Constant Surface Charge Density Research Articles

Electrophoresis of a weakly charged dielectric fluid droplet in a cylindrical pore.

Electrophoresis of a weakly charged dielectric droplet with constant surface charge density in a chargeless cylindrical pore is investigated theoretically in this study, focusing on the boundary confinement effect of the double layer, which in turn determines the ultimate motion of the droplet. A patched pseudo-spectral method based on the Chebyshev polynomial is adopted to solve the resulting governing fundamental electrokinetic equations. Mobility reversal, among other interesting phenomena, is observed when the droplet is in a narrow cylindrical pore. No such observation was made in the corresponding motion of a rigid particle. The droplet with a thick double layer may even move against the prediction based on the Coulomb electrostatic law, for instance, a positively charged droplet may move against the electric field. The significant enhancement of the motion-deterring double layer polarization due to the severe steric boundary confinement within a narrow cylindrical pore is found to be responsible for this seemingly peculiar phenomenon. Moreover, smaller droplets may move in the opposite direction of the larger ones. The results are useful in capillary electrophoresis involving droplets in particular and migration of droplets through narrow channels in general.

Constant Potential Thermodynamic Integration for Obtaining the Free Energy Profile of Electrochemical Reaction.

In this work, we advanced an efficient free energy sampling method based on constrained ab initio molecular dynamics (cAIMD) with a fully explicit solvent layer to depict the electrochemical reaction process at constant surface charge density, named the "Constant-Potential Thermodynamic Integration (CPTI)" method. For automatically adjusting surface charge density at different states, we built an "on-the-fly" procedure which is capable of managing all the necessary steps during cAIMD simulations, including the system pre-equilibrium, surface charge density updating, and force sampling. We applied it to predict the potential-dependent free energy profiles of CO2 adsorption on a single-atom catalyst. The results show that our method can not only account for changes in electrostatic potential energy associated with potential but also consider the potential-induced solvation effects. Our approach enables the accurate simulation of electrochemical environment by presenting the complete solid-liquid interface and efficient computation of electrocatalytic reaction energetics based on a robust potential descriptor.

Ion Concentration-Dependent Surface Charge Density Inside a Nanopore.

Surface charges shape the electrical double layer (EDL) structure at solid-liquid interfaces, critically influencing the performance of energy storage and micro/nanofluidic devices. However, accurately measuring surface charge density in nanoconfined spaces continues to be a challenge. Here, we introduce a methodology via solid-state nanopores that can investigate the dependence of surface charge density on salt concentrations and nanopore diameters. Measurements, complemented by a theoretical model, reveal that the surface charge density decreases as both the salt concentration in bulk solutions and the nanopore sizes are reduced. Notably, when the salt concentration in the bulk solution drops below 10-3 M, protons dominate ion conductance in a nanopore, resulting in a constant surface charge density. This study introduces an effective approach to surface charge characterization and may serve in the design of electrokinetically driven nanofluidic systems.

Solvation of Nanoions in Aqueous Solutions.

In recent years it has been increasingly recognized that different classes of large ions with multiple valency have effects conceptually similar to weakly solvated ions in the Hofmeister series, also labeled by the term chaotropic. The term "superchaotropic effect" has been coined because these effects are much more strongly pronounced for nanometer-sized ions, whose adsorption properties often resemble typical surfactants. Despite this growing interest in these nanometer-sized ions, a simple conceptual extension of the Hofmeister series toward nanoions has not been achieved because an extrapolation of the one-dimensional surface charge density scale does not lead to the superchaotropic regime. In this work, we discuss a generic model that is broadly applicable to ions of nearly spherical shape and thus includes polyoxometalates and boron clusters. We present a qualitative classification scheme in which the ion size appears as a second dimension. Ions of different sizes but the same charge density differ in their bulk solvation free energy. As the ions grow bigger at constant surface charge density, they become more stable in solution, but the adsorption behavior is still governed by the surface charge density. A detailed molecular dynamics simulation study of large ions that is based on a shifted Lennard-Jones potential is presented that supports the presented classification scheme.

Electrophoresis of a dielectric droplet with constant surface charge density.

