Finite Ion Size Research Articles

Corrected Debye–Hückel Analysis of Surface Complexation: II. A Theory of Surface Charging

A theory of surface charging of colloidal particles suspended in an electrolyte solution is presented. The charging at the particle surface is assumed to originate from the adsorption and desorption of protons and is therefore strongly dependent on the acidity of the solution. The surface binding of protons occurs locally at sites of occupancy zero or one that are described by a binding energy u 0 and a three-dimensional vibration of frequency ν. The diffuse screening of ions at the surface is described by the corrected Debye–Hückel analysis assuming linear response. The model contains a capacitor layer close to the charged surface and the finite size of the electrolyte ions is taken into account. The theory has been applied to titrated surface charge data on goethite (α-FeOOH) at NaClO 4 background concentrations ranging from 0.01 to 1.0 M. The protonation mechanism used in the modeling of these data corresponds to the 1-p K approach. A very good description of the experimental data was obtained at the highest ionic strength. Close to the pH pzc the theory also gave a good description at lower ionic strengths. However, at low salt concentrations and pH values far away from the pH pzc the electrostatic potential outside the capacitor layer becomes so high that nonlinear electrostatic effects become important and the theory therefore underestimates the surface charge. These results were compared with model calculations obtained using existing surface complexation models.

Charged cylindrical surfaces: effect of finite ion size

A simple statistical mechanical approach is applied to calculate the profile of the density of the number of particles and the profile of the electrostatic potential of an electric double layer formed by a charged cylindrical surface in contact with electrolyte solution. The finite size of particles constituting the electrolyte solution is considered by including the excluded volume effect within the lattice statistics while the electrostatic interactions are considered by means of the mean electrostatic field. It is shown that the excluded volume effect decreases the density of the number of counterions and increases the electrostatic potential near the charged cylindrical surface. The effect is more pronounced for high area densities of charge of the charged surface and for larger counterions. Further, it is shown that the ratio between the density of the number of the counterions near the charged cylindrical surface and the density of the number of counterions far from the charged surface reaches a plateau at large linear charge densities for ions of finite size, while no plateau is reached for dimensionless ions. The effective thickness of the electric double layer in cylindrical geometry is introduced. It is shown that the effective thickness increases with increasing counterion size while its dependence on the area density of charge of the charged surface exhibits a minimum. The theoretical approach presented in this work can be used for description of the electrostatics of the thin cylindrical structures in biological systems such as DNA, protein macromolecules and charged micro and nano tubes.

Consideration of Ion Size Effects in the Diffuse Double Layer on the Basis of Generalized Mean Spherical Approximation

The theory of the double layer is extended to obtain an analytical expression for the potential drop across the diffuse layer with consideration of the effects of finite ion size. The classical Gouy-Chapman results are compared with those of the hypernetted chain approximation (HNC) which gives non-analytical results. The HNC results are then used as a guide to the choice of correcting functions in a generalized mean spherical approximation (GMSA). The important quantities in the new description are the volume fraction for the ions and the reciprocal distance characterizing ion-ion interactions in the MSA. The results of the GMSA model are then compared with Monte-Carlo calculations for the case that the ions have a diameter of 300 pm.

Ion size effects on the current efficiency of narrow charged pores

The effects of ion size on the current efficiency (CE) of charged membranes with narrow pores are studied theoretically. The CE is a measure of the membrane permselectivity defined as the ratio between the counterion flux and the sum of the counterion and coion fluxes when an electric potential difference is applied between the two solutions bathing the membrane. It is studied here as a function of two relevant experimental parameters: the ratio between the ionic radius and the pore radius, and the ratio between the external salt concentration and the membrane fixed charge concentration. The ratio of the CE values corresponding to the point and finite size ions is also calculated as a function of the above two parameters. Ion size effects are introduced in the ionic diffusion coefficients (by means of the Renkin function) as well as in the ionic concentration profiles. It is shown that a significantly higher coion exclusion from the membrane and thus a higher CE is obtained for finite size ions compared to the case of point ions.

