Double Layer Potential Research Articles

A new methodology for computing ionic profiles and disjoining pressure in swelling porous media

A new two-scale computational model is proposed to construct the constitutive law of the swelling pressure which appears in the modified form of the macroscopic effective stress principle for expansive clays saturated by an aqueous electrolyte solution containing multivalent ionic species. The microscopic non-local nanoscale model is constructed based on a coupled Poisson-Fredholm integral equation arising from the thermodynamics of inhomogeneous fluids in nanopores (Density Functional Theory), which governs the local electric double layer potential profile coupled with the ion-particle correlation function in an electrolytic solution of finite size ions. The local problem is discretized by invoking the eigenvalue expansion of the convolution kernel in conjunction with the Galerkin method for the Gauss-Poisson equation. The discretization of the Fredholm equation is accomplished by a collocation scheme employing eigenfunction basis. Numerical simulations of the local ionic profiles in rectangular cell geometries are obtained showing considerable discrepancies with those computed with Poisson-Boltzmann based models for point charges, particularly for divalent ions in calcium montmorillonite. The constitutive law for the disjoining pressure is reconstructed numerically by invoking the contact theorem within a post-processing approach. The resultant computational model is capable of capturing ranges of particle attraction characterized by negative values of the disjoining pressure overlooked by the classical electric double layer theory. Such results provide further insight in the role the swelling pressure plays in the modified macroscopic effective stress principle for expansive porous media.

Dust ion acoustic solitary structures in presence of nonthermal electrons and isothermal positrons

The Sagdeev potential technique has been employed to study the dust ion acoustic solitary waves and double layers in an unmagnetized collisionless dusty plasma consisting of negatively charged static dust grains, adiabatic warm ions, and isothermally distributed electrons and positrons. A computational scheme has been developed to draw the qualitatively different compositional parameter spaces or solution spaces showing the nature of existence of different solitary structures with respect to any parameter of the present plasma system. The qualitatively distinct solution spaces give the overall scenario regarding the existence of different solitary structures. The present system supports both positive and negative potential double layers. The negative potential double layer always restricts the occurrence of negative potential solitary waves, i.e., any sequence of negative potential solitary waves having monotonically increasing amplitude converges to a negative potential double layer. However, there exists a parameter regime for which the positive potential double layer is unable to restrict the occurrence of positive potential solitary waves. As a result, in this region of the parameter space, there exist solitary waves after the formation of positive potential double layer, i.e., positive potential supersolitons have been observed. But the amplitudes of these supersolitons are bounded. A general theory for the existence of bounded supersolitons has been discussed analytically by imposing the restrictions on the Mach number. For any small value of positron concentration, there is no effect of very hot positrons on the dust ion acoustic solitary structures. The qualitatively different solution spaces are capable of producing new results for the formation of solitary structures.

43 years of fun basic plasma physics experiments

This paper presents an overview of some of the most enjoyable low-temperature plasma experiments I have carried out with my students over the course of my career. It begins with the connection between solitons and the Schrodinger equation, and continues with the discovery of cylindrical solitons. From moving structures, we progress to stable ones: sheaths and double layers. How can we measure double-layer potentials? This question leads to the development of emissive probe techniques. From there, we move on to the Bohm criterion and the first measurements of the full plasma potential variations of sheaths and presheaths in a single-species plasma, and later in two-ion-species plasmas. Finally, our recent measurements of double layers and presheaths in uniform helicon plasma are presented.

Phenomenological analysis of near-threshold periodic modulations of the proton timelike form factor

We have recently highlighted the presence of a periodically oscillating 10 \% modulation in the BABAR data on the proton timelike form factors, in the reaction $e^++e^-$ $\rightarrow$ $\bar{p}+p$. Here we deepen our previous data analysis, and confirm that in the case of several standard parametrizations it is possible to write the form factor in the form $F_0$ $+$ $F_{osc}$, where $F_0$ is a parametrization expressing the long-range trend of the form factor (for $q^2$ ranging from the $\bar{p}p$ threshold to 36 GeV$^2$), and $F_{osc}$ is a function of the form $\exp(-Bp)\cos(Cp)$, where $p$ is the relative momentum of the final $\bar{p}p$ pair. Error bars allow for a clean identification of the main features of this modulation for $q^2$ $<$ 10 GeV$^2$. Assuming this oscillatory modulation to be an effect of final state interactions between the forming proton and the antiproton, we propose a phenomenological model based on a double-layer imaginary optical potential. This potential is flux-absorbing when the distance between the proton and antiproton centers of mass is $\gtrsim$ 1.7-1.8 fm and flux-generating when it is $\lesssim$ 1.7-1.8 fm. The main features of the oscillations may be reproduced with some freedom in the potential parameters, but the transition between the two layers must be sudden (0-0.2 fm) to get the correct oscillation period. The flux-absorbing part of the $\bar{p}p$ interaction is due to the annihilation of $\bar{p}p$ pairs into multi-meson states. We interpret the flux-creating part of the potential as due to the creation of a $1/q$-ranged state when the virtual photon decays into a set of current quarks and antiquarks, including heavy ones, that may exist for a short time. The decay of these large mass states leads to an intermediate stage regeneration of the $\bar{p}p$ channel.