Electrophoresis of a dielectric fluid droplet with constant surface charge density is investigated theoretically in this study. A pseudo-spectral method based on Chebyshev polynomials is adopted to solve the governing electrokinetic equations. It is found, among other things, that the larger the electrolyte strength in the ambient solution is, the slower the droplet moves in general. This is due to the strong screening effect of the large amount of indifferent counterions in the neighborhood of the droplet, with no reinforcement of potential-determining ions adsorbing to the droplet surface. The droplet comes to a complete halt eventually. Critical points are discovered for highly charged droplets, at which the droplet surface becomes immobile and the interior fluid stops recirculating. The droplet moves like a rigid particle with constant mobility regardless of its viscosity, a situation referred to as the "solidification phenomenon." The deadlock between the spinning motions on the charged droplet surface induced by the electric driving force and the hydrodynamic driving force respectively is responsible for this peculiar phenomenon. This is also observed for a dielectric droplet with constant surface electric potential. We demonstrate here that it occurs in the constant surface charge density situation as well.

Numerical Investigation of Diffusioosmotic Flow in a Tapered Nanochannel.

Diffusioosmosis concerns ionic flow driven by a concentration difference in a charged nano-confinement and has significant applications in micro/nano-fluidics because of its nonlinear current-voltage response, thereby acting as an active electric gating. We carry out a comprehensive computation fluid dynamics simulation to investigate diffusioosmotic flow in a charged nanochannel of linearly varying height under an electrolyte concentration gradient. We analyze the effects of cone angle (), nanochannel length (l) and tip diameter (), concentration difference ( = 0–1 mM), and external flow on the diffusioosmotic velocity in a tapered nanochannel with a constant surface charge density (). External flow velocity (varied over five orders of magnitude) shows a negligible influence on the diffusioosmotic flow inside the tapered nanochannel. We observed that a cone angle causes diffusioosmotic flow to move towards the direction of increasing gap thickness because of stronger local electric field caused by the overlapping of electric double layers near the smaller orifice. Moreover, the magnitude of average nanoflow velocity increases with increasing . Flow velocity at the nanochannel tip increases when is smaller or when l is greater. In addition, the magnitude of diffusioosmotic velocity increases with increasing . Our numerical results demonstrate the nonlinear dependence of tapered, diffusioosmotic flow on various crucial control parameters, e.g., concentration difference, cone angle, tip diameter, and nanochannel length, whereas an insignificant relationship on flow rate in the low Peclet number regime is observed.

Diffusiophoresis of a highly charged dielectric fluid droplet induced by diffusion potential

Diffusiophoresis of a dielectric fluid droplet in electrolyte solutions is investigated theoretically, focusing on the electrophoresis component resulting from the induced diffusion potential in the electrolyte solution when the diffusivities of cations and anions there are different. The resultant electrokinetic equations are solved with a pseudo-spectral method based on the Chebyshev polynomials. We found, among other things, that the electrophoresis component dominates at a larger Debye length, whereas the chemiphoresis component at a smaller Debye length for a dielectric droplet of a constant surface charge density. The two components are of comparable magnitudes in the NaCl solution. The dual between the spinning electric driving force tangent to the droplet surface and the hydrodynamic drag force reinforced by the motion-deterring electrokinetic Maxwell traction from the surrounding exterior osmosis flow is crucial in the determination of the ultimate droplet motion. Unlike the chemiphoresis component, which is independent of the sign of charges carried by the droplet, the droplet moving direction as well as its magnitude in the electrophoresis component depends on the sign of charges carried by the droplet as well as the direction of the electric field induced by the diffusion potential. This gives the electrophoresis component excellent maneuverability in practical applications like drug delivery and enhanced oil recovery, where migration of droplets toward regions of higher solute concentrations is often desired.

Electrostatic energy of interaction between uniformly charged hemispherical surfaces

The purpose of this study is to gain insights on the nature of the electrostatic energy of interaction between two uniformly charged hemispherical surfaces with constant surface charge density. The present system of these two interacting bodies represents a very difficult problem to solve since hemispherical surfaces lack the spherical symmetry of a uniformly charged spherical surface or a solid sphere counterpart. For this reason, an exact analytical calculation of the electrostatic interaction energy is not possible for an arbitrary orientation. However, this work shows that analytical results are possible for scenarios where the system of the two interacting hemispherical surfaces manifests some form of inherent symmetry. We identify such situations where combination of suitable mathematical tools appropriate for the axial symmetry together with transformations that apply to a spherical system of coordinates can reduce the difficulty of the problem. We show that there are some special cases where this very difficult integral problem can be reduced to a simpler one of summation of an infinite series. The approach is illustrated by showing explicitly the calculation of the interaction energy between the two uniformly charged hemispherical surfaces when they are brought together so that they are touching each other across the “equator”.