Effects of ion size and valence on ion distribution in mixed counterion systems of rodlike polyelectrolyte solution. I. Mixed-size counterion systems with same valence

The influence of the counterion size is investigated in the mixed counterion systems of the salt-free polyelectrolyte solution in a cylindrical cell model. The mixtures of two species of the counterion having the same valence and different size are simulated systematically by means of the Monte Carlo method in the primitive model of the rodlike polyelectrolyte solution. The results of the free fractions and the selectivity coefficients are compared with the numerical solutions of the Poisson–Boltzmann equation. The observed differences between both methods are explained in terms of the ion–ion correlations and the effect of finite ion size.

Potential Distribution and Electrostatic Forces in Wedge-Shaped Geometries: Analytical and Numerical Results

The interparticle force due to electrostatic/ionic origin in thermal equilibrium is modeled for two particles in close contact in an ion-laden fluid. The space between the particles presents approximately a wedge-shaped geometry. Two methods are used to ascertain the value of the interparticle force: an analytical approximation (equivalent to the traditional Debye–Hückel (DH) method) and a simulation of the ionic fluid using a canonical Monte Carlo simulation. The analytical solution is obtained by traditional means, improving the double-layer solutions for single surfaces to fit the appropriate boundary conditions (constant potential or constant charge). Arbitrary curved surfaces can be treated with the same procedure. To investigate the physical effects not accounted for by the DH field theory (for instance, the finite ion size), canonical ensemble Monte Carlo simulations of a primitive electrolyte solution in a wedge-shaped geometry have been carried out, using the Metropolis method. For regions far removed from the top of the wedge, the two methods give the same answer; however, the contribution to the force of the ionic distribution close to the apex of the wedge is non-negligible, increasingly so for smaller angles. Full results are reported.

ASPEN: A Fully Kinetic, Reduced-Description Particle-in-Cell Model for Simulating Parametric Instabilities

A fully kinetic, reduced-description particle-in-cell (RPIC) model is presented in which deviations from quasineutrality, electron and ion kinetic effects, and nonlinear interactions between low-frequency and high-frequency parametric instabilities are modeled correctly. The model is based on a reduced description where the electromagnetic field is represented by three separate temporal envelopes in order to model parametric instabilities with low-frequency and high-frequency daughter waves. Because temporal envelope approximations are invoked, the simulation can be performed on the electron time scale instead of the time scale of the light waves. The electrons and ions are represented by discrete finite-size particles, permitting electron and ion kinetic effects to be modeled properly. The Poisson equation is utilized to ensure that space-charge effects are included. The RPIC model is fully three dimensional and has been implemented in two dimensions on the Accelerated Strategic Computing Initiative (ASCI) parallel computer at Los Alamos National Laboratory, and the resulting simulation code has been named ASPEN. We believe this code is the first particle-in-cell code capable of simulating the interaction between low-frequency and high-frequency parametric instabilites in multiple dimensions. Test simulations of stimulated Raman scattering, stimulated Brillouin scattering, and Langmuir decay instability are presented.

Adsorption Layer of Charged Surfaces: An Exact Nearest-Neighbor Model for the Linear Lattice

The effects of finite ion size and polyion concentration on the adsorption isotherms are examined within the context of a one-dimensional lattice restricted to nearest-neighbor interactions. The nearest-neighbor interactions include that of the bound ion with adjacent lattice sites as well as with a neighbor bound ion. Both screened and unscreened Coulombic interactions are employed, as they are shown to exhibit different behaviors under high salt concentrations. For the screened interactions, previously adsorbed ions are released, whereas for the unscreened Coulombic interaction charge reversal is observed for sufficiently separated lattice sites. The effect of polyion concentration is to provide a constant degree of adsorption for added salt concentrations less than the concentration ofcounterions released by the polyion. The model is applied to experimental data on monovalent and divalent salts.

