Planar Double Layer Research Articles

Electrostatic Interaction between Two Planar Double Layers: A Further Extension to Langmuir's Method

A simple method for calculating the interaction force and interaction free energy per unit area between parallel flat plates with high surface potential in the case of constant surface potential and constant surface charge is presented. It is a supplement to previous works (Langmuir, 17: 2167 (2001) and J. Colloid Interface Sci., 241: 81 (2001)). Although this approximation is essentially from simplifying the nonlinear PB equation in the case of high surface potentials, these approximate expressions work quite well for small plate separations for all values of the surface potentials.

Planar double layers in magnetized plasmas: Fine structures and their consequences

Using two‐dimensional electrostatic particle‐in‐cell code, we study the dynamical behavior of planar double layers (DLs) in magnetized plasmas nonuniformly distributed along an ambient magnetic field Bo. The plasma is driven by a magnetic field‐aligned (parallel) potential drop with a polarity appropriate for the return current auroral plasma; the low‐density hot plasma at the top of the simulation plane is positive with respect to the high‐density cold plasma at the bottom. In response to the applied potential drop, a planar double layer rapidly forms at the evolving interface between the cold and hot plasmas. The ions accelerated downward by the DL into the cold plasma generate low‐frequency electrostatic ion cyclotron (EIC) fluctuations; their nonlinear evolution fragments the DL horizontally into filamentary DLs having U‐ and V‐shaped equipotentials like in diverging electrostatic shocks (D‐shocks). The filamentary potential structures have density depletions bounded by high‐density plasma columns. The cold electrons accelerated by the parallel fields inside the DL substructures (DLSS) generate electron holes, which become the dominant feature of the plasma above the regions of the parallel fields. The initial filaments have the size of hot ion Larmor radius, but at latter times the transverse size of the filaments increases as they dynamically evolve both in shape and size with the transverse ion heating. During the process of fragmentation of the initial DL the cold ions undergo considerable transverse acceleration by the perpendicular electric fields in the developing substructures. Those initially cold ions which manage to get above the time‐dependent DLSSs and even the hot ions traversing the region above them are seen to undergo significant heating in both parallel and perpendicular directions by the sustained interactions with the upward moving large‐amplitude electron holes. Relevance of our findings to the physics of the auroral return current plasma is discussed.

Polysulfonylamine, CLXVII [1]. Wasserstoffbrücken in kristallinen Onium-dimesylamiden: Ein robustes Dreipunktmuster als Bestandteil eines 2D-, eines zweifach verwobenen 2D- und eines 3D-Wasserstoffbrücken-Netzwerks / Polysulfonylamines, CLXVII [1]. Hydrogen Bonding in Crystalline Onium Dimesylamides: A Robust Three-Point Pattern Integrated into a 2D, a Twofold Interwoven 2D, or a 3D Hydrogen-Bond Network

Abstract As an exercise in crystal engineering, preparations and low-temperature X-ray structures are reported for three ionic solids of general formula BH+(MeSO2)2N−, where BH+ is 2,4,6- triaminopyrimidinium (compound 1, triclinic, space group P1, Z = 2), 2,6-diaminopyridinium (2, monoclinic, C2/c, Z = 8), or 2,4-diaminopyrimidin-6(1H)-on-3-ium (3, monoclinic, P21/c, Z = 4). As a common feature, the onium cations in question exhibit a trifunctional hydrogen-bond donor sequence H-N-C-N(H)-C-N-H that is complementary to a W-shaped O-S-N-S-O fragment of the anion. Consequently, each structure displays a [DDD:AAA] three-point hydrogen-bond pattern formed by two lateral N-H···O bonds and a central N-H···N interaction. This grouping is integrated as a robust supramolecular synthon into two-dimensional (1, 2) or three-dimensional (3) hydrogen-bond networks, in which all good donors and all good acceptors are involved (excepting one S=O group in 2). In structure 1, the approximately planar cation-anion layers are perfect mosaics composed of 6-membered pyrimidine heterocycles and seven crystallographically independent types of 8-, 10-, 12- or 24-membered rings based upon hydrogen bonding. In contrast, the corresponding layers in structure 2 are marred by large 40-membered voids; in order to achieve dense packing, the imperfect layers adopt a strongly corrugated shape and interpenetrate to form twofold interwoven and nearly planar double-layers. Each structure features close C-H···O contacts consistent with weak hydrogen bonding; in the layer structures 1 and 2, some of these interactions serve as links between adjacent or interwoven layers.

