Helmholtz Smoluchowski Research Articles

Ion steric effects on electrophoresis of a colloidal particle

We calculate the electrophoretic mobility Me of a spherical colloidal particle, using modified Poisson–Nernst–Planck (PNP) equations that account for steric repulsion between finite sized ions, through Bikerman's mean-field model (Bikerman, Phil. Mag., vol. 33, 1942, p. 384). Ion steric effects are controlled by the bulk volume fraction of ions ν, and for ν = 0 the standard PNP equations are recovered. An asymptotic analysis in the thin-double-layer limit reveals at small zeta potentials (ζ < kBT/e ≈ 25 mV) Me to increase linearly with ζ for all ν, as expected from the Helmholtz–Smoluchowski (HS) formula. For larger ζ, however, it is well known that surface conduction of ions within the double layer reduces Me below the HS result. Crucially, however, in the PNP equations surface conduction becomes significant precisely because of the aphysically large and unbounded counter-ion densities predicted at large ζ. In contrast, ion steric effects impose a limit on the counter-ion density, thereby mitigating surface conduction. Hence, Me does not fall as far below HS for finite sized ions (ν ≠ 0). Indeed, at sufficiently large ν, ion steric effects are so dramatic that a maximum in Me is not observed for physically reasonable values of ζ(≤ 10 kBT/e), in stark contrast to the PNP-based calculations of O'Brien & White (J. Chem. Soc. Faraday Trans. II, vol. 74, 1978, p. 1607) and O'Brien (J. Colloid Interface Sci., vol. 92, 1983, p. 204). Finally, by calculating a Dukhin–Bikerman number characterizing the relative importance of surface conduction, we collapse Me versus ζ data for different ν onto a single master curve.

Estimation of zeta potentials of titania nanoparticles by molecular simulation

Non-equilibrium molecular dynamics (NEMD) simulations have been performed for static electric fields for a range of positively charged spherical rutile–titania nanoparticles with radii of 1.5 to 2.9 nm for two different salt concentrations in water, in order to simulate electrophoresis directly. Using the observed limiting drag velocities, Helmholtz–Smoluchowski (HS) theory was used to estimate their ζ potentials. These estimates were compared to values from numerical solution of the non-linear Poisson–Boltzmann (PB) equation for representative configurations of the nanoparticles, in addition to idealised analytic and Debye–Hückel (DH) solutions about spherical particles of the same geometry and charge state, for the given salt concentrations. It was found that reasonable agreement was obtained between the various approaches, with the NEMD-HS results some 15%–15% smaller than the numerical PB results for more highly charged nanoparticles.

Electroosmotic flow in micro/nanochannels with surface potential heterogeneity: An analysis through the Nernst–Planck model with convection effect

A complete two-dimensional model is discussed to analyze the electroosmotic flow near a step jump in surface charge distribution of a micro- or nanochannel. The present model does not require the core neutrality of the fluid and it can handle electrolyte of arbitrary ionic valence as well. The governing equations consists of the flow field, mass transfer and electric potential equations. A pressure correction-based iterative algorithm (SIMPLE) is employed for computation. An analytic solution is obtained near the surface heterogeneity based on the Debye–Hückel approximation. The EOF is investigated both for weak and strong electrolytes at different channel heights. The form of the vortical flow near the potential patch and the dependence of the vortex strength on the patch overpotential is analyzed. A linear pressure drop is observed above the vortex. Average streamwise velocity overshoots the Helmholtz–Smoluchowski velocity. A comparison of the present model with the model based on the equilibrium Boltzmann distribution of ions is made. The results due to the two models differ significantly in the region where the convection effect is strong.

Depletion of high molecular weight dextran from the red cell surface measured by particle electrophoresis

The reversible aggregation of human red blood cells (RBC) by proteins or polymers continues to be of biological and biophysical interest, yet the mechanistic details governing the process are still being explored. A depletion model has been proposed for aggregation by the neutral polyglucose dextran and its applicability at high molecular weights has been recently documented. In the present study the depletion of high molecular weight dextrans on the red cell surface was measured as a function of polymer molecular mass (40 kDa-28 MDa), ionic strength (5 and 15 mM NaCl) and polymer concentration (< or =0.9 g/dL). The experimental data clearly indicate an increasing depletion effect with increasing molecular weight: the effects of medium viscosity on RBC mobility were markedly overestimated by the Helmholtz-Smoluchowski relation, with the difference increasing with dextran molecular mass. These results agree with the concept of polymer depletion near the RBC surface and lend strong support to a "depletion model" mechanism for dextran-mediated RBC aggregation. Our findings provide important new insight into polymer-RBC interactions and suggest the usefulness of this model for fundamental studies of cell-cell affinity and for the development of new agents to stabilize or destabilize specific bio-fluids.

