Inverse Debye Length Research Articles

Derivation of Modified Reynolds Equation: A Porous Media Model With Effects of Electrokinetics

A lubrication theory that includes the effects of electrokinetics and surface microstructure is developed. A porous layer attached to the impermeable substrate is used to model the microstructure on a bearing surface. The Brinkman-extended Darcy equations and Stokes equations are modified by considering the electrical body force and utilized to model the flow in porous media and fluid film, respectively. The stress jump boundary conditions on the porous media/fluid film interface and the effects of viscous shear and electric double layer (EDL) are also considered when deriving the modified Reynolds equation. Under the usual assumptions of lubrication and Debye–Hückel approximation for low surface potential, the velocity distributions, the apparent viscosity, and the modified Reynolds equation are then derived. The apparent viscosity is expressed explicitly as functions of the Debye length, the electroviscosity, the charge density, the stress jump parameter, and the porous parameters (permeability, porosity, and porous film thickness). The considerations of EDL near the interface and the charge density of the flow in the porous media increase the apparent viscosity. The existence of porous film also increases the apparent viscosity as well. Both effects are important for flow within microspacing and lubrication problems. The apparent viscosity and the performance of 1D slider bearings are analyzed and discussed. The results show that the apparent viscosity and the load capacity increase as the permeability decreases, the stress jump parameter decreases, the charge density increases, the inverse Debye length decreases, or the porosity decreases.

Effects of electrokinetic slip flow on lubrication theory

A lubrication theory that includes the coupled effects of electric double layer (EDL) and boundary slip is developed. The consideration of EDL (electro-viscous effect) and boundary slip near the solid surface increases and decreases apparent viscosity, respectively. Both effects are important for flow within microscales and lubrication problems. Under the usual assumptions of lubrication and Debye—Hückel approximation for low surface potential, the Navier—Stokes equation with body force due to the electrical potential, as well as the Navier slip boundary conditions, is utilized to derive the velocity distributions, apparent viscosity, and modified Reynolds equation. The apparent viscosity is expressed explicitly as functions of the Debye length, the electro-viscosity, and the slip length. The coupled effects of EDL and boundary slip on the apparent viscosity and one-dimensional slider-bearing performance are analysed and discussed. Results show that the load capacity increases as a decrease in inverse Debye length ( K), a decrease in slip lengths ( B), or an increase in electro-viscosity.

Complete phase behavior of the symmetrical colloidal electrolyte

We computed the complete phase diagram of the symmetrical colloidal electrolyte by means of Monte Carlo simulations. Thermodynamic integration, together with the Einstein-crystal method, and Gibbs-Duhem integration were used to calculate the equilibrium phase behavior. The system was modeled via the linear screening theory, where the electrostatic interactions are screened by the presence of salt in the medium, characterized by the inverse Debye length, kappa (in this work kappasigma=6). Our results show that at high temperature, the hard-sphere picture is recovered, i.e., the liquid crystallizes into a fcc crystal that does not exhibit charge ordering. In the low temperature region, the liquid freezes into a CsCl structure because charge correlations enhance the pairing between oppositely charged colloids, making the liquid-gas transition metastable with respect to crystallization. Upon increasing density, the CsCl solid transforms into a CuAu-like crystal and this one, in turn, transforms into a tetragonal ordered crystal near close packing. Finally, we have studied the ordered-disordered transitions finding three triple points where the phases in coexistence are liquid-CsCl-disordered fcc, CsCl-CuAu-disordered fcc, and CuAu-tetragonal-disordered fcc.

