Effect of direct current dielectrophoresis on the trajectory of a non-conducting colloidal sphere in a bent pore

Dielectrophoresis has shown a wide range of applications in microfluidic devices. Force approximations utilizing the point-dipole method in dielectrophoresis have provided convenient predictions for particle motion by neglecting interactions between the particle and its surrounding electric and flow fields. The validity of this approach, however, is unclear when the particle size is comparable to the characteristic length of the channel and when the particle is in close proximity to the channel wall. To address this issue, we apply an accurate numerical approach based on the boundary-element method (BEM) to solve the coupled electric field, flow, and particle motion. This method can handle much closer particle-wall distances than the other numerical approaches such as the finite-element method. Using the BEM and integrating the Maxwell stress tensor, we simulate an electrokinetic, spherical particle moving within a bent cylindrical pore to investigate how the dielectrophoretic force affects the particle's trajectory. In the simulation, both the particle and the channel wall are non-conducting, and the electric double layers adjacent to the solid surfaces are assumed to be thin with respect to the particle radius and particle-wall gap. The results show that as the particle comes close to the wall, its finite size has an increasingly important effect on its own transient motion and the point-dipole approximation may lead to significant error.     

3-D transient electrophoretic motion of a spherical particle in a T-shaped rectangular microchannel

This paper considers the electrophoretic motion of a spherical particle in an aqueous electrolyte solution in a T-shaped rectangular microchannel, where the size of the channel is close to that of the particle. This is a complicated transient process where the electric field, the flow field, and the particle motion are coupled together. A theoretical model was developed to investigate the influences of the applied electric potentials, the zeta potentials of the channel and the particle, and the size of the particle on the particle motion. A direct numerical simulation method using the finite element method is employed. This method employs a generalized Galerkin finite element formulation that incorporates both equations of the fluid flow and equations of the particle motion into a single variational equation where the hydrodynamic interactions are eliminated. The ALE method is used to track the surface of the particle at each time step. The numerical results show that the electric field in the T-shaped microchannel is influenced by the presence of the particle, and that the particle motion is influenced by the applied electric potentials and the zeta potentials of the channel and the particle. The path of the particle motion is dominated by the local electric field and the ratio of the zeta potential of the channel to that of the particle. The particle's velocity is also dependent on its size in a small channel.     

Boundary effects on the electrophoretic motion of cylindrical particles: concentrically and eccentrically-positioned particles in a capillary

The bounded electrophoretic motion of a cylindrical particle in a circular cylindrical microchannel is explored for two cases: (1) the particle is located on the centerline of a channel (concentrically), with a symmetric wall boundary condition since gap width is constant throughout; and (2) the particle is at an eccentric location in the channel, with an asymmetric boundary condition set by the walls. The objective is to determine the effect of different boundary conditions, geometries, and physical properties on the velocity and orientation of the cylinder with respect to the boundary. A theoretical model for the motion of the cylinder is presented and the problem is solved numerically. The steady-state simulations show that the velocity of the cylinder is reduced at small gap widths for the concentric case, but the velocity is increased at small gap widths for the eccentric case. When the cylinder is angled with respect to the horizontal in the symmetric case or is near the boundary in the asymmetric case, vertical and rotational components of velocity are predicted. In such cases, transient simulations are appropriate for most accurately representing particle motion. Two such simulations are included herein and show both horizontal and vertical translation plus rotation of the particle as a function of time.     

Wall effects on electrophoretic motion of spherical polystyrene particles in a rectangular poly(dimethylsiloxane) microchannel

The wall effects on electrophoretic motion of spherical polystyrene particles in a rectangular poly(dimethylsiloxane) microchannel were studied experimentally. It is found that the particle electrophoretic velocity is insensitive to the trajectory between the channel sidewalls, consistent with the theoretical prediction. We also demonstrate that the electrophoretic motion of larger particles along the channel centerline is more viscously retarded by the sidewalls of a narrower channel. This observation is well predicted by incorporating the analytical models for the particle electrophoresis along the centerline of a slit channel and along the axis of a cylindrical pore.     

