Abstract
During the particle intrapore diffusion, the particle–pore hydrodynamic interaction enhances the drag on the particle, resulting in a particle diffusivity reduction. For charged particles submerged in an electrolytic solution, the distortion of electrical double layer caused by particle motion creates an additional drag: a phenomenon known as the relaxation effect. Such effect on intrapore diffusivities of rigid spherical particles confined in long cylindrical pores is determined by employing a perturbation scheme involving particle Peclet number. Comparison between present simulation results and previous results from perturbations involving both particle Peclet number and surface charge density indicates that the perturbation involving surface charge density overestimates the electrokinetic retardation. Our results demonstrate that, whereas a particle surface charge density increase always amplifies the excess drag due to relaxation, effects of pore surface charge density are more complicated and dependent on values of particle surface charge density and Debye length. If a highly charged particle is confined in a charged pore, this additional drag increases as a function of Debye length if the Debye length is small, but becomes constant despite the increasing Debye length if the Debye length is comparable or larger than the pore radius. For particles confined in uncharged pores, however, the Debye length increase always amplifies the electrokinetic retardation, causing the particle diffusivity reduction of a highly charged particle in an uncharged pore to be larger than that of a similar particle confined in a charged pore if the Debye length is comparable to or larger than the pore radius.
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References
Adler, P.M.: Influence of colloidal forces on a closely fitting sphere in a fluid-filled tube. Physicochem. Hydrodyn. 4(1), 1–10 (1983)
Agasanapura, B., Baltus, R.E., Tenneru, C.T., Chellam, S.: Effect of electrostatic interactions on rejection of capsular and spherical particles from porous membranes: theory and experiment. J. Colloid Interface Sci. 448, 492–500 (2015). https://doi.org/10.1016/j.jcis.2015.02.016
Ai, Y., Qian, S.: Electrokinetic particle translocation through a nanopore. Phys. Chem. Chem. Phys. 13(9), 4060–4071 (2011). https://doi.org/10.2217/17435889.2.6.875
Akinaga, T., Otani, H., Sugihara-seki, M.: The charge effect on the hindrance factors for diffusion and convection of a solute in pores: II. Fluid Dyn. Res. 44(6), 1–14 (2012). https://doi.org/10.1088/0169-5983/44/6/065504
Bhalla, G., Deen, W.M.: Effects of charge on osmotic reflection coefficients of macromolecules in porous membranes. J. Colloid Interface Sci. 333(1), 363–372 (2009)
Booth, F.: The cataphoresis of spherical, solid non-conducting particles in a symmetrical electrolyte. Proc. R. Soc. Lond. A 203(1075), 514–533 (1950). https://doi.org/10.1098/rspa.1950.0154
Booth, F.: Sedimentation potential and velocity of solid spherical particles. J. Chem. Phys. 22(12), 1956–1968 (1954). https://doi.org/10.1063/1.1739975
Brenner, H., Gaydos, L.J.: The constrained Brownian movement of spherical particles in cylindrical pores of comparable radius. J. Colloid Interface Sci. 58(2), 312–356 (1977). https://doi.org/10.1016/0021-9797(77)90147-3
Bungay, P.M., Brenner, H.: The motion of a closely fitting sphere in a fluid-filled tube. Int. J. Multiph. Flow 1(1), 25–56 (1973)
Chiu, H.C., Keh, H.J.: Electrophoresis of a colloidal sphere with double layer polarization in a microtube. Microfluid. Nanofluid. (2016). https://doi.org/10.1007/s10404-016-1728-z
Chiu, H.C., Keh, H.J.: Diffusiophoresis of a charged particle in a microtube. Electrophoresis 38, 2468–2478 (2017)
Dechadilok, P., Deen, W.M.: Hindrance factors for diffusion and convection in pores. Ind. Eng. Chem. Res. 45(21), 6953–6959 (2006). https://doi.org/10.1021/ie051387n
Dechadilok, P., Deen, W.M.: Electrostatic and electrokinetic effects on hindered diffusion in pores. J. Membr. Sci. 336, 7–16 (2009)