Electrokinetic Phenomena of Modified Polytetrafluoroethylene Membranes in the Oily Sewage from Oil Field

Experimental investigations of electrokinetic phenomena of modified polytetrafluoroethylene membranes in the oily sewage from oil field were performed by using the streaming potential method. The zeta potentials of the membranes in the oily waste water are estimated on the basis of Helmholtz—Smoluchowski equation. The experiment and calculation results show that the membranes are charged negatively, whose zeta poten-tials maintain at around -20mV. And the aperture of membranes, the temperature and the filtration flux have little influence on the streaming potentials and the zeta potentials of the membranes．Also the suspended particulates in the oily sewage are charged negatively. The membranes have strong ability to withhold the suspended substance and powerful antipollution competence because of the role of the charges on the mem-branes.

High-ionic-strength electroosmotic flows in uncharged hydrophobic nanochannels

We report molecular dynamics simulation results of high-ionic-strength electroosmotic flows inside uncharged nanochannels. The possibility of this unusual electrokinetic phenomenon has been discussed by Dukhin et al. [A. Dukhin, S. Dukhin, P. Goetz, Langmuir 21 (2005) 9990]. Our computed velocity profiles clearly indicate the presence of a net flow with a maximum velocity around 2 m/s. We found the apparent zeta potential to be − 29.7 ± 6.8 mV , using the Helmholtz–Smoluchowski relation and the measured mean velocity. This value is comparable to experimentally measured values in Dukhin et al. and references therein. We also investigate the orientations of water molecules in response to an electric field by computing polarization density. Water molecules in the bulk region are oriented along the direction of the external electric field, while their near-wall orientation shows oscillations. The computation of three-dimensional density distributions of sodium and chloride ions around each individual water molecule show that chloride ions tend to concentrate near a water molecule, whereas sodium ions are diffusely distributed.

Modeling on electrokinetic phenomena of charged porous membrane

Studies on streaming potential (SP), an important electrokinetic phenomena, is chiefly within the scope of microfiltration (MF) and ultrafiltration (UF) membrane processes based on the Helmholtz-Smoluchowski (HS) equation. However, the HS equation has some limitations, which is an adequate cause of the less study of SP for nanofiltration (NF) membranes. In this paper, experiments of SP for two commercial NF membranes (NTR7250 and ESNA 1-K) have been carried in four electrolytes (KCl, MgCl 2, Na 2SO 4 and MgSO 4) solutions. We proposed a new modified HS equation to analyze the experimental data and predict the electrical properties of NF membranes. Results show that the modified HS equation is more applicable than the classical HS equation in the evaluation of surface electrical properties of NF membranes. The established strategy to evaluate the SP of two typical NF membranes in the paper has important significance for other membranes processes.

Steady electro-osmotic flow of a micropolar fluid in a microchannel

We have formulated and solved the boundary-value problem of steady, symmetric and one-dimensional electro-osmotic flow of a micropolar fluid in a uniform rectangular microchannel, under the action of a uniform applied electric field. The Helmholtz–Smoluchowski equation and velocity for micropolar fluids have also been formulated. Numerical solutions turn out to be virtually identical to the analytic solutions obtained after using the Debye–Hückel approximation, when the microchannel height exceeds the Debye length, provided that the zeta potential is sufficiently small in magnitude. For a fixed Debye length, the mid-channel fluid speed is linearly proportional to the microchannel height when the fluid is micropolar, but not when the fluid is simple Newtonian. The stress and the microrotation are dominant at and in the vicinity of the microchannel walls, regardless of the microchannel height. The mid-channel couple stress decreases, but the couple stress at the walls intensifies, as the microchannel height increases and the flow tends towards turbulence.