Modeling and Simulation of Ionic Currents in Three-Dimensional Microfluidic Devices with Nanofluidic Interconnects

Electrokinetic fluid flow in nanocapillary array (NCA) membranes between vertically separated microfluidic channels offers an attractive alternative to using mechanical action to achieve fluidic communication between different regions of lab-on-a-chip devices. By adjusting the channel diameter, a, and the inverse Debye length, к, and applying the appropriate external potential, the nanochannel arrays, can be made to behave like digital fluidic switches, and the movement of molecules from one side of the array to the other side can be controlled. However, inherent differences in ionic mobility lead to non-equilibrium ion populations on the downstream side, which, in turn, shows up through transient changes in the microchannel conductance. Here we describe coupled calculations and experiments in which the electrical properties of a microfluidic–nanofluidic hybrid architecture are simulated by a combination of a compact model for the bulk electrical properties and iterative self-consistent solutions of the coupled Poisson, Nernst–Planck, and Navier–Stokes equations to recover the detailed ion motion in the nanopores. The transient electrical conductivity in the microchannel, after application of a forward bias pulse to the NCA membrane, is recovered in quantitative detail. The surface charge density of the nanopores and the capacitance of the membrane, which are critical determinants of electrokinetic flow through NCA, fall out of the analysis in a natural way, providing a clear mechanism to determine these critically important parameters.

A new and simpler approach for the solution of the electrostatic potential differential equation. Enhanced solution for planar, cylindrical and annular geometries

The motivation of the study performed in this project is focused on deriving a more effective, accurate, and mathematically friendly solution for the prediction of the electrostatic potential, commonly used on electrokinetic research and its related applications. In this contribution, based on the Debye–Hückel approximation, a new solution strategy for the differential equations of the electrostatic potential is proposed. In fact, a simple predictor–corrector calculation is developed to achieve more accurate predictions of electrostatic potential profiles. Furthermore, in this study the authors introduce the correction function f AO to the inverse Debye length, λ. The f AO function improves the Debye–Hückel approximation and it is a recursive function of the electrical potential. Once the inverse Debye length, λ, has been corrected by the f AO function and introduced in the simplified solution of the Poisson–Boltzmann equation (i.e., the linear approximation, due to Debye and Hückel), the electrostatic potential outcome little differs from the numerical solution of the complete (nonlinear) differential equation. This new approach embraces different geometries of interest, such as planar, cylindrical, and annular, with excellent results in all the cases and for a wide range of electrostatic potential values. This new predicting semi-analytical technique can be a useful tool on electrical field applications such as the separation of a mixture of macromolecules and the removal of contaminants in soil cleaning processes. Illustrative results are presented for the geometries identified above.

Influence of Surface Charge Density on the Electrochemically Derived Surface Roughness of Bi Electrodes

Electrical double-layer properties of variously treated Bi(001) and Bi(111) electrodes were studied in terms of the Debye length dependent roughness theory (i.e., nonlinear Poisson-Boltzmann theory), recently developed by Daikhin et al. The geometrical roughness factor, root mean square (rms) departure of the surface from flatness, the medium lateral correlation length values have been obtained. According to the experimental data and results of theoretical calculations, the surface roughness increases in the order of Bi electrodes: electrochemically polished single-crystal Bi(111) at the temperature of liquid nitrogen Bi(111) etched Bi(111) etched Bi(111) etched Bi(001) plane. This order of the electrode surface roughness is in agreement with the ex situ AFM data, i.e., with the rms roughness values. At small electrolyte concentrations the deviation of experimental roughness function vs. inverse Debye length curves from the theoretically calculated ones has been observed, which is explained by the influence of energetic inhomogeneity of the polycrystalline surfaces, i.e., the nonequal surface charge density values at the various homogeneous regions exposed at the macropolycrystalline electrode surface, on the interfacial capacitance values. © 2003 The Electrochemical Society. All rights reserved.

Thermal effects on electro-osmotic pumping of liquids in microchannels

In this paper we present a mathematical model predicting combined electro-osmotic- and pressure-driven flow behavior in a straight microchannel for the case when the system is under a non-isothermal condition. The distribution of the charge density is governed by a nonlinear, two-dimensional Poisson–Boltzmann equation, and a body force caused by the interaction between the charge density and the applied electrical potential field is included in the full Navier–Stokes equations. Under non-isothermal conditions, arising from heat sources (Joule heating) within the system and/or a temperature difference between the system and the ambient, the equation of energy conservation describing temperature distribution has to be considered as well. The governing equations, interlinked via temperature, are solved numerically using a finite difference method. The numerical results indicate that for a given cross-sectional mean velocity, there exists an optimal dimensionless parameter (κh), which is the inverse Debye length multiplied by the channel size, which gives the highest hydraulic head generated by the electro-osmotic force. It has also been demonstrated that the pumping performance predicted by the isothermal model deviates substantially from that predicted by the non-isothermal model when Joule heating is significant.

