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.     

Interaction between an ion-penetrable particle and a solid particle. 1. Electrical double-layer interaction

Expressions are derived for the force and potential energy of the electrical double layer interaction between two parallel plates of different nature, i. e., an ion-penetrable plate and an ion-impenetrable plate. The latter may have either constant surface potential or constant surface charge density. It is shown that when the ion-impenetrable plate has a constant surface potential, the interaction force may, under certain conditions, become attractive even if the surface potentials of the two plates at infinite separation are of the same sign. In contrast, when the ion-impenetrable plate has a constant surface charge density, the interaction force may, under certain conditions, become repulsive even if the two plates at infinite separation are of opposite sign. This means that an ion-penetrable plate shows a dual behavior. That is, under certain conditions, it behaves like a solid plate with constant surface potential or surface charge density, depending on whether it interacts with a solid plate having a constant surface potential or a constant surface charge density.     

Long-Range Electrostatic Interaction between a Charged Wall and Two Similarly Charged Colloidal Spheres at Low Surface Potentials

The long-range electrostatic interaction between a pair of similarly charged colloidal spheres and a charged planar wall at low surface potentials is theoretically investigated. The linear Poisson-Boltzmann equation (PBE) and the point charge approximation of the charged sphere are used. The electrical potential distribution in the electrolyte solution is found from the PBE at the constant surface potentials using the image charge method. The electrostatic forces acting on the spheres are then calculated. The results show that the repulsive interaction between a pair of similarly charged colloidal spheres clearly decreases when a charged wall appears nearby, but it is impossible for an attractive force to emerge at the scaled surface potentials less than 1. There is, however, an attractive force between the charged wall and the similarly charged colloidal spheres, when the surface potential zetap on the wall is sufficiently higher than the surface potential zetas on the spheres to make zetap > zetasexp(kappah) (h is the distance from the wall to the sphere center). In this case, there are negative surface charges on the spheres at positive surface potential zetas. It is these negative charges that produce the above attraction. Copyright 1999 Academic Press.     

Peak Force Infrared–Kelvin Probe Force Microscopy

Correlative scanning probe microscopy of chemical identity, surface potential, and mechanical properties provide insight into the structure–function relationships of nanomaterials. However, simultaneous measurement with comparable and high resolution is a challenge. We seamlessly integrated nanoscale photothermal infrared imaging with Coulomb force detection to form peak force infrared–Kelvin probe force microscopy (PFIR-KPFM), which enables simultaneous nanomapping of infrared absorption, surface potential, and mechanical properties with approximately 10 nm spatial resolution in a single-pass scan. MAPbBr₃ perovskite crystals of different degradation pathways were studied in situ. Nanoscale charge accumulations were observed in MAPbBr₃ near the boundary to PbBr₂. PFIR-KPFM also revealed correlations between residual charges and secondary conformation in amyloid fibrils. PFIR-KPFM is applicable to other heterogeneous materials at the nanoscale for correlative multimodal characterizations.     

Electrostatic interactions between double layers: influence of surface roughness, regulation, and chemical heterogeneities

Electrostatic interactions between two surfaces as measured by atomic force microscopy (AFM) are usually analyzed in terms of DLVO theory. The discrepancies often observed between the experimental and theoretical behavior are usually ascribed to the occurrence of chemical regulation processes and/or to the presence of surface chemical or morphological heterogeneities (roughness). In this paper, a two-gradient mean-field lattice analysis is elaborated to quantifying double layer interactions between nonplanar surfaces. It allows for the implementation of the aforementioned sources of deviation from DLVO predictions. Two types of ion-surface interaction ensure the adjustment of charges and potentials upon double layer overlap, i.e., specific ionic adsorption at the surfaces and/or the presence of charge-determining ions for the surfaces considered. Upon double layer overlap, charges and potentials are adjusted via reequilibrium of the different ion adsorption processes. Roughness is modeled by grafting asperities on supporting planar surfaces, with their respective positions, shapes, and chemical properties being assigned at will. Local potential and charge distributions are derived by numerically solving the nonlinear Poisson-Boltzmann equation under the boundary conditions imposed by the surface profiles and regulation mechanism chosen. Finite size of the ions is taken into account. A number of characteristic situations are briefly discussed. It is shown how the surface irregularities are reflected in the Gibbs energy of interaction.     

Peak Force Infrared–Kelvin Probe Force Microscopy

Correlative scanning probe microscopy of chemical identity, surface potential, and mechanical properties provide insight into the structure–function relationships of nanomaterials. However, simultaneous measurement with comparable and high resolution is a challenge. We seamlessly integrated nanoscale photothermal infrared imaging with Coulomb force detection to form peak force infrared–Kelvin probe force microscopy (PFIR‐KPFM), which enables simultaneous nanomapping of infrared absorption, surface potential, and mechanical properties with approximately 10 nm spatial resolution in a single‐pass scan. MAPbBr₃ perovskite crystals of different degradation pathways were studied in situ. Nanoscale charge accumulations were observed in MAPbBr₃ near the boundary to PbBr₂. PFIR‐KPFM also revealed correlations between residual charges and secondary conformation in amyloid fibrils. PFIR‐KPFM is applicable to other heterogeneous materials at the nanoscale for correlative multimodal characterizations.     

