Charge Regulation Boundary Condition Research Articles

Electrostatic interactions between charge regulated spherical macroions.

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte-thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Correlation between Electrostatic and Hydration Forces on Silica and Gibbsite Surfaces: An Atomic Force Microscopy Study.

The balance between hydration and Derjaguin–Landau–Verwey–Overbeek (DLVO) forces at solid–liquid interfaces controls many processes, such as colloidal stability, wetting, electrochemistry, biomolecular self-assembly, and ion adsorption. Yet, the origin of molecular scale hydration forces and their relation to the surface charge density that controls the continuum scale electrostatic forces is poorly understood. We argue that these two types of forces are largely independent of each other. To support this hypothesis, we performed atomic force microscopy experiments using intermediate-sized tips that enable the simultaneous detection of DLVO and molecular scale oscillatory hydration forces at the interface between composite gibbsite:silica–aqueous electrolyte interfaces. We extract surface charge densities from forces measured at tip–sample separations of 1.5 nm and beyond using DLVO theory in combination with charge regulation boundary conditions for various pH values and salt concentrations. We simultaneously observe both colloidal scale DLVO forces and oscillatory hydration forces for an individual crystalline gibbsite particle and the underlying amorphous silica substrate for all fluid compositions investigated. While the diffuse layer charge varies with pH as expected, the oscillatory hydration forces are found to be largely independent of pH and salt concentration, supporting our hypothesis that both forces indeed have a very different origin. Oscillatory hydration forces are found to be distinctly more pronounced on gibbsite than on silica. We rationalize this observation based on the distribution of hydroxyl groups available for H bonding on the two distinct surfaces.

Interaction between Charge-Regulated Metal Nanoparticles in an Electrolyte Solution

We present a theory which allows us to calculate the interaction potential between charge-regulated metal nanoparticles inside an acid-electrolyte solution. The approach is based on the recently introduced model of charge regulation which permits us to explicitly-within a specific microscopic model-relate the bulk association constant of a weak acid to the surface association constant for the same weak acid adsorption sites. When considering metal nanoparticles we explicitly account for the effect of the induced surface charge in the conducting core. To explore the accuracy of the approximations, we compare the ionic density profiles of an isolated charge-regulated metal nanoparticle with explicit Monte Carlo simulations of the same model. Once the accuracy of the theoretical approach is established, we proceed to calculate the interaction force between two charge-regulated metal nanoparticles by numerically solving the Poisson-Boltzmann equation with charge regulation boundary condition. The force is then calculated by integrating the electroosmotic stress tensor. We find that for metal nanoparticles the charge regulation boundary condition can be well approximated by the constant surface charge boundary condition, for which a very accurate Derjaguin-like approximation was recently introduced. On the other hand, a constant surface potential boundary condition often used in colloidal literature, shows a significant deviation from the charge regulation boundary condition for particles with large charge asymmetry.

Electroviscous Dissipation in Aqueous Electrolyte Films with Overlapping Electric Double Layers.

We use dynamic atomic force microscopy (AFM) to investigate the forces involved in squeezing out thin films of aqueous electrolyte between an AFM tip and silica substrates at variable pH and salt concentration. From amplitude and phase of the AFM signal we determine both conservative and dissipative components of the tip sample interaction forces. The measured dissipation is enhanced by up to a factor of 5 at tip–sample separations of ≈ one Debye length compared to the expectations based on classical hydrodynamic Reynolds damping with bulk viscosity. Calculating the surface charge density from the conservative forces using Derjaguin–Landau–Verwey–Overbeek (DLVO) theory in combination with a charge regulation boundary condition we find that the viscosity enhancement correlates with increasing surface charge density. We compare the observed viscosity enhancement with two competing continuum theory models: (i) electroviscous dissipation due to the electrophoretic flow driven by the streaming current that is generated upon squeezing out the counterions in the diffuse part of the electric double layer, and (ii) visco-electric enhancement of the local water viscosity caused by the strong electric fields within the electric double layer. While the visco-electric model correctly captures the qualitative trends observed in the experiments, a quantitative description of the data presumably requires more sophisticated simulations that include microscopic aspects of the distribution and mobility of ions in the Stern layer.

Modulation of Elasticity and Interactions in Charged Lipid Multibilayers: Monovalent Salt Solutions.

We have studied the electrostatic screening effect of NaCl solutions on the interactions between anionic lipid bilayers in the fluid lamellar phase using a Poisson–Boltzmann-based mean-field approach with constant charge and constant potential limiting charge regulation boundary conditions. The full DLVO potential, including the electrostatic, hydration and van der Waals interactions, was coupled to thermal bending fluctuations of the membranes via a variational Gaussian Ansatz. This allowed us to analyze the coupling between the osmotic pressure and the fluctuation amplitudes and compare them both simultaneously with their measured dependence on the bilayer separation, determined by the small-angle X-ray scattering experiments. High-structural resolution analysis of the scattering data revealed no significant changes of membrane structure as a function of salt concentration. Parsimonious description of our results is consistent with the constant charge limit of the general charge regulation phenomenology, with fully dissociated lipid charge groups, together with a 6-fold reduction of the membranes’ bending rigidity upon increasing NaCl concentration.