Detecting and identifying small molecules in a nanopore flux capacitor

A new method of molecular detection in a metallic-semiconductor nanopore was developed and evaluated with experimental and computational methods. Measurements were made of the charging potential of the electrical double layer (EDL) capacitance as charge-carrying small molecules translocated the nanopore. Signals in the charging potential were found to be correlated to the physical properties of analyte molecules. From the measured signals, we were able to distinguish molecules with different valence charge or similar valence charge but different size. The relative magnitude of the signals from different analytes was consistent over a wide range of experimental conditions, suggesting that the detected signals are likely due to single molecules. Computational modeling of the nanopore system indicated that the double layer potential signal may be described in terms of disruption of the EDL structure due to the size and charge of the analyte molecule, in agreement with Huckel and Debye’s analysis of the electrical atmosphere of electrolyte solutions.

Hypersingular meshless method using double-layer potentials for three-dimensional exterior acoustic problems.

Three-dimensional exterior acoustic problems with irregular domains are solved using a hypersingular meshless method. In particular, the method of fundamental solutions (MFS) is used to formulate and analyze such acoustic problems. It is well known that source points for MFS cannot be located on the real boundary due to the singularity of the kernel functions. Thus, the diagonal terms of the influence matrices are unobtainable when source points are located on the boundary. An efficient approach is proposed to overcome such difficulties, when the MFS is used for three-dimensional exterior acoustic problems. This work is an extension of previous research on two-dimensional problems. The solution of the problem is expressed in terms of a double-layer potential representation on the physical boundary. Three examples are presented in which the proposed method is compared to the MFS and boundary element method. Good numerical performance is demonstrated by the proposed hypersingular meshless method.

Mechanistic aspects of hydrazine-induced Pt colloid instability and monitoring aggregation kinetics with nanoparticle impact electroanalysis

Here we investigate the mechanistic aspects of Pt nanoparticle (NP) aggregation in solutions typically used for detecting NP/electrode impacts by electrocatalytic amplification (ECA). We previously proposed a general mechanism for Pt colloid destabilization that involved the participation of both the hydrazine redox probe and the pH buffer species as coagulants. Herein the Pt NP coagulation and aggregation mechanisms were further investigated with microscopic kinetic NP concentration monitoring and zeta potential measurements using nanoparticle tracking analysis (NTA), as well as open circuit potential experiments with a citrate-treated polycrystalline Pt surface to assess electrical double layer potential. After considering the combined results of these experiments we propose that the colloidal stability of citrate-capped platinum nanoparticles involves much more than the typical physicochemical interactions predicted by DLVO theory. A structure based on intermolecular H-bonding in the citrate capping layer is the most plausible explanation for the exceptional stability of large Pt NPs in high ionic strength buffers. Thus, the mechanism of Pt NP aggregation includes specific reactive contributions from hydrazine. The catalytic decomposition of hydrazine, in particular, is thought to occur to some extent at the citrate-coated Pt surface while the citrate remains adsorbed. Evolved gases such as ammonia and possible surface bound intermediates from Pt-catalyzed decomposition of hydrazine may disrupt the stability of the citrate layer, causing colloidal instability and thus promoting Pt NP coagulation. In the closing section, we demonstrate nanoparticle impact electroanalysis by ECA detection as a method to quantify Pt NP concentration with adequate time resolution for monitoring the kinetics of Pt NP coagulation.

Effect of eccentricity on the electroosmotic flow in an elliptic channel

The electroosmotic flow in an elliptic channel having constant surface potential (CSP) or charge density (CSCD) is considered at low potential and arbitrary double layer thickness. Analytical expressions for the flow velocity and the corresponding asymptotic results for thick double layers that are readily applicable to experimentalists are recovered. For the range of salt concentration usually encountered in practice, the mean flow velocity for the case of CSP differs both quantitatively and qualitatively from that for the case of CSCD. Using an equivalent circular channel to simulate an elliptic one is inappropriate, in general, neither is assuming electroneutrality on the channel axis even when double layer is ca. 1/3 of the equivalent channel radius.

