Smoluchowski Slip Velocity Research Articles

Electrophoresis of tightly fitting spheres along a circular cylinder of finite length

Electrophoresis of a tightly fitting sphere of radius $a$ along the centreline of a liquid-filled circular cylinder of radius $R$ is studied for a gap width $h_0=R-a\ll a$ . We assume a Debye length $\kappa ^{-1}\ll h_0$ , so that surface conductivity is negligible for zeta potentials typically seen in experiments, and the Smoluchowski slip velocity is imposed as a boundary condition at the solid surfaces. The pressure difference between the front and rear of the sphere is determined. If the cylinder has finite length $L$ , this pressure difference causes an additional volumetric flow of liquid along the cylinder, increasing the electrophoretic velocity of the sphere, and an analytic prediction for this increase is found when $L\gg R$ . If $N$ identical, well-spaced spheres are present, the electrophoretic velocity of the spheres increases with $N$ , in agreement with the experiments of Misiunas & Keyser (Phys. Rev. Lett., vol. 122, 2019, 214501).

Electrophoretic motion of a non-uniformly charged particle in a viscoelastic medium in thin electrical double layer limit

The electrophoretic motion of a non-uniformly charged particle in an Oldroyd-B fluid is analysed here in the limit of thin electrical double layers. To this end, we analytically derive expressions for the modified Smoluchowski slip velocity around the particle, carrying weak but otherwise arbitrary surface charge. Our analysis reveals that the modified Smoluchowski slip around a particle differs significantly in a viscoelastic medium as compared with Newtonian fluids. The flow field thus derived is applied to two special cases of non-uniformly charged particles to obtain a closed-form expression for their electrophoretic translational and rotational velocities. We show that the particle's velocity strongly depends on its size in a viscoelastic medium, even for weakly charged surfaces, which is in stark contrast to the well-established theory for Newtonian fluids for weakly charged particles with negligible surface conduction. We further demonstrate that the presence of non-uniform surface charge enhances the influence of the medium's viscoelasticity on the particle's translational as well as angular velocity and this effect strongly depends on the nature of surface charge distribution. Such a physical paradigm, which leads to a breaking of fore–aft symmetry that is unique to complex fluids despite operating in the regime of creeping flows. Our study provides new theoretical framework for understanding electrophoresis of charged entities (such as DNA or active matter) in complex fluids, including biologically relevant fluidic media.

Electroosmosis over non-uniformly charged surfaces: modified Smoluchowski slip velocity for second-order fluids

In the present paper we focus on deriving the modified Smoluchowski slip velocity of second-order fluids, for electroosmotic flows over plane surfaces with arbitrary non-uniform surface potential in the presence of thin electric double layers (EDLs). We employ matched asymptotic expansion to stretch the electric double layer and subsequently apply regular asymptotic expansions taking the Deborah number ($De$) as the gauge function. Modified slip velocities correct up to $O(De^{2})$ are presented. Two sample cases are considered to demonstrate the effects of viscoelasticity on slip velocity: (i) an axially periodic patterned potential and (ii) a step-change-like variation in the surface potential. The central result of our analysis is that, unlike Newtonian fluids, the electroosmotic slip velocity for second-order fluids does not, in general, align with the direction of the applied external electric field. Proceeding further forward, we show that the slip velocity in a given direction may, in fact, depend on the applied electric field strength in a mutually orthogonal direction, considering three dimensionality of the flow structure. In addition, we demonstrate that the modified slip velocity is not proportional to the zeta potential, as in the cases of Newtonian fluids; rather it depends strongly on the gradients of the interfacial potential as well. Our results are likely to have potential implications so far as the design of charge modulated microfluidic devices transporting rheologically complex fluids is concerned, such as for mixing and bio-reactive system analysis in lab-on-a-chip-based micro-total-analysis systems handling bio-fluids.

Electro-osmosis over inhomogeneously charged surfaces in presence of non-electrostatic ion-ion interactions