Electrostatic double-layer interaction between spherical particles inside a rough capillary

Predictions of electrostatic double-layer interaction forces between two similarly charged spherical colloidal particles inside an infinitely long "rough" capillary are presented. A simple model of a rough cylindrical surface is proposed, which assumes the capillary wall to be a periodic function of axial position. The periodic roughness of the wall is characterized by the wavelength and amplitude of the undulations. The electrostatic double-layer interaction force between two spherical particles located axially inside this rough capillary is determined by solving the nonlinear Poisson-Boltzmann equation employing finite element analysis. The effect of surface roughness of the cylindrical enclosure on the interaction force between two particles is extensively studied on the basis of this model. The simulations are carried out for dimensionless amplitudes (amplitude/particle radii) ranging from 0.05 to 0.15 and scaled wavelengths (wavelength/particle radii) ranging from 0.4 to 4.0. The interaction force between the particles is significantly modified by the proximity of the rough capillary wall. Generally, the interaction force for rough capillaries oscillates around the corresponding interaction force in a smooth capillary depending on the magnitudes of the scaled amplitude and wavelength of the roughness. The influence of roughness on the electrostatic interactions becomes more pronounced when the surface potential of the cylinder wall is different from the sphere surface potentials. When the cylinder and the particle surfaces have large potential differences, the axial force experienced by a particle is dominated by the capillary roughness. There are dramatic oscillations of the force, which alternately becomes repulsive and attractive as the particle moves from the crest to the trough of the rough capillary wall. These results suggest that manipulation of colloidal particles in narrow microchannels may be subject to significant force variations owing to the roughness inherent in microfabricated channels etched on metal films.     

Charge state of the fast gate in chloride channels: insights from electrostatic calculations in a schematic model

Fast gating is a unique property of chloride channels, where a permeating Cl(-) ion acts as its own ligand in opening the channel. The glutamate residue implicated in fast gating normally carries a unit negative charge. Whether this charge needs to be protonated to enable permeation of a Cl(-) ion is an important question that will affect how models of chloride channels are constructed. We investigate the energetic consequences of the charge state of this glutamate residue from continuum electrostatics using a schematic cylindrical channel model. Both analytical solutions of the Poisson equation for an infinite cylinder and numerical ones for a finite cylinder are employed in the calculations.     

Start‐up of electrophoresis of an arbitrarily oriented dielectric cylinder

An analytical study is presented for the transient electrophoretic response of a circular cylindrical particle to the step application of an electric field. The electric double layer adjacent to the particle surface is thin but finite compared with the radius of the particle. The time‐evolving electroosmotic velocity at the outer boundary of the double layer is utilized as a slip condition so that the transient momentum conservation equation for the bulk fluid flow is solved. Explicit formulas for the unsteady electrophoretic velocity of the particle are obtained for both axially and transversely applied electric fields, and can be linearly superimposed for an arbitrarily‐oriented applied field. If the cylindrical particle is neutrally buoyant in the suspending fluid, the transient electrophoretic velocity is independent of the orientation of the particle relative to the applied electric field and will be in the direction of the applied field. If the particle is different in density from the fluid, then the direction of electrophoresis will not coincide with that of the applied field until the steady state is attained. The growth of the electrophoretic mobility with the elapsed time for a cylindrical particle is substantially slower than for a spherical particle.     

Polymer translocation through a cylindrical channel

A formalism of polymer translocation through a cylindrical channel of finite diameter and length between two spherical compartments is developed. Unlike previous simplified systems, the finite diameter of the channel allows the number of polymer segments inside the channel to be adjusted during translocation according to the free energy of possible conformations. The translocation process of a Gaussian chain without excluded volume and hydrodynamic interactions is studied using exact formulas of confinement free energy under this formalism. The free energy landscape for the translocation process, the distribution of the translocation time, and the average translocation time are presented. The complex dependencies of the average translocation time on the length and diameter of the channel, the sizes of the donor and receptor compartments, and the chain length are illustrated.     

Search for a small hole in a cavity wall by intermittent bulk and surface diffusion

We study the search of a small round hole in the wall of a spherical cavity by a diffusing particle, which can reversibly bind to the cavity wall and diffuse on the surface being in the bound state. There are two channels for the particle first passage to the hole, through the bulk, and through the surface. We propose a coarse-grained model of the search process and use it to derive simple approximate formulas for the mean time required for the particle to reach the hole for the first time and for the probability of the first passage to the hole through the bulk channel. This is done for two distributions of the particle starting point: (1) Uniform distribution over the cavity volume and (2) uniform distribution over the cavity wall. We check the accuracy of the approximate formulas by comparing their predictions with the corresponding quantities found by solving the mixed bulk-surface diffusion problem numerically by the finite difference method. The comparison shows excellent agreement between the analytical and numerical results.     