Dynamic planar double layers: Filamentary substructures, electron holes and ion heating

[1] Using 2.5-D particle-in-cell (PIC) simulations, the formation of a double layer (DL) and its stability are studied. The ion beam accelerated by the DL generates fluctuations above the ion cyclotron frequency, triggering its fragmentation into substructures (DLSSs) like U- and V-shaped electrostatic shocks. The DL and DLSSs are seen to continually emit electron holes of varied shapes and sizes into the high-potential plasma above them, while below ions are transversely heated.

Molecular theory of an electrochemical double layer in a nanoporous carbon supercapacitor

We develop the replica RISM theory of electrolyte solution sorbed in a nanoporous carbon electrode. The model comprises carbon nanospheres forming a disordered network with the porosity, pores sizes and surface area fitted to carbonized polyvinylidene chloride (PVDC) material, and to activated carbon. We obtained the huge capacitance comparable to that achieved in supercapacitors, and found the higher capacitance per pores surface for carbonized PVDC material with uniform nanoporous texture than for activated carbon with micro- as well as nanopores. Unlike a planar electrochemical double layer with the voltage dominated by the inner layer, the nanoporous supercapacitor voltage is driven by the solvation chemical potentials of the sorbed ions.

Studies of interdiffusion in polymer blends by PALS

It is shown that positron annihilation parameters such as the average positron lifetime τ av and the ortho-positronium intensity I 3 respond to chemical inhomogeneities in disperse blends and their disappearance due to mutual diffusion of molecules of both phases, when Ps formation is inhibited by one of the constituents. This effect can be used to study mixing in initially demixed compatible blends, demixing in initially mixed incompatible blends, and phase diagrams. With a slow positron beam the concentration-distance profile in thin planar double layer films may be scanned systematically.

Parallel electric fields in the upward current region of the aurora: Numerical solutions

Direct observations of the parallel electric field by the Fast Auroral Snapshot satellite and the Polar satellite suggest that the ionospheric boundary of the auroral cavity is consistent with an oblique double layer that carries a substantial fraction (roughly 5% to 50%) of the auroral potential. A numerical solution to the Vlasov–Poisson equations of a planar, oblique double layer reproduces many of the properties of the observed electric fields, electron distributions, and ion distributions. The solutions indicate that the electron and ion distributions that emerge from the ionospheric side dominate the structure of the double layer. The ionospheric electron distribution includes scattered and reflected (mirrored) primaries, auroral secondaries, photoelectrons, and a cold population. A large fraction of the ionospheric electrons is reflected by the parallel electric field whereas the ionospheric ions are strongly accelerated. The steep density gradient between the ionosphere and the auroral cavity results in a highly asymmetric double layer, with a strong, localized positive charge layer on the ionospheric side and a moderate, extended negative charge layer on the auroral cavity side. This structure results in an asymmetric electric field, a feature also seen in the observations. The electric field observations, however, do not always support a planar double layer since the parallel and perpendicular signals are not always well correlated. Fully two-dimensional solutions are needed to better reproduce the observed features.