Diffusiophoresis of a highly charged dielectric fluid droplet

Diffusiophoresis of a dielectric fluid droplet with constant surface charge density in a symmetric binary electrolyte solution is investigated theoretically in this study, focusing on the chemiphoresis component, the very heart of diffusiophoresis. The resultant electrokinetic equations are solved with a pseudo-spectral method based on Chebyshev polynomial in the spirit of a computational fluid dynamic simulation. Reversions of moving directions are found for droplets less viscous than ambient solution when the electrolyte strength is increased due to the buildup of osmosis pressure in front of the moving droplets leading to an osmosis pressure gradient upon the droplet. The upward spouting effect of the spinning droplet surface is also responsible this buildup, which hinders the downward migration of ions and holds them up there. A solid particle may move faster than a gas bubble due to the energy consumption in the formation of an induced exterior vortex flow nearby surrounding the gas bubble. The less viscous the droplet is, the more severe this consumption is. A “solidification” phenomenon is observed where all the droplets move at the same speed with their surfaces and interior fluids motionless like rigid particles. Funnel-shape local extrema of mobility profiles provide solid evidence that the diffusion-induced double layer polarization is the very cause of the droplet motion in chemiphoresis. Excellent agreement with experimental data for a rigid particle is obtained. The study provides insights and guidelines in practical applications like drug delivery and other dead-end-pore types of operations such as EOR.

Analytical solution to dielectric droplet diffusiophoresis under Debye-Hückel approximation.

A simple analytical formula is obtained for the diffusiophoresis of a dielectric fluid droplet in symmetric binary electrolyte solutions under Debye-Hückel approximation valid for weakly charged droplets. The chemiphoresis is found to yield negative mobilities most of the time for droplets of constant surface charge density, which implies that the droplets tend to move away from the source releasing ionic chemicals. This is undesirable in some practical applications like drug delivery with liposomes in terms of conveying the drug-carrying liposomes to the desired area in the human body releasing specific ionic chemicals utilizing the self-guiding nature of diffusiophoresis. The further involvement of the electrophoresis component, however, may change the scenario via the oriented electric field generated by the induced diffusion potential. The lesson here is that while the impact of the chemiphoresis component is determined by nature and uncontrollable, the electrophoresis component serves as an artificially adjustable factor via choosing droplets with the surface charge of appropriate sign in practical applications. The results here have potential use in practical applications such as drug delivery. The portable simple analytical formula is a powerful asset to experimental researchers and design engineers in colloid science and technology to facilitate their works.

Electrostatic interaction between two spheroidal particles at large separations

Asymptotic expressions for the electrostatic interaction energy between two weakly charged spheroidal particles (prolate and oblate) with uniform surface potential or surface charge density at large separations in an electrolyte solution are derived on the basis of the linear superposition approximation. The electrostatic interaction between two spheroids is found to be the screened Coulomb interaction between two hypothetical point particles with orientation-dependent charges. Explicit interaction energy equations, which are expressed in terms of spheroidal wave functions, are given for four particular configurations of two similar spheroids, that is, side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates. Comparison is also made with results obtained using Derjaguin’s approximation. Electrostatic interaction between two spheroids • Asymptotic expressions for the electrostatic interaction energy between two spheroidal particles are derived. • Constant surface potential and charge density cases are considered. • Side-to-side prolates, end-to-end prolates, side-to-side oblates, and end-to-end oblates are considered.

Ionic Conductivity of Nanopores with Electrically Conductive Surface: Comparison Between 1D and 2D Models

AbstractNanoporous membranes with high ionic conductivity are advantageous in such electrochemical processes as (reverse) electrodialysis, capacitive deionization, and hydrogen energy conversion. In membranes with electrically conductive surface, the conductivity can be regulated by varying the surface potential. This work is devoted to the theoretical study of switchable ionic conductivity. The transport of ions is described by the 2D space charge model and 1D uniform potential model taking into account the Stern layer. The conductivity decreases with lowering the Stern layer permittivity due to enhanced screening of electronic surface charge. The growth of surface potential leads to the conductivity enhancement due to accumulation of more counter‐ions inside the nanopore. For nanopores with constant surface charge density, the ionic conductivity follows the bulk electrolyte conductivity at high ion concentrations, and becomes independent of concentration when the latter is low. In contrast, the nanopores with constant surface potential demonstrate a linear decrease of conductivity with lowering the logarithm of ion concentration. The deviation between 1D and 2D models becomes noticeable at higher values of Stern layer permittivity, pore radius, electrolyte concentration, and surface potential. The proposed models are verified by comparison with experimental data on pure water conductivity in charged porous matrix.