Drag of a Solid Particle Trapped in a Thin Film or at an Interface: Influence of Surface Viscosity and Elasticity

We propose a theoretical model for the motion of a spherical particle entrapped in a thin liquid film or in a monolayer of insoluble surfactant at the air/water interface. Both surface shear and dilational viscosity, surface diffusion, and elasticity of the film are taken into consideration. The drag force acting on the particle is analytically calculated and asymptotic expressions of the problem are provided. The relevance of the model is discussed by comparing the calculated "viscoelastic" drag, gamma(vel), to the one predicted by Saffman's theory, gamma(S), for cylindrical inclusions in membranes. Numerical analyses are performed to evaluate the contributions of the surface viscosity and the diffusion coefficient of the layer on the hydrodynamical resistance experienced by the particle. Copyright 2000 Academic Press.     

Electrophoretic motion of a charged porous sphere within micro- and nanochannels

Electrophoretic motion of a charged porous sphere within micro- and nanochannels is investigated theoretically. The Brinkman model and the full non-linear Poisson-Boltzmann equation are adopted to model the system, with the charged porous sphere resembling polyelectrolytes like proteins and DNA. General electrokinetic equations are employed and solved with a pseudo-spectral method. Key parameters of electrokinetic interest are examined for their respective effect as well as overall impact on the particle motion. We found, among other things, that the confinement effect of the channel can be so drastic that 75% reduction of particle mobility is observed in some situations for a poorly permeable particle. However, only 15% for the corresponding highly permeable particle due to the allowance of fluid penetration which alleviates the retarding shear stress significantly. In particular, an intriguing phenomenon is observed for the highly permeable particle: the narrower the channel is, the faster the particle moves! This was experimentally observed as well in the literature on DNA electrophoresis within nanostructures. The reason behind it is thoroughly explained here. Moreover, charged channels can exert electroosmosis flow so dominant that sometimes it may even reverse the direction of the particle motion. Comparison with experimental data available in the literature for some polyelectrolytes is excellent, indicating the reliability of this analysis. The results of this study provide fundamental knowledge necessary to interpret experimental data correctly in various microfluidic and nanofluidic operations involving bio-macromolecules, such as in biosensors and Lab-on-a-chip devices.     

Investigation of entrance and exit effects on liquid transport through a cylindrical nanopore

The entrance and exit effects on liquid transport through a nano-sized cylindrical pore under different solid wall-liquid interactions were studied by comparing molecular dynamics (MD) results of a finite length nanopore in a membrane with those of an infinite length one. The liquid transport through a finite length nanopore in a membrane was carried out by using a pressure-driven non-equilibrium molecular dynamics (NEMD) method proposed by Huang et al. [C. Huang, K. Nandakumar, P. Choi and L. W. Kostiuk, J. Chem. Phys., 2006, 124, 234701]. The fluid motion through an infinite length nanopore, which had the same cross-stream dimension as the finite length channel in the membrane, but with periodic boundary conditions in the stream-wise direction, was carried out by using the external-field driven NEMD approach [J. Koplik, J. R. Bavanar and J. F. Willemsen, Phys. Rev. Lett., 1988, 60, 1282]. The NEMD results show that the pressure and density distributions averaged over the channel in the radial direction in both finite and infinite length channels are similar, but the radial distributions of the stream-wise velocity were significantly different when the solid wall was repulsive. The entrance and exit effects lead to a decrease in flow rate at about 39% for the repulsive wall and 6% for the neutral-like wall.     

Thermophoretic motion of a solid spherical particle near a solid planar surface

The thermophoretic motion of a solid spherical aerosol particle directed normally to an infinite planar solid surface is analyzed. The solution is performed in a bispherical coordinate system with allowance for linear corrections in the Knudsen number. The finite thermal conductivity of a solid body is taken into account in the analysis.     