Smoluchowski hypernetted chain theory description of the dynamics of ions confined in charged micropores

The diffusion of an ion inside a charged planar slit micropore is analyzed from a stochastic point of view. Using the instantaneous relaxation approximation (IRA), the Fokker-Planck-Smoluchowski equation was used to calculate the survival probability ${G}_{i}(x,t),$ namely, the probability that an ion of species i remains inside the pore at time t given that it started to diffuse at a given position x. The ionic density profiles were obtained using the three-point extension hypernetted chain theory (HNC), which explicitly takes into account the finite ionic sizes, and the results are compared to those using the classical modified Gouy-Chapman (MGC) theory based on the Poisson-Boltzmann point-ion equation. Calculations were carried out for a variety of pore widths, electrolyte charges, surface potentials, and absorbing or reflecting boundary conditions. We also calculate the mean first passage time (MFPT), \ensuremath{\tau}, and the position-averaged MFPT, $\overline{\ensuremath{\tau}}.$ Our Smoluchowski HNC results show strong discrepancies with the classical Smoluchowski MGC theory. In particular, for small pores and doubly charge coions, we observed oscillations in the position-averaged MFPT, as a function of the pore size, which are absent in the classical theory.

An improved Space-Charge model for flow through charged microporous membranes

The Space-Charge model is modified to better analyze the steady-state electrohydrodynamic behavior of aqueous monovalent electrolytes in charged microporous membranes. The effects of changes in solvent dielectric constant near the wall, ion hydration effects, finite ion sizes, and charge regulating surface effects, are incorporated into the governing electrohydrodynamic equations (i.e., Navier-Stokes (NSE), Nernst-Planck (NPE), and Poisson-Boltzmann (PBE) equations). Their effect on streaming potential, pore conductivity, excess conductivity, and maximum energy conversion efficiency for electro-osmosis is illustrated. It is shown that the dielectric saturation and ion hydration effects cause significant changes in the electric potential field and ion concentration inside the capillary tubes. Quantitative comparisons of model results with measured electrokinetic data reveal better agreement when compared with the existing model.

Effect of Finite Ion Size on the “Overbeek Correction” to the Sogami−Ise Interaction Potential between Macroions of Like Charge

The Sogami−Ise (SI) interaction potential has a “long range attraction” tail that is not present in the DLVO potential. Overbeek suggested that an “error” was present in the SI theory and that the inclusion of the solvent contribution eliminated the source of this long range attraction. The model used by Overbeek assumed that the macroions, by virtue of being fixed in their equilibrium positions, did not contribute to the electrical free energy of the solution. It is shown herein that the “Overbeek correction” is not rigorously zero for arbitrary polyion concentrations, as it must be in order to eliminate the Gibbs minimum for actual experimental conditions. It is shown herein on the basis of standard statistical thermodynamics expressions that the presence of a minimum in the pairwise Gibbs electrical free energy occurs for all screened Coulombic expressions for the Helmholtz electrical free energy, and therefore the correction term of Overbeek is in error. It is concluded that the concepts of screened C...

The double layer at the interface between two immiscible electrolyte solutions—IV. Solvent effect

Differential capacitance of the electrical double layer at the water/nitrobenzene and water/1,2-dichloroethane interfaces was measured by an ac impedance technique. Potential difference across the diffuse layer in water, nitrobenzene and 1,2-dichloroethane and the corresponding capacitances, were calculated by using the modified Poisson-Boltzmann theory (version MPB4), which accounts for both the finite ion size and image forces. The two diffuse layers, one in water and the other in organic solvent, were assumed to be independent, ie they were associated only through an equal and opposite surface charge density. As compared with the Gouy-Chapman theory, the MPB theory provided a more reasonable description of the interface over the whole range of electrolyte concentrations and surface charge densities even in the system comprising the solvent of a low dielectric permittivity, such as 1,2-dichloroethane. By using the MPB theory, the interfacial potential difference and the inverse capacitance for both the systems studied were shown to split into two contributions, of which one is essentially independent of the electrolyte concentration and can be attributed to the presence of an ion-free inner (or compact) layer. The effect of the surface charge is to reduce the thickness of this layer, eg through the liquid phase interpenetration.