On slip velocity boundary conditions for electroosmotic flow near sharp corners

The Helmholtz–Smoluchowski (HS) slip velocity boundary condition is often used in computational models of microchannel flows because it allows the motion of the electric double layer (EDL) to be approximated without resolving the charge density profiles close to the walls while dramatically reducing the computational effort required to solve the flow model. The approximation works well for straight channel flows but breaks down in areas of high wall curvature such as sharp corners, where large nonphysical velocities are generated. Many microfluidic applications such as the on-chip focusing and separation of biomolecules rely on the interaction of electroosmosis and electrophoresis in complex channel geometries. In order for these effects to be properly treated using the slip velocity boundary condition, the errors introduced into the solution at corners must be understood. In this article, a complete model for the ion concentrations, electric field, and fluid flow in complex microchannel geometries is presented and is used to compute a pure electroosmotic flow in a two-dimensional microchannel cross slot. The full model solution near the corner at the edge of the EDL is compared to the approximate solution computed by using the HS boundary condition, and it is shown that the accuracy of the approximate solution may be greatly increased by “patching” the full solution as a boundary condition for the approximate solution at the edge of the double layer region. Finally, an empirically derived modified slip velocity boundary condition for electroosmotic flow is proposed. It is shown to improve the accuracy of the flow solution at sharp corners by about 60% when compared to the original boundary condition while also delivering a modest improvement in computational performance because of the elimination of a singularity in the velocity field.

Helmholtz–Smoluchowski velocity for viscoelastic electroosmotic flows

Many biofluids such as blood and DNA solutions are viscoelastic and exhibit extraordinary flow behaviors, not existing in Newtonian fluids. Adopting appropriate constitutive equations these exotic flow behaviors can be modeled and predicted reasonably using various numerical methods. However, the governing equations for viscoelastic flows are not easily solvable, especially for electroosmotic flows where the streamwise velocity varies rapidly from zero at the wall to a nearly uniform velocity at the outside of the very thin electric double layer. In the present investigation, we have devised a simple method to find the volumetric flow rate of viscoelastic electroosmotic flows through microchannels. It is based on the concept of the Helmholtz–Smoluchowski velocity which is widely adopted in the electroosmotic flows of Newtonian fluids. It is shown that the Helmholtz–Smoluchowski velocity for viscoelastic fluids can be found by solving a simple cubic algebraic equation. The volumetric flow rate obtained using this Helmholtz–Smoluchowski velocity is found to be almost the same as that obtained by solving the governing partial differential equations for various viscoelastic fluids.

Drag reduction in electro-osmosis of polymer solutions

Electro-osmosis is the preferred transport mechanism in microfluidic systems. Drag reduction in electro-osmosis of polymer solutions is observed due to polymer depletion in the electric double layer (EDL). The well-known Helmholtz-Smoluchowski (HS) equation indicates that the electro-osmosis mobility is inversely proportional to the solution viscosity. For low molecular weight the polymer size (R) is smaller than the EDL thickness (λ) and the HS equation is valid. For high molecular weight (R&amp;gt;λ) the chains in the EDL are partially sheared and the effective viscosity is smaller than the solution viscosity. Salt addition reduces λ and can enhance drag reduction substantially.

Conditions for similitude and the effect of finite Debye length in electroosmotic flows

Under certain conditions, the velocity field is similar to the electric field for electroosmotic flow (EOF) inside a channel. There was a disagreement between investigators on the necessity of the infinitesimal-Reynolds-number condition for the similarity when the Helmholtz–Smoluchowski relation is applied throughout the boundaries. What is puzzling is a recent numerical result that showed, contrary to the conventional belief, an evident Reynolds number dependence of the EOF. We show here that the notion that the infinitesimal-Reynolds-number condition is required originates from the misunderstanding that the EOF is the Stokes flow. We point out that the EOF becomes the potential flow when the Helmholtz–Smoluchowski relation is applied at the boundaries. We carry out a numerical simulation to investigate the effect of finiteness of the Debye length and the vorticity layer inherently existing at the channel wall. We show that the Reynolds number dependence of the previous numerical simulation resulted from the finiteness of the Debye length and subsequent convective transport of vorticity toward the bulk flow. We discuss in detail how the convection of vorticity occurs and what factors are involved in the transport process, after carrying out the simulation for different Reynolds numbers, Debye lengths, corner radii, and geometries.

On the use of electrokinetic phenomena of the second kind for probing electrode kinetic properties of modified electron-conducting surfaces