Electrostatic potential around charged finite rodlike macromolecules: nonlinear Poisson-Boltzmann theory

In this paper, we show that the far field electrostatic potential created by a highly charged finite size cylinder within the nonlinear Poisson-Boltzmann (PB) theory, is remarkably close to the potential created within the linearized PB approximation by the same object at a well-chosen fixed potential. Comparing the nonlinear electrostatic potential with its linear counterpart associated to a fixed potential boundary condition (called the effective surface potential), we deduce the effective charge of the highly charged cylinder. Values of the effective surface potential are provided as a function of the bare surface charge and Debye length of the ionic solution. This allows to compute the anisotropic electrostatic interaction energy of two distant finite rods.     

Electrical double-layer interaction between oppositely charged dissimilar oxide surfaces with charge regulation and Stern-Grahame layers

The force of double-layer interaction between dissimilar flat oxide surfaces charged oppositely at infinite separation and with Stern-Grahame layers was calculated as a function of the surface separation under the conditions of charge regulation. The interaction force showed a monotonic attraction with respect to the surface separation, lying between constant surface potential and constant surface charge conditions. The variations of surface potentials and surface charge densities of respective double layers were also presented as functions of the surface separation. When the negative diffuse-layer potential at infinite separation changed its sign to positive as a result of specific adsorption of cations on the negatively changed surface, the double-layer interaction force showed a repulsive force barrier followed by an attraction at some particular separation.     

Influence of atomistic physics on electro-osmotic flow: an analysis based on density functional theory

Molecular density profiles and charge distributions determined by density functional theory (DFT) are used in conjunction with the continuum Navier-Stokes equations to compute electro-osmotic flows in nanoscale channels. The ion species of the electrolyte are represented as centrally charged hard spheres, and the solvent is treated as a dense fluid of neutral hard spheres having a uniform dielectric constant. The model explicitly accounts for Lennard-Jones interactions among fluid and wall molecules, hard sphere repulsions, and short range electrical interactions, as well as long range Coulombic interactions. Only the last of these interactions is included in classical Poisson-Boltzmann (PB) modeling of the electric field. Although the proposed DFT approach is quite general, the sample calculations presented here are limited to symmetric monovalent electrolytes. For a prescribed surface charge, this DFT model predicts larger counterion concentrations near charged channel walls, relative to classical PB modeling, and hence smaller concentrations in the channel center. This shifting of counterions toward the walls reduces the effective thickness of the Debye layer and reduces electro-osmotic velocities as compared to classical PB modeling. Zeta potentials and fluid speeds computed by the DFT model are as much as two or three times smaller than corresponding PB results. This disparity generally increases with increasing electrolyte concentration, increasing surface charge density and decreasing channel width. The DFT results are found to be comparable to those obtained by molecular dynamics simulation, but require considerably less computing time.     

The interaction energy between two parallel plates with constant surface charge density

On the basis of Langmuir's suggestion we simplify the Poisson-Boltzmann equation and derive the relation of surface potential, potential midway, and the plate distance. Thus we obtain the interaction force and energy equations between two dissimilar plates in the case of constant surface charge density. Agreement with the exact numerical values of the interaction of dissimilar plates is good. This method may not only apply to the cases of high constant potential but to the case of high constant charge density.     

Effects of image charges, interfacial charge discreteness, and surface roughness on the zeta potential of spherical electric double layers

We investigate the effects of image charges, interfacial charge discreteness, and surface roughness on spherical electric double layer structures in electrolyte solutions with divalent counterions in the setting of the primitive model. By using Monte Carlo simulations and the image charge method, the zeta potential profile and the integrated charge distribution function are computed for varying surface charge strengths and salt concentrations. Systematic comparisons were carried out between three distinct models for interfacial charges: (1) SURF1 with uniform surface charges, (2) SURF2 with discrete point charges on the interface, and (3) SURF3 with discrete interfacial charges and finite excluded volume. By comparing the integrated charge distribution function and the zeta potential profile, we argue that the potential at the distance of one ion diameter from the macroion surface is a suitable location to define the zeta potential. In SURF2 model, we find that image charge effects strongly enhance charge inversion for monovalent interfacial charges, and strongly suppress charge inversion for multivalent interfacial charges. For SURF3, the image charge effect becomes much smaller. Finally, with image charges in action, we find that excluded volumes (in SURF3) suppress charge inversion for monovalent interfacial charges and enhance charge inversion for multivalent interfacial charges. Overall, our results demonstrate that all these aspects, i.e., image charges, interfacial charge discreteness, their excluding volumes, have significant impacts on zeta potentials of electric double layers.     