The electrostatic Gibbs energy of finite-size ions near a planar boundary between two dielectric media

We have calculated analytically the electrostatic Gibbs energy profile for transfer of an ion between two bulk dielectric phases separated by a planar boundary. The profile incorporates the finite ionic size and dielectric image interactions. There is no discontinuity as the ion traverses the boundary and the model is well represented by a simple point charge when the ionic centre is beyond about half an ionic radius from the boundary. The profile is in line with observations based on interfacial inner-layer capacitances that ions can partly penetrate the boundary between two immiscible liquids at equilibrium. It represents an inner-layer capacitance contribution when the distance of closest approach is different for cations and anions.

Ion binding to charged lipid monolayers: The role of double layer and ion binding models

We present a new approach to the analysis of monolayer surface potential experiments, based on a lattice model for binding of ions to the monolayer and two models for the electrical double layer: Gouy-Chapman-Debye-Hückel (GCDH) theory and a recently developed model that takes into account the finite polarizability of solvent dipoles and finite ion sizes. We demonstrate this approach by reanalyzing two recent monolayer surface potential experiments. For highly charged monolayers (Lakhadar-Ghazal, F., Tichadou, J. L., and Tocanne, J. F., Eur. J. Biochem., 134, 531 (1985)) we find that the new double layer model yields a description of surface properties which is distinctly different from that of GCDH theory, while at low surface charge densities the two models are effectively the same. The results of S. Ohki and R. Kurland ( Biochim. Biophys. Acta, 645, 170 (1981)), fall in the latter regime, so our analysis of these data serves to contrast different binding models. Our results suggest that interactions between bound ions, or between lipid molecules in different ionization states, may be important in understanding surface properties when there is a high degree of ion binding.

The charging properties of monodisperse colloidal silica in symmetrical quaternary ammonium ion solutions

The zeta potentials of monodisperse colloidal silica were measured as a function of pH and as a function of the concentration of tetramethyl-, tetrapropyl-, and tetrapentylammonium bromide in aqueous solution. The variation of the potential with pH was explained by a simple, point ion adsorption model from which a dissociation constant and Gibbs free energy of dissociation were obtained. By contrast, the results obtained with tetraalkyl ammonium ions could not be explained using this simple model. However, incorporation of a hydrophobic adsorption term and finite ion size in a more sophisticated model gave good agreement with experimental measurements. The validity of the model is supported by the reasonable dissociation constants which were obtained on fitting the experimental data. Dissociation constants and the corresponding Gibbs free energies were calculated for both hydrophobic and site binding adsorption for each tetraalkyl ammonium ion. The stability of colloidal silica in solutions of these ions can now be satisfactorily explained in terms of the electrostatic repulsion between particles rather than by a hydrophobic/hydrophilic solvation effect previously proposed.

On the role of ion size in coagulation

It is a well-established fact that the coagulation of colloidal particles suspended in an aqueous medium is affected by the size of the dissolved ions. In general the larger the ions, the more salt is needed to coagulate a suspension. The explanation most commonly given is that the finite size of ions somehow increases the double layer thickness, resulting in a larger critical coagulation concentration (CCC), or more convincingly that the ions adsorb onto colloidal particles and affect the ζ-potential. Without invoking these effects it can be proven that the CCC will increase with ion size because the diffusion and friction coefficients of charged particles are affected by the presence of electrical double layers around them. The reduction in the diffusion coefficient and the increase in the friction coefficient depends explicitly on the size of the ions in the double layer. Provided the ζ-potential is not affected by the size of the ion it is predicted that the CCC increases linearly with the scaled friction coefficient of the ions, in agreement with literature data.

Two-dimensional classical electron gas in a periodic field: Delocalization and dielectric-plasma transition.