A Fast Adaptive Multipole Algorithm for Calculating Screened Coulomb (Yukawa) Interactions

The screened Coulomb (Yukawa or Debye–Hückel) potential, Φ=exp(−κr)/r, whereris the separation distance and κ is the Debye–Hückel screening parameter, gives a good description of the electrostatic interactions in a variety of biologically and physically important charged systems. It is well known that the direct calculation of the energy and forces due to a collection ofNcharged particles involves the pairwise summation of all charged particle interactions and exhibits anO(N2) computational complexity which severely restricts maximum problem size. This has prompted the development of fast summation algorithms that allow the electrostatic energy and forces to be obtained in onlyO(NlogN) operations. To date, however, practically all such implementations have been limited exclusively to pure Coulombic potentials (κ=0), and the central contribution of the present method is to extend this capability to the entire range of the inverse Debye length, κ≥0. The basic formulation and computational implementation of the spherical modified Bessel function-based multipole expansions appropriate for the screened Coulomb kernel are first presented. Next, a simple model system consisting of a single source charged particle is studied to show that the maximum electrostatic energy error incurred by anM-order multipole expansion for the Yukawa potential is bounded above by the error of the equivalent multipole expansion for the Coulombic potential. Finally, timing and accuracy studies are presented for a variety of charged systems including polyelectrolyte chains, random distributions of charges inside a cube, and face-centered-cubic lattice charge configurations containing up to 103,823 charges.

Surface roughness of bismuth, antimony and cadmium electrodes

Electric double layer properties of variously treated Bi, Sb and Cd electrodes were studied in terms of the Debye length dependent roughness theory, recently developed by Daikhin et al. The geometrical roughness factor, root mean square departure of the surface from flatness, height of surface roughness and the medium lateral correlation length values have been obtained. According to the experimental data and results of theoretical simulation the surface roughness rises in the order of Bi electrodes: electrochemically polished single crystal Bi(111) plane<cut at the temperature of liquid nitrogen Bi(111) plane<Bi solid drop electrode with additionally remelted surface<cut at the room temperature Bi(111) plane<electrochemically etched Bi(111) plane<chemically etched Bi(111) plane. This order of the electrode surface roughness was in agreement with the ex situ AFM data. At small electrolyte concentrations ( c el≤0.01 M) the deviation of experimental roughness function–inverse Debye length curves from the theoretically simulated ones has been observed, explained by the influence of energetic inhomogeneity of polycrystalline surfaces on the non-equal surface charge density at various homogeneous regions, exposed at the macro-polycrystalline electrode surface.

Boundary Effects on Electrophoretic Motion of Spherical Particles for Thick Double Layers and Low Zeta Potential

The electrophoretic motion of a charged sphere in the presence of a rigid boundary is analyzed for low surface ζ potentials but arbitrary κa, whereais the particle radius and κ is the inverse Debye length. The boundary configurations considered are a single flat wall, a slit, and a long cylindrical tube. Using a method of reflections, we obtain the particle velocity for a constant applied electric field in powers of λ up toO(λ6), where λ is the ratio of the particle radius to the distance from the boundary. This analysis is valid as long as the double layer around the particle does not overlap significantly with the double layer at the boundary. The effect of finite κais to enhance the viscous retardation of the particle, although for large separations the first effect due to the proximity of the boundary is still atO(λ3) in all cases. When the applied field is parallel to the boundary, the electrophoretic velocity is not proportional to the difference in zeta potential between the particle and the boundary (as occurs for κa→ ∞), and the proximity of the boundary may increase the particle velocity or change its direction. An important result of the analysis is that the hindrance to the electrophoretic velocity of a particle in a cylindrical pore increases significantly as κais reduced below 10.