Integral Equations of Diffraction Problems with Unbounded Smooth Obstacles

The paper is devoted to the boundary integral equations method for the diffraction problems on obstacles D in \({\mathbb{R}^{n}}\) with smooth unbounded boundaries for Helmholtz operators with variable coefficients. The diffraction problems are described by the Helmholtz operators $$\mathcal{H}u(x)=\left( \rho (x)\nabla \cdot \rho ^{-1}(x)\nabla+a(x)\right) u(x),\quad x\in \mathbb{R}^{n}$$ where \({\rho, a}\) belong to the space of the infinitely differentiable functions on \({\mathbb{R}^{n}}\) bounded with all derivatives. We introduce the single and double layer potentials associated with the operator \({ \mathcal{H}}\), and reduce by means of these potentials the Dirichlet, Neumann, and Robin problems to pseudodifferential equations on the infinite boundary \({\partial D}\). Applying the limit operators method we study the Fredholm properties and the invertibility of the boundary pseudodifferential operators in the Sobolev spaces \({H^{s}(\partial D),s\in \mathbb{R}}\).

Sharp norm estimates of layer potentials and operators at high frequency

In this paper, we investigate single and double layer potentials mapping boundary data to interior functions of a domain at high frequency λ 2 → ∞ . In the high frequency regime the key problem is the dependence of mapping norms on the parameter λ . For single layer potentials, we find that the L 2 ( ∂ Ω ) → L 2 ( Ω ) norms decay in λ . The rate of decay depends on the curvature of ∂Ω: The norm is λ − 3 / 4 in piecewise smooth domains and λ − 5 / 6 if the boundary ∂Ω is positively curved. The double layer potential, however, displays uniform L 2 ( ∂ Ω ) → L 2 ( Ω ) bounds independent of curvature. By various examples, we show that all our estimates on layer potentials are sharp. Appendix A by Galkowski gives bounds L 2 ( ∂ Ω ) → L 2 ( ∂ Ω ) for the single and double layer operators at high frequency that are sharp modulo log ⁡ λ . In this case, both the single and double layer operator bounds depend upon the curvature of the boundary.

Explicit description of generalized double-layer potentials for the Lamé system

Matrix kernels of generalized double-layer potentials representing homogeneous functions of zero degree are described in explicit form in terms of elastic moduli and the roots of the characteristic equation in the upper half-plane (more precisely, in terms of the sum and product of these roots). The cases of orthotropic and isotropic media are considered separately, for which these formulas simplify substantially.

On the free terms of the dual BIE for N-dimensional Laplace problems

Dual boundary integral equations for the N-dimensional Laplace problems with a smooth boundary are derived by using the contour approach surrounding the singularity. The potentials resulted from the four kernel functions in the dual formulation have different properties across the smooth boundary. For the generalization, we focus on the N-dimensional Laplace equation. The Hadamard principal value (H.P.V.) is derived naturally and is composed of two parts, the Cauchy principal value (C.P.V.) and an unbounded boundary term. The hypersingular integral is not a divergent integral since we can collect the C.P.V. and the unbounded term together. Besides, the weighting of the free term contributed by different kernels is also examined. Finally, a special case of the four-dimensional Laplace equation is implemented and the free term, for any dimension are obtained. The contributions of the free terms for the boundary normal derivative of potential due to the single (L kernel) and the double (M kernel) layer potentials are 1/N and (N−1)/N, respectively. It is an interesting phenomenon that the hypersingular kernel contributes more than the singular kernel, and, in addition, the former also yields an unbounded boundary term.

Arbitrary amplitude slow electron-acoustic solitons in three-electron temperature space plasmas

We examine the characteristics of large amplitude slow electron-acoustic solitons supported in a four-component unmagnetised plasma composed of cool, warm, hot electrons, and cool ions. The inertia and pressure for all the species in this plasma system are retained by assuming that they are adiabatic fluids. Our findings reveal that both positive and negative potential slow electron-acoustic solitons are supported in the four-component plasma system. The polarity switch of the slow electron-acoustic solitons is determined by the number densities of the cool and warm electrons. Negative potential solitons, which are limited by the cool and warm electron number densities becoming unreal and the occurrence of negative potential double layers, are found for low values of the cool electron density, while the positive potential solitons occurring for large values of the cool electron density are only limited by positive potential double layers. Both the lower and upper Mach numbers for the slow electron-acoustic solitons are computed and discussed.