The electrokinetic features of electron-conducting substrates, as measured in a conventional thin-layer electrokinetic cell, strongly depend on the extent of bipolar faradaic depolarisation of the interface formed with the adjacent electrolytic solution. Streaming potential versus applied pressure data obtained for metallic substrates must generally be interpreted on the basis of a modified Helmholtz-Smoluchowski equation corrected by an electronic conduction term-non linear with respect to the lateral potential and applied pressure gradient-that stems from the bipolar electrodic behavior of the metallic surface. In the current study, streaming potential measurements have been performed in KNO(3) solutions on porous plugs made of electron-conducting grains of pyrite (FeS(2)) covered by humic acids. For zero coverage, the extensive bipolar electronic conduction taking place in the plug-depolarized by concomitant and spatially distributed oxidation and reduction reactions of Fe(2+) and Fe(3+) species-leads to the complete extinction of the streaming potential over the entire range of applied pressure examined. For low to intermediate coverage, the local electron-transfer kinetics on the covered regions of the plug becomes more sluggish. The overall bipolar electronic conduction is then diminished which leads to an increase in the streaming potential with a non-linear dependence on the pressure. For significant coverage, a linear response is observed which basically reflects the interfacial double layer properties of the humics surface layer. A tractable, semi-analytical model is presented that reproduces the electrokinetic peculiarities of the complex and composite system FeS(2)/humics investigated. The study demonstrates that the streaming potential technique is a fast and valuable tool for establishing how well the electron transfer kinetics at a partially or completely depolarised bare electron-conducting substrate/electrolyte solution interface is either promoted (catalysis) or blocked (passivation) by the presence of a discontinuous surface layer.

Influence of the Dukhin and Reynolds numbers on the apparent zeta potential of granular porous media

The Helmholtz–Smoluchowski (HS) equation is widely used to determine the apparent zeta potential of porous materials using the streaming potential method. We present a model able to correct this apparent zeta potential of granular media of the influence of the Dukhin and Reynolds numbers. The Dukhin number represents the ratio between the surface conductivity (mainly occurring in the Stern layer) and the pore water conductivity. The Reynolds number represents the ratio between inertial and viscous forces in the Navier–Stokes equation. We show here that the HS equation can lead to serious errors if it is used to predict the dependence of zeta potential on flow in the inertial laminar flow regime without taking into account these corrections. For indifferent 1:1 electrolytes (such as sodium chloride), we derived two simple scaling laws for the dependence of the streaming potential coupling coefficient (or the apparent zeta potential) on the Dukhin and Reynolds numbers. Our model is compared with a new set of experimental data obtained on glass bead packs saturated with NaCl solutions at different salinities and pH. We find fairly good agreement between the model and these experimental data.

Characterisation of microporous tubular membranes by tangential streaming potential

Streaming potential measurements can beperformed in two different ways: by flowthrough the membrane pores (transmembranestreaming potential) or by flow across the topsurface of the membrane (tangential streamingpotential, TSP). The se cond procedure is usuallyused for the characterisation of asymmetric orcomposite membranes because it has the advan-tage of avoiding undesirable effects such as thecontribution of both supporting layer(s) to themeasured signal and the membrane potentialinduced by the concentration difference acrossthe active layer. Up to now the tangential tech-nique was applied to flat membranes but notto mono- or multi-channels tubular membranesprobably due to the largeness of channels insidewhich the solution flows. Indeed, in these condi-tions, the flow becomes rapidly turbulent andthe classical Hagen-Poiseuille equation cannotbe used anymore in the derivation of the stream-ing current expression. Although the flow is notwholly laminar, it is show n that the electrokinet-ics theory can be used to convert TSP data intoz-potentials provided that the electrical doublelayer lies within a laminar sublayer near thechannel walls. In this study, we present and test an electro-kinetic setup for the determination of z-potentialof tubular membranes from tangential streamingpotential measurements under conditions of tur-bulent flow. The study was performed with athree-channels tubular membrane close to theNF range over a range of pH. z-potentials deter-mined by combining TSP with electrical con-ductance measurements were compared withthose calculated from the classical Helmholtz–Smoluchowski (H–S) equation.

Deviation of linear relation between streaming potential and pore fluid pressure difference in granular material at relatively high Reynolds numbers

Abstract We conducted streaming potential measurements on the packing of glass beads, and investigated the deviation of streaming potential from the Helmholtz-Smoluchowski (H-S) equation. The H-S equation was originally derived on the assumption of laminar flows. Studies using a capillary have shown that the H-S equation is valid for turbulent flows in so far as the viscous sublayer is thicker than the electrical double layer and the entrance effect is negligible. Although the streaming potential in porous media has been reported to deviate from the H-S equation for turbulent flows, its mechanism is still poorly understood. We measured the fluid flux and the streaming potential as a function of the pore fluid pressure difference. The fluid flux begins to deviate from Darcy’s law at Reynolds number &gt;3, and the streaming potential begins to deviate from the linear relation at larger Reynolds numbers. When the flow is fast, the fluid inertia separates the boundary layer from the solid surface and induces the counter flows. The fluid in the counter-flow region is separated from the circulating fluid, and ions there cannot contribute to the convection current. We think that this results in a lower streaming potential than expected from the H-S equation.