Electrostatic potentials and fields in the vicinity of engineered nanostructures

We have developed a method for calculating the electrostatic potentials and fields in the vicinity of geometrically complex engineered nanostructures composed of varying materials in electrolytes of arbitrary pH and ionic strength. The method involves direct summation of charged Debye-Hückel spheres composing the nanostructural surfaces and, by including charge redistribution on the surface of conducting materials held at constant potential, is applicable to mixed boundary conditions. The method is validated by comparison to analytical solutions for an infinite plane (Gouy-Chapman), an infinite cylinder (Bessel functions), and an infinite plane which contains a hole and which is held at constant potential. Excellent agreement between the potentials obtained by our numerical method and the closed form solutions is found for these conditions. The method is applied to the calculation of the electric field enhancement in the vicinity of a nanomembrane whose pore wall is held at constant charge and whose membrane surfaces are held at constant potential. The electric field is found to be enhanced by the charge buildup in the rim of the hole of the nanomembrane; the buildup results from the potential being held constant in the conducting region. Ion concentrations are also calculated. Positive ion rejection is found to be enhanced by this charge buildup in the region of the rim when a constant positive potential is applied.     

A boundary element method for molecular electrostatics with electrolyte effects

A boundary element method is developed to compute the electrostatic potential inside and around molecules in an electrolyte solution. A set of boundary integral equations are derived based on the integral formulations of the Poisson equation and the linearized Poisson-Boltzmann equation. The boundary integral equations are then solved numerically after discretizing the molecular surface into a number of flat triangular elements. The method is applied to a spherical molecule for which analytical solutions are available. Use is made of both constant and linearly varying unknowns over the boundary elements, and the method is tested for various values of parameters such as the dielectric constant of the molecule, ionic strength, and the location of the interior point charge. The use of the boundary integral method incorporating the nonlinear Poisson-Boltzmann equation is also briefly discussed.     

Calculation of the microscopic and macroscopic linear and nonlinear optical properties of liquid acetonitrile. II. Local fields and linear and nonlinear susceptibilities in quadrupolar approximation

A discrete model based on the multipolar expansion including terms up to hexadecapoles was employed to describe the electrostatic interactions in liquid acetonitrile. Liquid structures obtained form molecular dynamics simulations with different classical, nonpolarizable potentials were used to analyze the electrostatic interactions. The computed average local field was employed for the determination of the environmental effects on the linear and nonlinear electrical molecular properties. Dipole-dipole interactions yield the dominant contribution to the local field, whereas higher multipolar contributions are small but not negligible. Using the effective in-phase properties, macroscopic linear and nonlinear susceptibilities of the liquid were computed. Depending on the partial charges describing the Coulomb interactions of the force field employed, either the linear properties (refractive index and dielectric constant) were reproduced in good agreement with experiment or the nonlinear properties [third-harmonic generation (THG) and electric field induced second-harmonic (EFISH) generation] and the bulk density but never both sets of properties together. It is concluded that the partial charges of the force fields investigated are not suitable for reliable dielectric properties. New methods are probably necessary for the determination of partial charges, which should take into account the collective and long-range nature of electrostatic interactions more precisely.     

On the nature of liquid junction and membrane potentials

Whenever a spatially inhomogeneous electrolyte, composed of ions with different mobilities, is allowed to diffuse, charge separation and an electric potential difference is created. Such potential differences across very thin membranes (e.g. biomembranes) are often interpreted using the steady state Goldman equation, which is usually derived by assuming a spatially constant electric field. Through the fundamental Poisson equation of electrostatics, this implies the absence of free charge density that must provide the source of any such field. A similarly paradoxical situation is encountered for thick membranes (e.g. in ion-selective electrodes) for which the diffusion potential is normally interpreted using the Henderson equation. Standard derivations of the Henderson equation appeal to local electroneutrality, which is also incompatible with sources of electric fields, as these require separated charges. We analyse self-consistent solutions of the Nernst-Planck-Poisson equations for a 1 : 1-univalent electrolyte to show that the Goldman and Henderson steady-state membrane potentials are artefacts of extraneous charges created in the reservoirs of electrolyte solution on either side of the membrane, due to the unphysical nature of the usual (Dirichlet) boundary conditions assumed to apply at the membrane-electrolyte interfaces. We also show, with the aid of numerical simulations, that a transient electric potential difference develops in any confined, but initially non-uniform, electrolyte solution. This potential difference ultimately decays to zero in the real steady state of the electrolyte, which corresponds to thermodynamic equilibrium. We explain the surprising fact that such transient potential differences are well described by the Henderson equation by using a computer algebra system to extend previous steady-state singular perturbation theories to the time-dependent case. Our work therefore accounts for the success of the Henderson equation in analysing experimental liquid-junction potentials.     