McMillan–Mayer theory for solvent effects in inhomogeneous systems: Calculation of interaction pressure in aqueous electrical double layers

We demonstrate how to use the McMillan–Mayer theory to include solvent effects in effective solute–solute interactions for inhomogeneous systems, extending a recent derivation [S. Marčelja, Langmuir 16, 6081 (2000)] for symmetric planar double layers to the general case. In the exact treatment, the many-body potential of mean force between the solute molecules can be evaluated for an inhomogeneous reference system in equilibrium with pure bulk solvent. The reference system contains only solvent and a finite number, n, of fixed solute molecules and it has an external potential that in some cases is different from that of the original system. It is discussed how the n-body potential of mean force between the ions for the relevant cases of large n values can be approximated by a sum of effective singlet and pair interactions evaluated in the presence of, on average, all n ions, i.e., at finite concentration. In examples considered in this work we use effective interionic pair potentials evaluated from bulk electrolyte calculations at finite electrolyte concentrations. We calculate the contribution to the double layer interaction pressure arising from the interaction between ions dissolved in aqueous electrolyte. In cases of moderate or high surface charge, calculations show several new effects. At small surface separations one finds attractive and then strongly repulsive contributions. For surface charge density around one negative charge per 70 Å2 the full results for pressures resemble “secondary hydration force” measured in classical experiments in 1980s. When there is a tendency for ions to adsorb at the surfaces there is a marked change in behavior. The force is then oscillatory, reminiscent of results obtained with the surface force apparatus at low electrolyte concentration.

Exact Description of Aqueous Electrical Double Layers

We describe the exact steps needed to reduce the problem of planar electrical double layers in aqueous solvent to two simpler problems: (i) a reference system consisting of solvent between the surfaces and (ii) ions interacting with other ions and with surfaces via the potentials of mean force evaluated in the reference system. The mapping is achieved by extending the McMillan−Mayer transformation to the nonuniform fluid between the surfaces. The results indicate approximate additivity of short- and long-range effects in ion density and pressure, as observed earlier in numerical work. After two approximation steps, we arrive at a practical and accurate method for calculating properties of planar aqueous double layers.

Leader discharge over a water surface in a Lichtenberg figure geometry

A multichannel leader discharge over a water surface is investigated in a Lichtenberg figure geometry. It is established that the Ohmic conductivity of water causes nonlinearity of the R(t)C discharge circuit. A mutual one-to-one correspondence between the channel lengths and the currents flowing in them is established during the discharge, and the discharge has a selfconsistent character. A mechanism is proposed for the initiation of initial channels by maxima which arise in the charge structure of the planar double layer on the water surface during the development of Rayleigh-Benard instability in the layer after the pulsed corona from the anode reaches the water.

Dressed ion theory for electric double layer structure and interactions; an exact analysis

Dressed ion theory, an exact statistical mechanical formalism for electrolyte systems in the primitive model, is extended to electric double layer systems in various geometries. In this theory an exact equation that has the same form as the linearized Poisson–Boltzmann (PB) equation is set up, in which ‘dressed ions’ take the same role in the exact theory as the bare ions have in the PB approximation. Various distribution functions are expressed in terms of the dressed ion charges. Exact asymptotic results for large particle separations are obtained for the distribution functions, the average electrostatic potential and the interaction free energies in colloid dispersions and for planar double layer systems. A practical method is derived for evaluating the effective surface charge densities of the particles. The question of whether double layer interactions between equally charged colloid particles must be repulsive at large separations (as suggested by the PB approximation) or whether they can be attractive there is treated in some detail.

Role of electrostatic interactions in particle adsorption

The role of the electrostatic double-layer interactions in adsorption of colloid particles at solid/liquid interface was reviewed. The phenomenological formulation of the governing PB equation was presented with the expressions for the pressure tensor enabling one to calculate forces, torques and interaction energies between particles in electrolyte solutions. Then, the limiting analytical results for an isolated double-layer (both spherical and planar) were discussed in relation to the effective surface potential concept. The range of validity of the approximate expression connecting the surface potential and the effective surface potential with surface charge for various electrolytes was estimated. The results for double-layer systems were next presented including the case of two planar double-layers and two dissimilar spherical particles. Limiting solutions for short and long distances as well as for low potentials (linear HHF model) were discussed. The approximate models for calculating interactions of spheres, i.e., the extended Derjaguin summation method and the linear superposition approach (LSA) were also introduced. The results stemming from these models were compared with the exact numerical solution obtained in bispherical coordinate system. Possibilities of describing interactions of nonspherical particles (e.g., spheroids) in terms of the Derjaguin and the equivalent sphere methods were pointed out. In further part of the review the role of these electrostatic interactions in adsorption of colloid particles was discussed. Theoretical predictions were presented enabling a quantitative determination of both the initial adsorption flux for low surface coverages and the surface blocking effects for larger surface coverages. Possibility of bilayer adsorption for dilute electrolytes was mentioned. The theoretical results concerning both the adsorption kinetics and structure formation were then confronted with experimental evidences obtained in the well-defined systems, e.g., the impinging-jet cells and the packed-bed columns of monodisperse spherical particles. The experiments proved that the initial adsorption flux was considerably increased in dilute electrolytes whereas the monolayer coverages were considerably decreased due to lateral interactions among particles. It was then concluded that the good agreement between experimental and theoretical data confirmed the thesis of an essential role of the electrostatic interactions in adsorption phenomena of colloid particles.