Grand Canonical Ensemble Monte Carlo Simulations of Ion Distribution Functions and of the Electric Double Layer in Spherical Charged or Uncharged Pores with Restricted Primitive Model 1:1 Electrolytes at Moderate Concentrations

Abstract A number of Grand Canonical Ensemble simulations have been performed for restricted primitive model 1:1 electrolytes in spherical hard-wall pores with the same dielectric permittivity in the pores and in the wall. The electrolyte concentrations are moderate corresponding to ka = ca. 0.9 and 1.4, where a = ionic diameter and K is the inverse Debye length. The pores have a continuously distributed surface charge from 0 to 10 elementary charges and radii from 2 to 9 ionic diameters. Mean ionic distribution functions and the electric potential distributions in the spherical double layers are investigated. The mean ionic distribution functions seem to strike a balance between “attractive” hard sphere-hard wall effective forces and a repulsive force Wall. etermined by the Debye length (1/K) and due to the lack of symmetry in the ion clouds around ions near the wall. However, there is also in small pores a general depression of the mean ionic distribution functions, which increases with the applied exte...

The Effect of Hydrodynamic Flow Field on Colloidal Stability

Abstract Colloid-colloid interactions are important in understanding the macroscopic properties of flowing suspensions. In many processes of technological and biological interest, it is important to identify conditions which promote or inhibit colloidal aggregation. In this paper, we use trajectory analysis to understand the effect of hydrodynamic and nonhydrodynamic forces on colloidal stability. All linear hydrodynamic flows can be represented in a finite region of the tr L2-deT L plane, where L is the normalized velocity gradient tensor with constant magnitude. We calculate the stability ratio W for different flow types, specified by tr L2 and det L. For purely attractive interparticle potentials, a small region around simple shear flow (tr L2 = 0, det L = 0) shows uniquely high stability. Small changes in tr L2 or det L, which are equivalent to changes in the relative magnitude of vorticity or the relative orientation between vorticity and extension, cause a great decrease in stability. Away from simple shear flow, W is independent of changes in flow type for the entire class of linear flows with open streamlines (tr L2 ⩾ 0). Interparticle potentials with primary and secondary minima exhibit the same stability as purely attractive potentials as long as the Debye screening length is less than a critical value. Greater Debye lengths lead to complete stability (W → ∞). The critical Debye length depends on fluid flow type and the value of inverse critical Debye length correlates with the strength of the flow field. The magnitude of the particle surface potential has little effect on the stability ratio. Taken together, the results show the type of hydrodynamic flow to be an important determinant of the aggregation behavior of colloidal particles. Furthermore, aggregation in simple shear flow is different than that in other linear flows, and we caution against extrapolating aggregation behavior in simple shear to more complex fluid flow situations.

Salt Rejection in a Sinusoidal Capillary Tube

Electrokinetic models for electrolyte transport in narrow sinusoidal capillary tubes were used to study salt rejection. Two approaches were adopted. The first formulation made use of Gouy-Chapman model of the double layer coupled with the creeping flow and Nernst-Planck equations. The elliptic nature of the formulation was retained. The second formulation was an adaptation of Dresner's model where Boltzmann distribution was assumed to hold in the radial direction. The salt rejection as evaluated by the two formulations was fairly close. However, the details of the electrokinetic field variables obtained from the two formulations were not in close agreement. The dimensionless inverse Debye length, κa, based on the average tube radius, a, was found to play a major role in the salt rejection. When the electrical double layers overlap, i.e., κa is small, the salt rejection is governed by the surface potential, flow Peclet number, and the amplitude of the tube surface. In addition, for high Peclet numbers, the tube surface potential becomes the sole parameter controlling the salt rejection. When the electrical double layers do not overlap, i.e., κa is large, the salt rejection becomes a strong function of the tube geometry as well as the surface potential and Peclet number.

