Enabling the enhancement of electroosmotic flow over superhydrophobic surfaces by induced charges. Colloids Surf A 376(1–3):85–88CrossRefGoogle Scholar
  47. Stein D, Kruithof M, Dekker C (2004) Surface-charge-governed ion transport in nanofluidic channels. Phys Rev Lett 93(3):035901CrossRefGoogle Scholar
  48. Tandon V, Bhagavatula SK, Nelson WC, Kirby BJ (2008) Zeta potential and electroosmotic mobility in microfluidic devices fabricated from hydrophobic polymers: 1. The origins of charge. Electrophoresis 29(5):1092–1101CrossRefGoogle Scholar
  49. Yang M, Yang X, Wang K, Wang Q, Fan X, Liu W, Liu X, Liu J, Huang J (2015) Tuning transport selectivity of ionic species by phosphoric acid gradient in positively charged nanochannel membranes. Anal Chem 87(3):1544–1551CrossRefGoogle Scholar
  50. Yaroshchuk AE (2000) Dielectric exclusion of ions from membranes. Adv Colloid Interface Sci 85(2–3):193–230CrossRefGoogle Scholar
  51. Yeh L-H, Zhang M, Hu N, Joo SW, Qian S, Hsu J-P (2012a) Electrokinetic ion and fluid transport in nanopores functionalized by polyelectrolyte brushes. Nanoscale 4(16):5169–5177CrossRefGoogle Scholar
  52. Yeh L-H, Zhang M, Qian S, Hsu J-P (2012b) Regulating DNA translocation through functionalized soft nanopores. Nanoscale 4(8):2685–2693CrossRefGoogle Scholar
  53. Yeh L-H, Zhang M, Qian S, Hsu J-P, Tseng S (2012c) Ion concentration polarization in polyelectrolyte-modified nanopores. J Phys Chem C 116(15):8672–8677CrossRefGoogle Scholar
  54. Yusko EC, An R, Mayer M (2009) Electroosmotic flow can generate ion current rectification in nano-and micropores. ACS Nano 4(1):477–487CrossRefGoogle Scholar
  55. Zhang XH, Zhang XD, Lou ST, Zhang ZX, Sun JL, Hu J (2004) Degassing and temperature effects on the formation of nanobubbles at the mica/water interface. Langmuir 20(9):3813–3815CrossRefGoogle Scholar
  56. Zhang X, Wang L, Levänen E (2013) Superhydrophobic surfaces for the reduction of bacterial adhesion. RSC Adv 3(30):12003–12020CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

Personalised recommendations