Charge regulation: A generalized boundary condition?

The three most commonly used boundary conditions for charged colloidal systems are constant charge (insulator), constant potential (conducting electrode) and charge regulation (ionizable groups at the surface). It is usually believed that the charge regulation is a generalized boundary condition that reduces in some specific limits to either constant-charge or constant-potential boundary conditions. By computing the disjoining pressure between two symmetric planes for these three boundary conditions, both numerically (for all inter-plate separations) and analytically (for small inter-plate separations), we show that this is not, in general, the case. In fact, the limit of charge regulation is a separate boundary condition, yielding a disjoining pressure with a different characteristic separation scaling. Our findings are supported by several examples demonstrating that the disjoining pressure at small separations for the charge regulation boundary condition depends on the details of the dissociation/association process.

Orthokinetic aggregation of charged colloidal particles in the presence of repulsive double layer force: A trajectory analysis with the solution of non-linear Poisson–Boltzmann equation

Abstract We studied turbulent coagulation rates of well-characterized carboxyl-terminated latex particles as a function of pH at different KCl concentrations. The negative charge of the studied particles increases and thus develops the electrical double layer repulsion with increasing pH. The pH-dependent coagulation rate was analyzed on the basis of the trajectory analysis with the Derjaguin, Landau, Verwey, and Overbeek theory using the solution of non-linear Poisson–Boltzmann (PB) equation under the charge regulation boundary condition, in which the electrostatic boundary conditions such as surface potential depends on surface separation distance. It should be noted that this is the first attempt to compare the calculated results with the experimental data of the effect of surface charge on turbulent coagulation rates. The calculation using the trajectory analysis can well describe the experimental data. However, to obtain the good agreement between the experimental data and the calculation, we need to decrease the Hamaker constant A H with increasing KCl concentration. The dependence of A H on salt concentration is probably ascribed to the fact that the origin of the van der Waals attraction is essentially electrostatic dipole–dipole interaction between molecules. That is, we consider that the van der Waals attraction is increasingly screened by the charge of ions with increasing salt concentration.

Consequences of shifted ion adsorption equilibria due to nonelectrostatic interaction potentials in electrical double layers

Abstract Consistent incorporation of nonelectrostatic interactions in the chemical potential for electrical double layers at interfaces interacting across electrolytes under the constant potential or charge regulation boundary conditions has recently been presented for the case of ionic dispersion potentials. This gives rise to shifted adsorption equilibria, and thereby a shift in the predicted surface electrostatic potential. It also results in an additional component previously unaccounted for in the total double layer interaction force. The new force component due to ionic dispersion can be repulsive or attractive and can in some cases exceed entropic repulsion in magnitude. The altered force leads to the need to recalibrate electrostatic surface potentials and equilibrium constants when fitting to experimental force data. We explore the implications and consequences for model systems of mica surfaces, to illustrate the effect of salt concentration, and cellulose surfaces to illustrate the effect of pH. The final example is the more complex, asymmetric system of crude oil and glass interacting across salt solutions of varying concentration and pH. The DLVO theory augmented by ionic dispersion potentials is used to fit measured data for ζ-potentials of oil and glass, from which the calculated disjoining pressure isotherms are compared to measured macroscopic oil–glass adhesion data.

The effect of surface charge regulation on conductivity in fluidic nanochannels

The precise electrostatic potential distribution is very important for the electrokinetic transport in fluidic channels. This is especially valid for small nanochannels where the electric double layers formed at the walls are comparable to the channel width. It can be expected that due to the large surface to volume ratio in such systems, they will exhibit properties that are not detectable in larger channels, capillaries and pores. We present a detailed numerical analysis of the current transport in fluidic nanochannels. It is based on solving the Poisson–Boltzmann equation with charge regulation boundary conditions that account for the surface-aqueous solution chemical equilibria. The focus is on studying the effect of the pH on the current transport. The pH is varied by adding either HCl or KOH. The analysis predicts non-monotonous and sometimes counterintuitive dependence of the conductivity on the pH. The channel conductivity exhibits practically no change over a range of pH values due to a buffering exerted by the chemical groups at the walls. An unexpected drop of the conductivity is observed around the wall isoelectric point and also in the vicinity of pH=7 even though the concentration of ions in the channel increases. These observations are explained in the framework of charge regulation theory.