Discrete charge effects on the Donnan potential and surface potential of a soft particle

The Donnan potential and surface potential of soft particles (i.e., polyelectrolyte-coated hard particles) in an electrolyte solution play an essential role in their electric behaviors. These potentials are usually derived via a continuum model in which fixed charges inside the surface layer are distributed with a continuous charge density. In this paper, for a plate-like soft particle consisting of a cubic lattice of fixed point charges, on the basis of the linearized Poisson–Boltzmann equation, we derive expressions for the electric potential distribution in the regions inside and outside the surface layer. This expression is given in terms of a sum of the screened Coulomb potentials produced by the point charges within the surface layer. We show that the deviation of the results of the discrete charge model from those of the continuous charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large.     

Hydrophobic and ionic interactions in nanosized water droplets

A number of situations such as protein folding in confined spaces, lubrication in tight spaces, and chemical reactions in confined spaces require an understanding of water-mediated interactions. As an illustration of the profound effects of confinement on hydrophobic and ionic interactions, we investigate the solvation of methane and methane decorated with charges in spherically confined water droplets. Free energy profiles for a single methane molecule in droplets, ranging in diameter (D) from 1 to 4 nm, show that the droplet surfaces are strongly favorable as compared to the interior. From the temperature dependence of the free energy in D = 3 nm, we show that this effect is entropically driven. The potentials of mean force (PMFs) between two methane molecules show that the solvent separated minimum in the bulk is completely absent in confined water, independent of the droplet size since the solute particles are primarily associated with the droplet surface. The tendency of methanes with charges (M(q+) and M(q-) with q(+) = |q(-)| = 0.4e, where e is the electronic charge) to be pinned at the surface depends dramatically on the size of the water droplet. When D = 4 nm, the ions prefer the interior whereas for D < 4 nm the ions are localized at the surface, but with much less tendency than for methanes. Increasing the ion charge to e makes the surface strongly unfavorable. Reflecting the charge asymmetry of the water molecule, negative ions have a stronger preference for the surface compared to positive ions of the same charge magnitude. With increasing droplet size, the PMFs between M(q+) and M(q-) show decreasing influence of the boundary owing to the reduced tendency for surface solvation. We also show that as the solute charge density decreases the surface becomes less unfavorable. The implications of our results for the folding of proteins in confined spaces are outlined.     

Rapid grid-based construction of the molecular surface and the use of induced surface charge to calculate reaction field energies: applications to the molecular systems and geometric objects

This article describes a number of algorithms that are designed to improve both the efficiency and accuracy of finite difference solutions to the Poisson-Boltzmann equation (the FDPB method) and to extend its range of application. The algorithms are incorporated in the DelPhi program. The first algorithm involves an efficient and accurate semianalytical method to map the molecular surface of a molecule onto a three-dimensional lattice. This method constitutes a significant improvement over existing methods in terms of its combination of speed and accuracy. The DelPhi program has also been expanded to allow the definition of geometrical objects such as spheres, cylinders, cones, and parallelepipeds, which can be used to describe a system that may also include a standard atomic level depiction of molecules. Each object can have a different dielectric constant and a different surface or volume charge distribution. The improved definition of the surface leads to increased precision in the numerical solutions of the PB equation that are obtained. A further improvement in the precision of solvation energy calculations is obtained from a procedure that calculates induced surface charges from the FDPB solutions and then uses these charges in the calculation of reaction field energies. The program allows for finite difference grids of large dimension; currently a maximum of 571(3) can be used on molecules containing several thousand atoms and charges. As described elsewhere, DelPhi can also treat mixed salt systems containing mono- and divalent ions and provide electrostatic free energies as defined by the nonlinear PB equation.     

Ion-specific forces between a colloidal nanoprobe and a charged surface

We investigate the effect of ion-specific potentials on the force between a nanoprobe attached to a cantilever tip, and a charged surface. The probe is treated as a spherical nanoparticle with constant charge. A modified Poisson-Boltzmann equation in bispherical coordinates is used to address this problem in a more quantitative way. We predict that the ion-specific series of measured forces depend on the sign and magnitude of surface charge densities.     

平板型高电位胶粒双电层的相互作用

利用线性迭加法,提出了平行平板型高电位颗粒之间的弱相互作用的近似表达式.结合文献[3]给出的强相互作用表达式,对高电位平行平板型颗粒的相互作用给出了完整的描述,和精确数值解吻合相当好.强弱相互作用的接合点在κh=4,误差在接合点处最大,～10％.根据Derjaguin法和改进的Derjaguin法,求出了高电位球颗粒在恒电位条件下的相互作用能.