Calculation of the force between planar electrical double layers containing counterion mixtures

The distribution of ions about a planar surface of fixed charge density can be solved within the hypernetted chain (HNC)/mean spherical approximation (MSA) by a simplest yet very accurate method based on the calculus of variations. Here we present an application of that work, the calculation of the potential of mean force and the net pressure for two charged planar surfaces immersed in a salt solution. The expression for the potential of mean force is derived from the Ornstein-Zernike relation for a homogenous fluid by taking the planar geometry limit and employing the conventional HNC approximation. This method requires as input only the distribution of ions about an isolated planar double layer. Though this simple method of pressure calculation is more approximate than some other schemes utilizing the HNC approximation, under many circumstances the agreement among the different theories is excellent. We present results for 1:1 and 1:2 salts as well as mixtures of monovalent and divalent counterions for planar double layers with fixed surface charge densities. Under identical conditions of charge density and bulk coion concentration, the interaction between the surfaces is repulsive at all distances for the case of a monovalent counterion, while regions of attraction are observed for the case of an electrolyte with divalent counterions. It is found that salt solutions with a mixture of monovalent and divalent counterions show properties intermediate to the pure monovalent and divalent cases but the present of only a small amount of divalent ion is sufficient to cause large changes in the interaction between the surfaces. Systems containing a moderate fraction of divalent ions show behaviour nearly identical to the pure divalent counterion case.

A fast pipelined trigger for the H1 experiment based on multiwire proportional chamber signals

Based on a total of 1920 pads from two cylindrical and one planar double layer multiwire proportional chamber, the vertex position of ep collisions along the beam axis of the HERA storage ring is reconstructed. This allows discrimination between ep events, originating from the nominal interaction region, and background, introduced mainly by proton beam pipe and proton rest gas interactions. The trigger works in a pipelined manner, delivering a delayed decision every 96 ns, corresponding to the accelerator bunch crossing frequency. It employs custom designed gate arrays as well as XILINX logic cell arrays and a 16 Mbyte static RAM lookup table. Computer controlled debugging procedures take advantage of the reprogrammability of XILINX chips.

Variational approach to the electrostatic free energy in charged colloidal suspensions: general theory for open systems

A general variational technique for calculating the electrostatic free energy of a system of interacting double layers is presented. A stationarity principle for a large class of electrical potential- and position-dependent ion distribution functions and boundary conditions is derived and shown to be unique. The functional satisfying this principle for an open system is identified with the corresponding grand canonical excess surface free energy Ω. The special case of a functional quadratic in the potential ψ(i.e. the Debye–Huckel approximation for the ion distributions together with generalized linear boundary conditions) is shown to permit reduction of the free energy calculation to the evaluation of surface integrals. Illustrations of the approach are presented in the Debye–Huckel approximation for two interacting planar double layers and in the non-linear Poisson–Boltzmann approximation for an isolated planar interface under both constant surface charge density and constant surface potential conditions.The variational approach is straightforward to implement for a variety of double-layer geometries and surface charge conditions. Moreover, it directly and unambiguously identifies the appropriate electrostatic free energy. We believe the variational technique to be powerful and versatile method for calculating interactions between colloidal particles.