Electrokinetic phenomena of a polyethylene microfiltration membrane in single salt solutions of NaCl, KCl, MgCl2, Na2SO4, and MgSO4

Abstract The streaming potential measurements of a polyethylene microfiltration (PE–MF) membrane were carried out under aqueous solutions of five single salts (NaCl, KCl, MgCl 2 , Na 2 SO 4 , and MgSO 4 ); the zeta potential and surface charge density of the membrane were estimated on the basis of the Helmholtz–Smoluchowski equation and the Gouy–Chapmann theory. The results show that the PE–MF membrane has a weak negative charge due to the specific sorption of ions, and behaves almost as a certain zeta potential in a very dilute salt solution and as almost a certain surface charge density in a solution with a concentration more than 4 mol·m −3 . However, the streaming potential, the zeta potential and the surface charge density of the membrane depend strongly on the salt concentration and the type and valence of ions.

Electro-osmosis at inhomogeneous charged surfaces: Hydrodynamic versus electric friction

Electrokinetic methods are efficient in probing the electrostatic surface properties of charged systems. However, anomalies observed in experiments indicate that the classical electrokinetic theory should be reconsidered. Using Green's function methods and hydrodynamic simulations, we investigate electro-osmosis driven by electric-field-induced ion motion near a charged planar substrate with smooth or rough boundary. First, a reformulation of electro-osmotic theory for planar charged surfaces employing Green's functions shows that the Helmholtz-Smoluchowski (HS) relation between electrostatic potential and solvent velocity is exact for smooth surfaces, even in the presence of ion correlations. Deviations from HS theory are caused by combined hydrodynamic and electric surface friction, as our hydrodynamic simulations of ions at smooth and corrugated charged surfaces in lateral electric fields demonstrate. Within the simulations, hydrodynamic interactions are treated in the continuum limit and the presence of a no-slip boundary condition at the surface is taken into account. While electrofriction is relevant in highly charged system and/or for multivalent ions, hydrodynamic friction is dominant in systems with moderate surface charge density and/or low ionic valency. We also derive the effective electrokinetic surface charge from the electro-osmotic solvent profiles, which is substantially reduced when compared with the bare value and shows qualitative agreement with the experimental tendency.

Modeling reaction-transport processes in a microcapillary biosensor for detection of human IgG

Mathematical modeling of electrokinetic transport processes and antigen–antibody interactions in a microfluidic biosensor for heterogeneous immunoassay is described. Two mathematical models of externally controlled electroosmotic transport are constructed – the slip model based on Helmholtz–Smoluchowski approximation and the non-slip model based on the use of Poisson equation. It was found that the use of the slip model is restricted by concentration of the carrier electrolyte and dimensions of the system. The antigen–antibody interaction cannot be described by the slip model. Dynamics of the heterogeneous immunoassay in the proposed biosensor was also studied. Human-IgG–Protein A was used as model system. Parametric studies revealed strong sensitivity of the biosensor to the antibody charge number and to the character of surface heterogeneity in the position of immobilized antigen. This sensitivity can be one of the reasons why the practical applications of heterogeneous protein chips are still rare.

Electroosmotic flow in microchannels with arbitrary geometry and arbitrary distribution of wall charge

General solutions are developed for direct current (DC) and alternating current (AC) electroosmotic flows in microfluidic channels with arbitrary cross-sectional geometry and arbitrary distribution of wall charge (zeta potential). The applied AC electric field can also be of arbitrary waveform. By proposing a nondimensional time scale ϖ defined as the ratio of the diffusion time of momentum across the electric double-layer thickness to the period of the applied electric field, we demonstrate analytically that the Helmholtz–Smoluchowski electroosmotic velocity is an appropriate slip condition for AC electroosmotic flows in typical microfluidic applications. With this slip condition approach, electroosmotic flows in rectangular and asymmetric trapezoidal microchannels with nonuniform wall charge, as examples, are investigated. The unknown constants in the proposed general solutions are numerically determined with a least-squares method through matching the boundary conditions. We find that the wall charge affects significantly the electroosmotic flow while the channel geometry does not. Moreover, the flow feature is characterized by another nondimensional time scale Ω defined as the ratio of the diffusion time of momentum across the channel hydraulic radius to the period of the applied electric field. The onset of phase shift between AC electroosmotic velocity and applied electric field is also examined analytically.