Dechadilok, P., Intum, C., Manaratha, S., Sathanon, U.: Effect of relaxation on drag forces and diffusivities of particles confined in rectangular channels. J. Fluids Eng. 138(12), 121105-1–121105-15 (2016). https://doi.org/10.1115/1.4033914
Deckert, T., Feldt-Rasmussen, B., Djurup, R., Deckert, M.: Glomerular size and charge selectivity in insulin-dependent diabetes mellitus. Kidney Int. 33(1), 100–106 (1998)
Deen, W.M.: Hindered transport of large molecules in liquid-filled pores. AIChE J. 33(9), 1409–1425 (1987). https://doi.org/10.1002/aic.690330902
Einstein, A.: Investigations on the Theory of the Brownian Movement. Dover, New York (1956)
Giddings, J.C., Kucera, E., Russell, C.P., Myers, M.N.: Statistical theory for the equilibrium distribution of rigid molecules in inert porous networks. Exclusion chromatography. J. Phys. Chem. 72, 4397–4408 (1968). https://doi.org/10.1021/j100859a008
Guasch, A., Deen, W.M., Myers, B.D.: Charge selectivity of the glomerular filtration barrier in healthy and nephrotic humans. J. Clin. Investig. 92(5), 2274–2282 (1993)
Happel, P., Brenner, H.: Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Martinus Nijhoff Publishers, The Hague (1983)
Healy, K., Schiedt, B., Morrison, A.P.: Solid-state nanopore technologies for nanopore-based DNA analysis. Nanomedicine 2(6), 875–897 (2007). https://doi.org/10.1039/C0CP02267E
Higdon, J.J.L., Muldowney, G.P.: Resistance functions for spherical particles, droplets and bubbles in cylindrical tubes. J. Fluid Mech. 298, 193–210 (1995)
Howorka, S., Cheley, S., Bayley, H.: Sequence-specific detection of individual DNA strands using engineered nanopores. Nat. Biotech. 19(7), 636–639 (2001)
Hsu, J.P., Liu, B.: Electrical interaction energy between two charged entities in an electrolyte solution. J. Colloid Interface Sci. 217(2), 219–236 (1999). https://doi.org/10.1016/S0927-7757(00)00819-0
Hsu, J.P., Chen, Z.S.: Electrophoresis of a sphere along the axis of a cylindrical pore: effects of double-layer polarization and electroosmotic flow. Langmuir 23(11), 6198–6204 (2007). https://doi.org/10.1021/la070079m
Hsu, J.P., Hsu, W.L., Chen, Z.S.: Boundary effect on diffusiophoresis: spherical particle in a spherical cavity. Languir 25(3), 1772–1784 (2009). https://doi.org/10.1021/la803334a
Hsu, J.P., Ko, I.F., Tseng, S.: Importance of boundary effect on the diffusiophoretic behavior of a charged particle in an electrolyte medium. J. Phys. Chem. C 116(7), 4455–4464 (2012). https://doi.org/10.1021/jp210927k
Keh, H.J., Ding, J.M.: Sedimentation velocity and potential in concentrated suspensions of charged spheres with arbitrary double-layer thickness. J. Colloid Interface Sci. 227(2), 540–552 (2000). https://doi.org/10.1006/jcis.2000.6918
Keh, H.J., Hsieh, T.H.: Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions. Langmuir 23(15), 7928–7935 (2007). https://doi.org/10.1021/la7004002
Keh, H.J., Cheng, T.F.: Sedimentation of a charged colloidal sphere in a charged cavity. J. Chem. Phys. 135(21), 214706 (2011). https://doi.org/10.1063/1.3663380
Lee, E., Yen, C.-B., Hsu, J.-P.: Sedimentation of a nonconducting sphere in a spherical cavity. J. Phys. Chem. B 104(29), 6815–6820 (2000). https://doi.org/10.1021/jp993706s
Lee, S.Y., Yalcin, S.E., Joo, S.W., Baysal, O., Qian, S.: Diffusiophoretic motion of a charged spherical particle in a nanopore. J. Phys. Chem. B 114(19), 6437–6446 (2010). https://doi.org/10.1021/jp9114207
O’Brien, R.W., White, L.R.: Electrophoretic mobility of a spherical colloidal particle. J. Chem. Soc. Faraday Trans. 2 74, 1607–1626 (1978). https://doi.org/10.1039/F29787401607
Ohsima, H.: Electrophoretic mobility of a highly charged soft particle: relaxation effect. Colloids Surf. A Physicochem. Eng. Asp. 36(1–3), 72–75 (2011). https://doi.org/10.1016/j.colsurfa.2010.09.012
Ohshima, H.: Simple approximate analytic expression for the electrophoretic mobility of a spherical colloidal particle in a mixed solution of 1:1 and 2:1 electrolytes. Colloid Polym. Sci. 292(6), 1457–1461 (2014). https://doi.org/10.1007/s00396-014-3193-0