Results are reported of extensive molecular-dynamics simulations of a two-dimensional Coulomb gas made up of finite-size ions held fixed on the sites of a triangular lattice, and of classical electrons moving in the periodic field of the ions. The fixed-ion model maps the dielectric-plasma (or Kosterlitz-Thouless) transition of the Coulomb gas onto a delocalization transition of the electrons. The transition is characterized by a number of static and dynamic ``diagnostics.'' As the temperature is increased in the dielectric phase, electron self-diffusion and electrical conductivity set in at a density-dependent threshold temperature ${T}_{1}$. The breakup of ion-electron pairs is signaled by a sharp peak in the specific heat at a temperature ${T}_{2}$>${T}_{1}$. As ${T}_{1}$ is approached from above in the plasma phase, the screening length diverges. In the high-temperature plasma, the frequency of the long-wavelength collective charge oscillation (plasmon) mode decreases with T and the mode becomes overdamped by ion-electron recombination well before the threshold ${T}_{1}$ is reached. The dispersion \ensuremath{\omega}(k) of the mode exhibits an unexpected oscillatory behavior. The temperatures ${T}_{1}$ and ${T}_{2}$ increase as the density decreases and there is strong evidence that the low-density limit of the reduced temperatures ${T}_{1}^{\mathrm{*}}$=${k}_{B}$T/${e}^{2}$ and ${T}_{2}^{\mathrm{*}}$ is 1/2 as compared with the value (1/4 expected for the mobile-ion case.

CH overtones in acetophenone and benzaldehyde: aryl and methyl local modes

Even though there is great practical benefit from evidence that ideal theories are reasonable approximations at small separations (10-30 A) where these relations are expected to fail, we should not be lulled into complacency. Why do not we observe major deviations from ideality due to structure of the solvent, finite ion size, correlations, and fluctuations? What is the origin(s) of the short-range repulsion and its decay at large separations? Furthermore, does the exponential form for the close-range repulsion hold at very close proximity? The power-law attraction should overwhelm the repulsion to cause a precipitous collapse. Clearly, stiff-steric interactions must exist which are of ultimate importance

The double layer at the interface between two immiscible electrolyte solutions: Part III. Capacitance of the water/1,2-dichloroethane interface

From ac impedance and galvanostatic pulse measurements the capacitance of the interface between solutions of LiCl in water and tetrabutylammonium tetraphenylborate in 1,2-dichloroethane has been evaluated at various electrolyte concentrations. The experimental data have been interpreted on the basis of the modified Verwey-Niessen model, according to which a layer of oriented solvent molecules (inner layer) separates two space charge regions (diffuse double layer). As for the water/nitrobenzene interface, the interfacial potential difference is spread mainly within the diffuse double layer. In the sequence nitrobenzene < 1,2-dichloroethane, the effects of the ion association, finite ion size and image forces, which modify the double-layer structure, become more pronounced. The latter two effects have been estimated by means of the weak-coupling and hypernetted chain theories, respectively. No evidence for the specific adsorption of ions or ion pairs was found.

Electrolyte friction on charged spherical macroparticles: Beyond the Debye–Hückel limit

The contribution to the friction coefficient of a charged spherical polyion diffusing in an ionic solution due to its interactions with the electrolyte ions is calculated, in the absence of hydrodynamic interactions, starting from a formal expression for ζel previously derived by Medina-Noyola and Vizcarra-Rendón. It is shown that such a formal expression can be given a particularly simple form in terms of the equilibrium distribution neqi(r) of the small ions around the tracer, which allows us to study effects ignored in Debye–Hückel level theories. We consider, for example, the effect of a finite size of the small ions and nonlinear contributions to ζel on the polyion charge. We show that the correct Debye–Hückel limit differs from Schurr’s result for ζel, and the source of the disagreement is clearly exposed. The results derived here for ζel are not restricted in the number of species of electrolyte ions or in their charges or mobilities.