Transient electrophoresis of a conducting spherical particle embedded in an electrolyte-saturated Brinkman medium.

In this study, the time-dependent electrophoretic motion of a conducting spherical particle embedded in an arbitrary electrolyte solution saturated porous medium is investigated. The porous medium is uniformly charged and the embedded hard particle is charged with constant -potential or constant surface charge density. The unsteady modified Brinkman equation with an electric force term, which governs the fluid velocity field, is used to model the porous medium and is solved by Laplace's transform technique. An analytical expression for the electrophoretic velocity of the spherical particle is obtained in Laplace transform domain as a function of the relevant parameters, and its inversion is obtained through numerical techniques. Also, in this study, the steady-state electrophoretic velocity is obtained analytically as linear functions of -potential (or surface density charge) and the fixed charge density. The steady-state electrophoretic velocity is displayed graphically for various relevant parameters and compered with the available data in the literature. Also, the numerical values of the transient electrophoretic velocity are plotted versus the nondimensional elapsed time and discussed for different values of the Debye length parameter, density ratio, permeability of the porous medium, and for high and nonconducting particles.

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers.

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.

Electrospun PVDF-TrFE/MXene Nanofiber Mat-Based Triboelectric Nanogenerator for Smart Home Appliances.

Understanding of the triboelectric charge accumulation from the view of microcapacitor formation plays a critical role in boosting the output performance of the triboelectric nanogenerator (TENG). Here, an electrospun nanofiber-based TENG (EN-TENG) using a poly(vinylidene fluoride-trifluoroethylene) (PVDF-TrFE)/MXene nanocomposite material with superior dielectric constant and high surface charge density is reported. The influence of dielectric properties on the output performance of the EN-TENG is investigated theoretically and experimentally. The fabricated EN-TENG exhibited a maximum power density of 4.02 W/m2 at a matching external load resistance of 4 MΩ. The PVDF-TrFE/MXene nanocomposite improved the output performance of the EN-TENG fourfold. The EN-TENG successfully powered an electronic stopwatch and thermo-hygrometer by harvesting energy from human finger tapping. Moreover, it was utilized in smart home applications as a self-powered switch for controlling electrical home appliances, including fire alarms, fans, and smart doors. This work presents an effective and innovative approach toward self-powered systems, human-machine interfaces, and smart home applications.

Electroosmotic thrusters in soft nanochannels for space propulsion

Space propulsion of electroosmotic thrusters (EOTs) with a soft charged nanochannel is investigated considering the Navier slip boundary and constant surface charge density on the walls of slit channels. The soft nanochannel is characterized by a wall-grafted ion-penetrable charged polyelectrolyte layer (PEL). The Poisson–Boltzmann equation is solved to give the electric potential distribution based on the assumption of the Debye–Hückel linearization for the low electric potential. An analytical solution of the electroosmotic velocity through the soft channel is obtained. The thrust, specific impulse, and total input power of EOTs produced by the electroosmotic flow are presented, and then, two significant physical quantities, thruster efficiency and thrust-to-power ratio, are described. It is found that these performance curves strongly depend on the slip length, surface charge density on the walls, drag coefficient, equivalent electric double layer thickness, PEL thickness, and density ratio of the PEL to the electrolyte solution layer. By analyzing and optimizing these design parameters, the simulated EOTs can deliver the thrust from 0 μN to 10 µN as well as the specific impulse from 40 s to 100 s, and the thruster efficiency up to 87.22% is realized. If more thrust control and kinetic energy are needed for different space missions, an array composed of thousands of single EOT emitters is constructed and maintains high thruster efficiency. Moreover, during mission operation, the total potential can be simply varied to optimize the performances of thrusters at any moment.