Effect of Nonelectrostatic Ion Interactions on Surface Forces Involving Ion Adsorption Equilibria

The chemical, or chemisorption, part of colloidal interaction free energy is revisited. Consistent incorporation of nonelectrostatic interactions in the chemical potential for the constant potential and charge regulation boundary conditions is developed. This gives rise to shifted adsorption equilibria, and thereby a shift in the predicted surface electrostatic potential. It also results in an additional component previously unaccounted for in the total double layer interaction force. The altered force leads to the need of recalibrating electrostatic surface potentials and equilibrium constants when fitting to experimental force data. A numerical illustration is presented using ionic dispersion potentials for mica surfaces interacting across NaCl at various concentrations. The new force component due to ionic dispersion is typically repulsive and exceeds entropic repulsion in magnitude. These results suggest that the effect of ionic dispersion is more profound than previously believed, even at low electrolyte concentrations.

Electrokinetics in nanochannels: Part II. Mobility dependence on ion density and ionic current measurements

In the first of this two-paper series, a new model was developed for calculating the electric potential field in a long, thin nanochannel with overlapped electric double layers. The model takes into account the dependence of ion mobility on local ion densities and pH. This model is used here to study and demonstrate the effect of ion density and pH on ionic current measurements. A comparison is shown of predictions based on each of three boundary conditions, as studied in Part I. The model developed in Part I is validated by comparing simulations with measurements of ionic current as a function of sodium borate concentration. Results show that predictions based on extended Debye–Hückel theory for ion mobility significantly improve the accuracy of simulations, but that these do not predict exact scaling behavior. A simple bulk conductivity measurement used as input parameter for the simulations, in place of the predicted bulk conductivity ( K 0 ), guarantees agreement with data in the thin EDL region. Results also indicate that the charge regulation boundary condition, complemented with an adequate bulk electrolyte model, provides better agreement with experimental trends than the specified zeta potential or specified surface net charge boundary conditions. Further, it is shown that currents due to advection (by electroosmotic flow) are in all cases studied less than 25% of the total current in the system.

A Simple Algorithm for Calculating Electrical Double Layer Interactions in Asymmetric Electrolytes—Poisson–Boltzmann Theory

A simple, general, and numerically robust algorithm is presented for calculating the disjoining pressure and interaction free energy per unit area between two identically charged flat plates due to electrical double layer interactions according to the nonlinear Poisson–Boltzmann theory. The result is applicable to electrolytes with any number of ionic species having any combination of valencies as well as to constant potential, constant charge, or charge regulation boundary conditions on the plates. The algorithm is very simple to implement on commonly available numerical software environments and is therefore particularly suitable for use in data analysis.

Charge Regulation and Electrostatic Interactions for a Spherical Particle in a Cylindrical Pore

Previous theoretical analyses of electrostatic interactions for proteins in porous media have assumed that the protein (and pore) surface maintains either constant charge or constant potential during the interaction; however, the actual surface charge is determined by the extent of the surface ionization and binding reactions, both of which are altered when the protein enters the pore due to the change in the local ionic environment caused by the distortion of the electrical potential field. Theoretical calculations for the electrostatic potential are performed for a spherical particle in a cylindrical pore, accounting for this charge regulation phenomenon using a linearized form of the charge regulation boundary condition. The equilibrium partition coefficient in the pore is then evaluated from the free energy of interaction. Specific calculations are provided for the protein bovine serum albumin, with the charge regulation parameters evaluated from a model for the detailed protein charge characteristics. Model predictions are compared with experimental data for the bovine serum albumin sieving coefficient at different pH and ionic strength. These results provide important insights into the effects of charge regulation on the magnitude of the electrostatic interactions between charged proteins and charged pores.

Interaction Free Energy between Plates with Charge Regulation: A Linearized Model

The linearized Poisson-Boltzmann theory is used to calculate the electrical double layer interaction free energy between two parallel charged plates for the case in which charge regulation due to the dissociation of surface groups may be modelled by the linearized regulation model that specifies a linear relationship between the surface charge and the surface potential. This charge regulation model is characterized by a constant—termed the regulation capacitance of the surface. Analytic expressions for the force per unit area, the interaction free energy per unit area as well as the interaction free energy between two nonidentical spheres in the Deryaguin limit are given for the general case of nonidentical surfaces. An expression for the interaction free energy, applicable to any geometry, is obtained by thermodynamic arguments. Numerical comparisons for the case of identical amphoteric surfaces show that linearizing the charge regulation boundary conditions produces little error in the resultant interaction free energy.

Electrostatic interactions between two spheres: Solutions of the debye-hückel equation with a charge regulation boundary condition

The electrostatic interaction between a pair of spherical colloidal particles is analyzed using a charge regulation model to span behavior ranging from constant potential to constant charge boundary conditions. Electrostatic interparticle forces are calculated through a multipole expansion, the linear superposition approximation, and, for strongly overlapping thin double layers, the “Derjaguin approximation.” By comparing the forces calculated via the different methods, we establish the range of surface parameters for which the Derjaguin approximation and linear superposition are valid in a manner similar to that employed by A. B. Glendinning and W. B. Russel ( J. Colloid Interface Sci. 93, 95 (1983)) . It is shown that the Derjaguin approximation furnishes an accurate representation over a wide range of surface conditions for the charge regulation model.