A model for investigating the behaviour of non-spherical particles at interfaces

This paper introduces a simple method for modelling non-spherical particles with a fixed contact angle at an interface whilst also providing a method to fix the particles orientation. It is shown how a wide variety of particle shapes (spherical, ellipsoidal, disc) can be created from a simple initial geometry containing only six vertices. The shapes are made from one continuous surface with edges and corners treated as smooth curves not discontinuities. As such, particles approaching cylindrical and orthorhombic shapes can be simulated but the contact angle crossing the edges will be fixed. Non-spherical particles, when attached to an interface can cause large distortions in the surface which affect the forces acting on the particle. The model presented is capable of resolving this distortion of the surface around the particle at the interface as well as allowing for the particle's orientation to be controlled. It is shown that, when considering orthorhombic particles with rounded edges, the flatter the particle the more energetically stable it is to sit flat at the interface. However, as the particle becomes more cube like, the effects of contact angle have a greater effect on the energetically stable orientations. Results for cylindrical particles with rounded edges are also discussed. The model presented allows the user to define the shape, dimensions, contact angle and orientation of the particle at the interface allowing more in-depth investigation of the complex phenomenon of 3D film distortion around an attached particle and the forces that arise due to it.     

Dielectrophoretic choking phenomenon of a deformable particle in a converging‐diverging microchannel

The translational motion of small particles in an electrokinetic fluid flow through a constriction can be enhanced by an increase of the applied electric potential. Beyond a critical potential, however, the negative dielectrophoresis (DEP) can overpower other forces to prevent particles that are even smaller than the constriction from passing through the constriction. This DEP choking phenomenon was studied previously for rigid particles. Here, the DEP choking phenomenon is revisited for deformable particles, which are ubiquitous in many biomedical applications. Particle deformability is measured by the particle shear modulus, and the choking conditions are reported through a parametric study that includes the channel geometry, external electric potential, and particle zeta potential. The study was carried out using a numerical model based on an arbitrary Lagrangian‐Eulerican (ALE) finite‐element method.     

Dynamic shape and wall correction factors of cylindrical particles falling vertically in a Newtonian liquid

Results of numerical calculations of dynamic shape and wall correction factors for the flow of a Newtonian fluid over a vertically oriented cylindrical particle in a cylindrical tube are reported. Mathematical model of the flow was solved using the finite element method by means of the COMSOL Multiphysics software. Dependences of the shape factor on the cylinder aspect ratio and of the wall correction factor, F _W , on the ratio of the cylindrical particle diameter to the tube diameter, and on the aspect ratio were obtained. Numerical dependences were approximated by simple relationships suitable for dynamic shape and wall correction factors prediction.     

On the limiting electrophoretic mobility of a highly charged colloidal particle in an electrolyte solution

The electrophoretic mobility of a spherical charged colloidal particle in an electrolyte solution with large kappaa (where kappa= Debye-Hückel parameter and a= particle radius) tends to a nonzero constant value in the limit of high zeta potential. It is demonstrated that this is caused by the fact that counterions condensed near the highly charged particle surface do not contribute to the electrophoretic mobility and only co-ions govern the mobility. A simple method to derive the limiting electrophoretic mobility expression is given. The present method is also applied to cylindrical particles, showing that the leading term of the limiting electrophoretic mobility of a cylindrical particle in a transverse field with large kappaa is the same as that of a spherical particle. The electrophoretic mobility of a cylindrical particle in a tangential field, on the other hand, is proportional to the particle zeta potential and does not exhibit a constant limiting value for high zeta potentials.     

Electrophoresis of concentrically and eccentrically positioned cylindrical particles in a long tube

We study analytically and numerically the electrophoretic motion of cylindrical particles translating slowly in long tubes filled with an electrolyte solution and subjected to axial electric fields. Both thin and thick double layers are considered. Of particular interest is the case when the particle's and tube's radii are of the same order of magnitude. The model accounts for the flow induced by the particle's motion (the particle acts as a leaky piston) and the electroosmotic flow in the tube. The electrophoretic velocity of the particle and the forces and torques acting on it are determined as functions of the particle's radius, length, and position, the particle's and tube's zeta potentials, the tube's length, and the externally imposed pressures. When the particle is positioned off center, it rotates and its trajectory traces an oscillatory path.