In this study, we attempt to bring out a generalized formulation for electro-osmotic flows over inhomogeneously charged surfaces in presence of non-electrostatic ion-ion interactions. To this end, we start with modified electro-chemical potential of the individual species and subsequently use it to derive modified Nernst-Planck equation accounting for the ionic fluxes generated because of the presence of non-electrostatic potential. We establish what we refer to as the Poisson-Helmholtz-Nernst-Planck equations, coupled with the Navier-Stokes equations, to describe the complete transport process. Our analysis shows that the presence of non-electrostatic interactions between the ions results in an excess body force on the fluid, and modifies the osmotic pressure as well, which has hitherto remained unexplored. We further apply our analysis to a simple geometry, in an effort to work out the Smoluchowski slip velocity for thin electrical double layer limits. To this end, we employ singular perturbation and develop a general framework for the asymptotic analysis. Our calculations reveal that the final expression for slip velocity remains the same as that without accounting for non-electrostatic interactions. However, the presence of non-electrostatic interactions along with ion specificity can significantly change the quantitative behavior of Smoluchowski slip velocity. We subsequently demonstrate that the presence of non-electrostatic interactions may significantly alter the effective interfacial potential, also termed as the “Zeta potential.” Our analysis can potentially act as a guide towards the prediction and possibly quantitative determination of the implications associated with the existence of non-electrostatic potential, in an electrokinetic transport process.

Electroosmosis of viscoelastic fluids over charge modulated surfaces in narrow confinements

In the present work, we attempt to analyze the electroosmotic flow of a viscoelastic fluid, following quasi-linear constitutive behavior, over charge modulated surfaces in narrow confinements. We obtain analytical solutions for the flow field for thin electrical double layer (EDL) limit through asymptotic analysis for small Deborah numbers. We show that a combination of matched and regular asymptotic expansion is needed for the thin EDL limit. We subsequently determine the modified Smoluchowski slip velocity for viscoelastic fluids and show that the quasi-linear nature of the constitutive behavior adds to the periodicity of the flow. We also obtain the net throughput in the channel and demonstrate its relative decrement as compared to that of a Newtonian fluid. Our results may have potential implications towards augmenting microfluidic mixing by exploiting electrokinetic transport of viscoelastic fluids over charge modulated surfaces.

Electroosmotic flow of a viscoplastic material through a slit channel with walls of arbitrary zeta potential

Electroosmotic (EO) flow is known to have a nearly uniform velocity profile, but such a plug-flow velocity can be considerably diminished if the fluid is a viscoplastic material having a yield stress. This paper aims to investigate the reduction of EO velocity (also known as Smoluchowski slip velocity) due to a yield stress as a function of the material rheological parameters and the zeta potential. Three rheological models are considered: Casson, Herschel–Bulkley, and Bingham fluids. In the absence of pressure forcing and without the Debye–Hückel approximation, the problems of EO flow of these materials in a slit channel with walls uniformly charged with an arbitrary zeta potential are analytically solved. Analytical expressions are deduced for the reduced Smoluchowski velocity under the limiting conditions of very small and very large zeta potentials. It is shown that qualitatively different asymptotic behaviors will be exhibited by materials of different models.

Electrokinetic motion of a rectangular nanoparticle in a nanochannel

This article presents a theoretical study of electrokinetic motion of a negatively charged cubic nanoparticle in a three-dimensional nanochannel with a circular cross-section. Effects of the electrophoretic and the hydrodynamic forces on the nanoparticle motion are examined. Because of the large applied electric field over the nanochannel, the impact of the Brownian force is negligible in comparison with the electrophoretic and the hydrodynamic forces. The conventional theories of electrokinetics such as the Poisson–Boltzmann equation and the Helmholtz–Smoluchowski slip velocity approach are no longer applicable in the small nanochannels. In this study, and at each time step, first, a set of highly coupled partial differential equations including the Poisson–Nernst–Plank equation, the Navier–Stokes equations, and the continuity equation was solved to find the electric potential, ionic concentration field, and the flow field around the nanoparticle. Then, the electrophoretic and hydrodynamic forces acting on the negatively charged nanoparticle were determined. Following that, the Newton second law was utilized to find the velocity of the nanoparticle. Using this model, effects of surface electric charge of the nanochannel, bulk ionic concentration, the size of the nanoparticle, and the radius of the nanochannel on the nanoparticle motion were investigated. Increasing the bulk ionic concentration or the surface charge of the nanochannel will increase the electroosmotic flow, and hence affect the particle’s motion. It was also shown that, unlike microchannels with thin EDL, the change in nanochannel size will change the EDL field and the ionic concentration field in the nanochannel, affecting the particle’s motion. If the nanochannel size is fixed, a larger particle will move faster than a smaller particle under the same conditions.