Binding of blood coagulation factor V to planar phospholipid double layers

The binding of coagulation factor V to planar phospholipid double layers is investigated by ellipsometry. It is found that native factor V binds with high affinity ( k on > 10 8 M −1 s −1; k off = 1.5 × 10 −3 s −1; K d < 1.5 × 10 −11 M) to mixtures of 20% dioleoylphosphatidylserine and 80% dioleoylphosphatidylcholine. At 20°C, however, the protein in buffer solution undergoes a rapid ( t 1/2 ∼ 25 min) spontaneous denaturation which destroys its binding capacity. Due to this effect, factor V shows an apparently reversible binding isotherm from which overestimations of K d with several orders of magnitude may be obtained. Conflicting results in the literature can thus be explained. Experiments with radiolabeled factor V show that the adsorbed protein does not show net desorption after dilution of the protein solution but readily exchanges with unlabeled factor V. This phenomenon was observed earlier for several proteins and has resulted in specific assumptions about macromolecular exchange mechanisms. For factor V, however, this phenomenon is a consequence of the values of the binding parameters and fits completely within the framework of classical binding theory.

Consequences of misfit and threading dislocations on PV device design

The relative weights of surface recombination, misfit dislocation density and threading dislocation density in the performance of a photovoltaic device are ascertained in the generation-recombination operating limit of a planar HgCdTe double layer heterostructure. In this limit, surface recombination is found to be much less important than the dislocations, while the balance between misfit and threading dislocations is determined by the relative size of the average misfit segment length and the device lateral dimension.

Interaction of planar double layers in the modified Gouy-Chapman approximation

The force between two charged planar surfaces containing an electrolyte solution is calculated. The calculation is done for a 1-1 electrolyte with size-asymmetric ions using a Modified Gouy-Chapman theory. It is shown that at least part of the explanation for the sharp rise in the force between charged surfaces at small separations seen in experimental data may be related to finite-sized ion effects in the double layer. An interesting effect of size-asymmetric ions is the prediction of a force between uncharged surfaces.

The effect of ionic sizes on the potential distribution in the double layer in the absence of specific adsorption

By use of the linearized Poisson-Boltzmann equation, electrostatic calculations have been performed for a model of a planar double layer in which ionic sizes are considered explicitly in the absence of specific adsorption for surface charge densities near the potential of zero charge. The results of the calculations show that the differences in the ionic sizes become important above 0.05 M aqueous solution of 1:1 electrolytes at 25°C. The model qualitatively predicts a shift of the potential of point of zero charge with concentration, a reversal of potential sign, and a maximum in the potential function within the inner diffuse part of the double layer. These phenomena have been usually attributed to specific adsorption of ions. A consistent electrostatic approach to the double-layer theory is used in order to avoid the usual two-capacitor-in-series model, and suggestions for further development of this phenomenological approach are made.

The simulation of plasma double-layer structures in two dimensions

Electrostatic plasma double layers are numerically simulated by means of a magnetized 2½-dimensional particle-in-cell method, periodic in one direction and bounded by reservoirs of Maxwellian plasma in the other. The investigation of planar double layers indicates that these one-dimensional potential structures are susceptible to periodic disruption by plasma instabilities. A slight increase in the double-layer thickness with an increase in its obliqueness to the magnetic field is observed. It is noted that weak magnetization results in the double-layer electric-field alignment of particles accelerated by these potential structures and that strong magnetization results in their magnetic-field alignment. Electron-beam-excited electrostatic electron cyclotron waves and ion-beam-driven electrostatic turbulence are present in the plasmas adjacent to the double layers. The numerical simulations of spatially periodic two-dimensional double layers also exhibit cyclical instability. A morphological invariance in two-dimensional double layers with respect to the degree of magnetization implies that the potential structures scale with Debye lengths rather than with gyroradii. Ion-beam-driven electrostatic turbulence and electron-beam-driven plasma waves are again detected. A simplified one-dimensional model of oblique plasma double layers, using water-bag velocity distribution functions, is presented in an appendix.