Calcium Concentration Dependence of the Intergranular Film Thickness in Silicon Nitride

High‐resolution electron microscopy and nano‐beam analytical electron microscopy have been used to characterize both the intergranular silicate film thickness and its local composition in a series of high‐purity Si3N4 ceramics doped with 0–450 at. ppm Ca. Calcium was detected at both two‐grain junctions and triple junctions, even in the 80‐ppm‐Ca‐doped specimen. The thickness of the intergranular film at two‐grain junctions was found to depend sensitively on Ca content. In undoped material, the thickness was 1.0 ± 0.1 nm. With increasing Ca content, the thickness decreased in the dilute region (80 ppm Ca), but then increased. The variation in film thickness can be qualitatively understood in terms of the balance of three long‐range forces acting normal to the film, namely the van der Waals dispersion force, a structural “steric” force, and an electrical‐double‐layer force. By comparing the measured thicknesses to those predicted, estimates for the structural correlation length and the inverse Debye length can be made. These estimates have values of ∼ 0.22 nm and approximately 0.3–0.5 nm, respectively, for the calcia‐free and 80 ppm calcia materials.

Extrapolation formulas for dimensions of branched polyelectrolytes

Simple extrapolation formulas are derived for the mean-square radius of gyration and structure factor of any branched polyelectrolytes for arbitrary values of molecular weight and various parameters of the screened electrostatic and excluded-volume interactions. In general, the mean-square radius of gyration b is proportional to k -4/5 L 6/5 and L 2 in the strong and weak Coulombic screening limits, respectively, where k is the inverse Debye length, L is the total chain length, and the proportionality coefficients depend in the nature and details of the branching

Multipolar electrolyte solution models. III. Free energy of the charged and dipolar hard sphere mixture

Helmholtz free energies, internal energies, and pressures of mixtures of charged and dipolar hard spheres are obtained from Monte Carlo computer simulation. Ionic and solvent contributions to these properties of this electrolyte solution model are reported for a large number of states, with variations in total density, ionic concentration, and the ratio of ionic charge to solvent polarity. From this data we show that, in contrast to solution theories based on replacement of the solvent by a continuum with dielectric constant ε0, there is a correspondence in properties of states at fixed κ02=κ2ε0, where κ is the inverse Debye length.

ION DYNAMICAL CORRECTIONS TO THE HOLTSMARK THEORY OF SPECTRAL LINE BROADENING

Summary A method is described whereby earlier work by Kogan and Griem on ion dynamical corrections to the Holtsmark theory of spectral line shapes may be generalized to include the effects of Debye shielding and ion-ion correlations. Expressions for the dynamical corrections to the Holtsmark profile of a single Stark component are obtained to second order in the inverse Debye length. The correction terms are evaluated in the case of a high-density plasma (Ne ∼ 1017 cm−3), and the results used to discuss the central structure of the hydrogen Balmer β line profile. On the basis of our model, it is concluded that allowance for ion motion, with or without shielding, cannot account for outstanding discrepancies between the calculated and measured central structure of this line, under these conditions.

Equilibrium Properties of a System of Charged Particles

Motivated by the formal identification that can be made between the electronic charge e and a range parameter γ that we have previously studied, we have developed high-temperature, high-density expansions of the Helmholtz free energy and potentials of average force for systems of ions with repulsive cores. It appears that the natural ordering parameter for our free energy expansion is κR, where κ is the inverse Debye length and R an effective core diameter, and as a result the expansion is useful for either sufficiently high temperatures at any density or sufficiently low densities at any temperature. We find the terms in this expansion through order (κR)7. We have also found effects that will serve to modify the shielding length at high densities so that it is no longer just κ−1. One such effect depends upon the disparity in the core sizes of the positive and negative ions, and it vanishes only when the disparity does. A higher-order modification of the shielding length that remains in this symmetric case is also discussed.

Contribution of Interionic Forces to the Thermal Conductivity of Dilute Electrolyte Solutions

The contribution λion of interionic forces to the thermal conductivity of dilute electrolyte solutions is calculated on the basis of a model previously proposed and is found to vary as k3 or (concentration)3/2 according to the expression λion=13κ(e2/εT) ∑ βzβ2cβDβ, where k is the inverse Debye length, e is the electronic charge, ε is dielectric constant of solvent, z is charge number, cβ is concentration of dissolved ion β in units of number of ions per cubic centimeter, and Dβ is the self-diffusion coefficient of the ion.