Approximate analytic expressions for the electrophoretic mobility of spheroidal particles

Approximate analytic expressions are derived for the electrophoretic mobility of spheroidal particles (prolate and oblate) carrying low zeta potential in an electrolyte solution under an applied tangential or transverse electric field. The present approximation method, which is based on the observation that the electrophoretic mobility of a particle is determined mainly by the distortion of the applied electric field by the presence of the particle. The exact expression for the equilibrium electric potential distribution around the particle, which can be expressed as an infinite sum of spheroidal wave functions, is not needed in the present approximation. The electrophoretic mobility values calculated with these approximate expressions for spheroidal particles with constant surface potential or constant surface charge density are in excellent agreement with the exact numerical results of previous reports with the relative errors less than about 4%.

Features of Electrical Double Layers Formed Around Strongly Charged Nanoparticles Immersed in an Electrolyte Solution. The Effect of Ion Sizes

Using the modified Poisson–Boltzmann (PB) theory, which includes restrictions on the maximum attainable concentration of ionic species in a solution Cmax determined by their effective sizes, the distributions of electrostatic potential φ(r) and ion concentration near a spherical nanoparticle with radius a immersed in a 1 : 1 electrolyte solution have been studied under the conditions of constant surface charge density σs. For weakly charged particles, the φ(r) profiles are almost independent of Cmax and coincide with the profile obtained in terms of the classical PB model. Surface potential |φs| gradually increases with a rise in |σs|. Far from the particle, when the potential becomes lower than its thermal value $${{\varphi }_{{\text{T}}}} = \frac{{kT}}{e}$$ (k is the Boltzmann constant, T is the temperature, and e is the elementary charge), the potential decreases exponentially irrespective of the counterion sizes: $$\psi \left( r \right) = \frac{{\varphi \left( r \right)}}{{{{\varphi }_{{\text{T}}}}}} = {{\psi }_{{{\text{eff}}}}}\frac{a}{r}\exp \left( { - \kappa \left( {r - a} \right)} \right),$$ where r is the distance from the particle center and κ is the reciprocal Debye radius. According to the classical PB theory, the growth of the surface charge leads to the saturation of $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|~\left( { \to 4} \right),$$ with the value of $$\left| {{{\psi }_{{\text{s}}}}} \right|$$ obtained by solving the nonlinear PB equation being higher than $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|.$$ In the modified PB theory, which takes into account the size of electrolyte ions in the simplest form, this effect of saturation is absent. Now, $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|$$ depends on both the value of the surface charge and the sizes of counterions. Moreover, at a large size of counterions, $$\left| {{{\psi }_{{{\text{eff}}}}}} \right|$$ substantially exceeds the corresponding value obtained by solving the nonlinear modified PB equation. The difference between the electrical double layer properties obtained by solving the classical and modified PB equations directly follows from the fact that the modified theory predicts the appearance of a condensed layer at a particle surface, with the concentration of counterions in this layer being equal to Cmax. Therewith, the thickness of the layer grows with increasing |σs| (at a constant size of the ions) and the size of the ions (at constant σs).

Analysis and spline approximation of surface charge and potential at planar electrochemical interfaces

In this paper, we consider the Poisson‐Boltzmann theory to model the electrostatic potential of a bulk electrolyte containing a single planar charged surface. In the case of a constant surface charge density, we address the problem as a ODE system by using the stability theory for autonomous dynamical systems. By stating that the surface charge density and the surface potential are non‐linearly related through the Grahame equation, we give a description of the stable manifold of the system. To solve the Grahame equation and obtain an approximation for the stable manifold, we propose a smoothing collocation method based on cubic splines, including implementation details and numerical results.

Steady-state voltage distribution in three-dimensional cusp-shaped funnels modeled by PNP.

We study here the bulk electro-diffusion properties of micro- and nanodomains containing a cusp-shaped structure in three-dimensions when the cation concentration dominates over the anions. To determine the consequences on the voltage distribution, we use the steady-state Poisson-Nernst-Planck equation with an integral constraint for the number of charges. A non-homogeneous Neumann boundary condition is imposed on the boundary. We construct an asymptotic approximation for certain surface charge distribution that agree with numerical simulations. Finally, we analyze the consequences of several piecewise constant non-homogeneous surface charge densities, motivated by designing new nanopipettes. To conclude, when electro-neutrality is broken at the scale of hundreds of nanometers, the geometry of cusp-shaped domains influences the voltage profile, specifically inside the cusp structure. The main results are summarized in the form of new three-dimensional electrostatic laws for non-electroneutral electrolytes. These formula provide a refined characterization of voltage distribution at steady-state in neuronal microdomains such as dendritic spines, but can also be used to design nanometric patch-pipettes.