Generalized double-layer potentials in anisotropic elasticity on the plane

Generalized double-layer potentials in anisotropic elasticity on the plane

Reductive Hydrogenation of the Aqueous Rutile TiO2(110) Surface

At a semiconducting oxide water interface, two double layers (space charge layer and electric double layer) can develop near the interface. The double layer potentials can vary against the voltage and pH because they depend on the density of surface charges that have both electronic and protonic components. While the surface protonic charge density is determined by the acid-base equilibrium between the surface and the solution, electronic charges can also develop owing to the presence of electronic surface states that serve as traps for electrons and holes. A further complication is that the protonation and reduction (or deprotonation and oxidation) can couple at certain surface sites through proton coupled electron transfer (PCET) reactions. This is believed to be the cause of the observed (non-)Nernstian relation between the changes in electrode potentials and the pH, an approximately linear relationship with varying slopes depending on the nature of oxide surfaces. With the aim of addressing this issue, we present a first principles study for computation of the PCET energies of the reductive hydrogenation of the aqueous TiO2 surface. Our calculation shows that excess electrons and protons don’t interact chemically at the interface, suggesting that the coupling between them is mainly electrostatic in nature. Our previous finding, on the other hand, showed that the electronic holes can chemically couple with protons through the terminal water. Combining these two results, we conclude that TiO2 may show different dependences of the electrode potentials on the pH at the cathodic and anodic voltage conditions.

L q -solution of the Robin Problem for the Oseen System

We define Oseen single layer and double layer potentials and study their properties. Using the integral equation method we prove the existence and uniqueness of an Lq-solution of the Robin problem for the Oseen system.

Existence domains of slow and fast ion-acoustic solitons in two-ion space plasmas

A study of large amplitude ion-acoustic solitons is conducted for a model composed of cool and hot ions and cool and hot electrons. Using the Sagdeev pseudo-potential formalism, the scope of earlier studies is extended to consider why upper Mach number limitations arise for slow and fast ion-acoustic solitons. Treating all plasma constituents as adiabatic fluids, slow ion-acoustic solitons are limited in the order of increasing cool ion concentrations by the number densities of the cool, and then the hot ions becoming complex valued, followed by positive and then negative potential double layer regions. Only positive potentials are found for fast ion-acoustic solitons which are limited only by the hot ion number density having to remain real valued. The effect of neglecting as opposed to including inertial effects of the hot electrons is found to induce only minor quantitative changes in the existence regions of slow and fast ion-acoustic solitons.

Symmetric singular boundary method for potential problems with mixed boundary conditions

The paper presents a symmetric formulation of the singular boundary method (SBM) for potential problems, whose resultant interpolation matrix is symmetric irrespective of boundary conditions. The SBM is a recent strong-form boundary discretization numerical technique, which is meshless, integration-free, and easy-to-implement. In stark contrast to the method of fundamental solution that requires the fictitious boundary outside physical domain to avoid the singularity of fundamental solution, the SBM can place source points, which coincide with collocation points, on the real physical boundary via the concept of origin intensity factor. The proposed symmetric formulation preserves all the merits of the SBM. The novel idea behind this study is that the unknown solution is represented by both the single and double layer potentials compared with the single layer potential in the original SBM. Numerical solution of three-dimensional potential and two-dimensional fluid flow and heat transfer problems demonstrates the utility and validity of the present SBM symmetric formulation.

Integral equations requiring small numbers of Krylov-subspace iterations for two-dimensional smooth penetrable scattering problems

This paper presents a class of boundary integral equations for the solution of problems of electromagnetic and acoustic scattering by two-dimensional homogeneous penetrable scatterers with smooth boundaries. The new integral equations, which, as is established in this paper, are uniquely solvable Fredholm equations of the second kind, result from representations of fields as combinations of single and double layer potentials acting on appropriately chosen regularizing operators. As demonstrated in this text by means of a variety of numerical examples (that resulted from a high-order Nyström computational implementation of the new equations), these “regularized combined equations” can give rise to important reductions in computational costs, for a given accuracy, over those resulting from previous iterative boundary integral equation solvers for transmission problems.

A Theorem on the Surface Traction Field in Potential Representations of Stokes Flow

A characterization theorem for moments of the Stokes traction field on the bounding surface of a three-dimensional flow domain is stated and proved. Whereas the single-layer Stokes potentials lead to a weakly singular representation of the traction field on the boundary, the double-layer potentials lead to a hyper-singular representation, which is less convenient for analysis and numerics. However, in various applications, pointwise values of the traction field up to the boundary are not of direct interest, but rather moments of the traction with respect to given weighting functions. It is shown that such moments can be characterized in terms of weakly singular integrals for both the single- and double-layer potentials, and moreover, the characterization does not involve derivatives of the potential densities or the moment weighting functions. The result for the double-layer potentials can be viewed as a generalization and enhancement of the classic Lyapunov--Tauber theorem and may make these potentials easier to use. An example application to the modeling of immersed flexible bodies is discussed.