Ohshima, H., Healy, T.W., White, L.R., O’Brien, R.W.: Sedimentation velocity and potential in a dilute suspension of charged spherical colloidal particles. J. Chem. Soc. Faraday Trans. 2 Mol. Chem. Phys. 80(10), 1299–1317 (1984). https://doi.org/10.1039/F29848001299
Otani, H., Akinaga, T., Sugihara-seki, M.: The charge effect on the hindrance factors for diffusion and convection of a solute in pores: I. Fluid Dyn. Res. 43(6), 1–12 (2011). https://doi.org/10.1088/0169-5983/43/6/065505
Pujar, N.S., Zydney, A.L.: Electrostatic and electrokinetic interactions during protein transport through narrow pore membranes. Ind. Eng. Chem. Res. 33, 2473–2482 (1994). https://doi.org/10.1021/ie00034a032
Pujar, N.S., Zydney, A.L.: Boundary effects on the sedimentation and hindered diffusion of charged particles. AIChE J. 42(8), 2101–2111 (1996). https://doi.org/10.1002/aic.690420802
Pujar, N.S., Zydney, A.L.: Charge regulation and electrostatic interactions for a spherical particle in a cylindrical pore. J. Colloid Interface Sci. 192(2), 338–349 (1997). https://doi.org/10.1006/jcis.1997.5017
Reiner, E.S., Radke, C.J.: Electrostatic interactions in colloidal suspensions: test of pairwise additivity. AIChE J. 37(6), 805–824 (1991)
Rohani, M.M., Zydney, A.L.: Role of electrostatic interactions during protein ultrafiltration. Adv. Colloid Interface Sci. 160, 40–48 (2010). https://doi.org/10.1016/j.cis2010.07.02
Sharp, K.A., Honig, B.: Calculating total electrostatic energies with the nonlinear Poisson–Boltzmann equation. J. Phys. Chem. 94, 7684–7692 (1990)
Smith, F.G., Deen, W.M.: Electrostatic double-layer interactions for spherical colloids in cylindrical pores. J. Colloid Interface Sci. 78(2), 444–465 (1980)
Smith, F.G., Deen, W.M.: Electrostatic effects on the partition of spherical colloids between dilute bulk solution and cylindrical pores. J. Colloid Interface Sci. 91(2), 571–590 (1983). https://doi.org/10.1016/0021-9797(83)90371-5
Stigter, D.: Sedimentation of highly charged colloidal spheres. J. Phys. Chem. 84(21), 2758–2762 (1980). https://doi.org/10.1021/j100458a018
Van de Ven, T.G.M.: Colloidal Hydrodynamics. Academic Press, San Diego (1989)
Valino, V.M., San Roman, F., Ibanez, R., Ortiz, I.: Improved separation of bovine serum albumin and lactoferrin mixtures using charged ultrafiltration membranes. Sep. Purif. Technol. 125, 163–169 (2014)
Verwey, E.J.W., Overbeek, J.T.: Theory of the Stability of Lyophobic Colloids. Elsevier Publishing Company. Inc., Amsterdam (1948)
Vilker, V.L., Colton, C.K., Smith, K.A.: The osmotic pressure of concentrated protein solutions: effect of concentration and pH in saline solutions of bovine serum albumin. J. Colloid Interface Sci. 79(2), 548–566 (1981). https://doi.org/10.1016/0021-9797(81)9016-5
Yalcin, S.E., Lee, S.Y., Joo, S.W., Baysal, O., Qian, S.: Electrodiffusiophoretic motion of a charged spherical particles in a nanopore. J. Phys. Chem. B 114(11), 4082–4093 (2010). https://doi.org/10.1021/jp100784p
Zeman, L.J., Zydney, A.L.: Microfiltration and Ultrafiltration: Principles and Applications. Marcel Dekker, New York (1996)
Acknowledgements
K. Yooprasertchuti was supported by the Development and Promotion of Science and Technology Talents Project, The Institute for the Promotion of Teaching Science and Technology, Thailand. This work was funded by Ministry of Science and Technology of Thailand (Grant No. SCH-NR-2009-09-01) and Faculty of Science, Chulalongkorn University.
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contains an estimation of the Reynolds number characterizing the system of charged particles diffusing through long pores, the calculation of the electrostatic potential energy of interaction and the expansion of field variables using the surface charge density (PDF 403 kb)
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Yooprasertchuti, K., Dechadilok, P. Relaxation Effect on Intrapore Diffusivities of Highly Charged Colloidal Particles Confined in Porous Membranes. Transp Porous Med 123, 341–366 (2018). https://doi.org/10.1007/s11242-018-1046-x
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DOI: https://doi.org/10.1007/s11242-018-1046-x