Electrokinetic motion of a deformable particle: Dielectrophoretic effect

Electrokinetics-induced motion and deformation of a hyperelastic particle confined in a slit microchannel has been numerically investigated for the first time with a full consideration of the fluid-particle-electric field interactions and the dielectrophoretic (DEP) effect. When the initial orientation of a cylindrical particle with respect to the applied electric field, θ(p0), is 90°, the particle tends to curl up as a "C" shape when moving from left to right. The electrokinetics-induced particle deformation is due to the joint effects of the shear force arising from the non-uniform Smoluchowski slip velocity on the particle surface and the asymmetric DEP force with respect to the center of the deformed particle arising from the spatially non-uniform electric field surrounding the particle. The electrokinetics-induced particle deformation is opposite to that of a particle moving in the same direction subjected to a pressure-driven flow. When the initial particle orientation is 0<θ(p0) <90°, a net torque arising from the DEP effect progressively rotates and aligns the particle with its longest axis parallel to the applied electric field, thus decreasing the non-uniformity of the electric field and accordingly the particle deformation. The numerical predictions are in qualitative agreement with our previous experimental observation. The results show that the DEP effect is significant and must be taken into account in the modeling of electrokinetic motion of a deformable particle in microfluidics.

On the Effect of Induced Electro-Osmosis on a Cylindrical Particle Next to a Surface

The effect of induced electro-osmosis on a cylindrical particle positioned next to a planar surface (wall) is studied theoretically both under the thin double layer approximation utilizing the Smoluchowski slip velocity approximation and under thick electric double layer conditions by solving the Poisson-Nernst-Planck (PNP) equations. The imposed, undisturbed electric field is parallel to the planar surface. The induced hydrodynamic and electrostatic forces are calculated as functions of the particle's and the medium's dielectric constants and the distance between the particle and the surface. The resultant force acting on the particle is directed normal to and away from the wall. The presence of such a repulsive force may adversely affect the interactions between macromolecules suspended in solution and wall-immobilized molecules and may be significant to near-wall particle imaging velocimetry (PIV) in electrokinetic flows.

Control of Particle-Deposition Pattern in a Sessile Droplet by Using Radial Electroosmotic Flow

In this technical note, we report an experimental investigation of radial electroosmotic flow (EOF) as an effective means for controlling particle-deposition pattern inside an evaporating droplet, which has a potential application to biochemistry and analytical chemistry especially for sample preparation steps. Using the microelectrode, which consists of the circular electrode around the rim of droplet and the point electrode at the center of the droplet, we generate the radial electric field at the bottom of the electrolyte droplet. The electric field developed between the center electrode and the circular electrode causes a radial EOF in the vicinity of the bottom of the droplet. By changing the applied voltage, the strengths and directions of the radial EOF are controlled at one's own discretion, and thus, we can modify the solute distribution inside the droplet during evaporation. When the radial EOF compensates the natural outward flow at a suitable choice of electrical voltage, the particles are uniformly distributed at the entire droplet spot. Moreover, with strong radial EOF, all the particles are deposited at the center rather than at the rim. We also carry out a simple theoretical investigation of flow field inside the droplet with Smoluchowski slip velocity condition to show how the particles travel during evaporation.

Microfluidic Mixing by dc and ac Nonlinear Electrokinetic Vortex Flows

A novel micromixing strategy is proposed for microfluidic applications. High-intensity nonlinear electroosmotic microvortices, with angular speeds in excess of 1 cm/s, are generated around a small (∼1 mm) conductive ion-exchange granule when moderate dc and ac electric fields (30−125 V/cm) are applied across a miniature chamber smaller than 10 μL. Coupling between these microvortices and the fast electrophoretic motion (∼1 cm/s) of the granule in low frequency (between 0.3 and 1.0 Hz) ac fields and a slower backflow vortex (velocity ∼1 mm/s) in dc fields produces an intense chaotic micromixing action. Two segregated dye samples, normally requiring nearly 0.5 h to mix by diffusion, are observed to mix within 30 s in the mixing chamber. The effective diffusivity scales as E2 and is measured to be 2 orders of magnitude higher than molecular diffusivity at reasonable field strengths and optimal frequencies. The main vortices are generated by nonlinear versions of the Smoluchowski slip velocity on the granule ...

Nonlinear Smoluchowski slip velocity and micro-vortex generation

When an electric field is applied across a conducting and ion-selective porous granule in an electrolyte solution, a polarized surface layer with excess counter-ions is created. The depth of this layer and the overpotential V across this layer are functions of the normal electric field j on the granule surface. By transforming the ionic flux equations and the Poisson equation into the Painlevé equation of the second type and by analysing the latter's asymptotic solutions, we derive a linear universal j–V correlation at large flux with an electrokinetic slip length β. The flux-induced surface polarization produces a nonlinear Smoluchowski slip velocity that can couple with the granule curvature to produce micro-vortices in micro-devices. Such vortices are impossible in irrotational electrokinetic flow with a constant zeta-potential and a linear slip velocity.