Nonlinear Poisson-Boltzmann Theory Research Articles

Osmotic swelling behavior of surface-charged ionic microgels.

In recent years, ionic microgels have garnered much attention due to their unique properties, especially their stimulus-sensitive swelling behavior. The tunable response of these soft, permeable, compressible, charged colloidal particles is increasingly attractive for applications in medicine and biotechnologies, such as controlled drug delivery, tissue engineering, and biosensing. The ability to model and predict variation of the osmotic pressure of a single microgel with respect to changes in particle properties and environmental conditions proves vital to such applications. In this work, we apply both nonlinear Poisson-Boltzmann theory and molecular dynamics simulation to ionic microgels (macroions) in the cell model to compute density profiles of microions (counterions, coions), single-microgel osmotic pressure, and equilibrium swelling ratios of spherical microgels whose fixed charge is confined to the macroion surface. The basis of our approach is an exact theorem that relates the electrostatic component of the osmotic pressure to the microion density profiles. Close agreement between theory and simulation serves as a consistency check to validate our approach. We predict that surface-charged microgels progressively deswell with increasing microgel concentration, starting well below close packing, and with increasing salt concentration, in qualitative agreement with experiments. Comparison with previous results for microgels with fixed charge uniformly distributed over their volume demonstrates that surface-charged microgels deswell more rapidly than volume-charged microgels. We conclude that swelling behavior of ionic microgels in solution is sensitive to the distribution of fixed charge within the polymer-network gel and strongly depends on bulk concentrations of both microgels and salt ions.

Curvature effects in interfacial acidity of amphiphilic vesicles.

We analyze the changes in the vicinal acidity (pH) at a spherical amphiphilic membrane. The membrane is assumed to contain solvent accessible, embedded, dissociable, charge-regulated moieties. Basing our approach on the linear Debye-Hückel approximation, as well as on the nonlinear Poisson-Boltzmann theory, together with the general Frumkin-Fowler-Guggenheim adsorption isotherm model of the charge-regulation process, we analyze and review the dependence of the local pH on the position, as well as bulk electrolyte concentration, bulk pH, and curvature of the amphiphilic single membrane vesicle. With appropriately chosen adsorption parameters of the charge-regulation model, we find a good agreement with the available experimental data.

Electrostatic interactions control the adsorption of extracellular vesicles onto supported lipid bilayers

Communication between cells located in different parts of an organism is often mediated by membrane-enveloped nanoparticles, such as extracellular vesicles (EVs). EV binding and cell uptake mechanisms depend on the heterogeneous composition of the EV membrane. From a colloidal perspective, the EV membrane interacts with other biological interfaces via both specific and non-specific interactions, where the latter include long-ranged electrostatic and van der Waals forces, and short-ranged repulsive “steric-hydration” forces. While electrostatic forces are generally exploited in most EV immobilization protocols, the roles played by various colloidal forces in controlling EV adsorption on surfaces have not yet been thoroughly addressed. In the present work, we study the adsorption of EVs onto supported lipid bilayers (SLBs) carrying different surface charge densities using a combination of quartz crystal microbalance with dissipation monitoring (QCM-D) and confocal laser scanning microscopy (CLSM). We demonstrate that EV adsorption onto lipid membranes can be controlled by varying the strength of electrostatic forces and we theoretically describe the observed phenomena within the framework of nonlinear Poisson-Boltzmann theory. Our modelling results confirm the experimental observations and highlight the crucial role played by attractive electrostatics in EV adsorption onto lipid membranes. They furthermore show that simplified theories developed for model lipid systems can be successfully applied to the study of their biological analogues and provide new fundamental insights into EV-membrane interactions with potential use in developing novel EV separation and immobilization strategies.

Theoretical description of the electrical double layer for a mixture of n ionic species with arbitrary size and charge asymmetries. I. Spherical geometry.

In this work, we propose a theoretical finite element description of the ionic profiles of a general mixture of n species of spherical charged particles dissolved in an implicit solvent, with arbitrary size and charge asymmetries, neutralizing a spherical macroion. This approach aims to close the gap between the nano- and the micro-scales in macroion solutions, taking into account the ion correlations and ionic excluded volume effects consistently. When these last two features are neglected, the classical non-linear Poisson-Boltzmann theory for n ionic species-with different ionic closest approach distances to the colloidal surface-is recovered as a limit case. As a proof of concept, we study the electrical double layer of an electroneutral mixture of oppositely charged colloids and small microions, with an asymmetry 1:333 in size and 1:10 in valence, in salt-free and added salt environments. Our theoretical approach displays a good agreement regarding the ionic profiles, the integrated charge, and the mean electrostatic potential obtained from molecular dynamics simulations with explicit-sized microions. Although the non-linear Poisson-Boltzmann colloid-colloid and colloid-microion profiles differ notably from those obtained via molecular dynamics simulations with explicit small-sized ions, the associated mean electrostatic potential agrees well with the corresponding explicit microion simulations.

Size and/or charge asymmetry effects in coulombic fluids in the presence of external fields: From simple electrolytes to molten salts.

Charged fluids are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of charged colloidal solutions are of outstanding relevance in nature and in diverse biomedical applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. When an electrode or a charged colloidal particle is immersed in a Coulombic fluid, an ionic cloud or electrical double layer forms. One of the most simple theoretical descriptions of the electrical double layer can be provided via the non-linear Poisson-Boltzmann theory of point-ions. In planar geometry, it is possible to derive an analytical description of the electrical double layer for z:z electrolytes. In spite of its simplicity, this mean field approach neglects important characteristics of charged fluids such as ion correlations, ionic excluded volume, polarization effects, etc. Thus, in this review article I would like to discuss some consequences of breaking the symmetry in the ionic size and/or valence of charged fluids in the presence of an external field when ion correlations and ionic excluded volume effects are included consistently. The external field can be produced by a charged electrode or an electrified colloidal particle, and the associated electrical double layer is studied via integral equations theory, and/or molecular simulations beyond the classical non-linear Poisson Boltzmann theory of point ions.

On the non-dominance of counterions in the 1:z planar electrical double layer of point-ions

According to the dominance prescription of point-ions in the non-linear Poisson-Boltzmann theory, proposed by Valleau and Torrie almost 40 years ago, the microscopic and thermodynamic properties of an asymmetric binary electrolyte converge asymptotically to those of a completely symmetric electrolyte, in the limit of an infinite surface charge density of a planar electrode, if the properties of the counterions are the same in both instances. By using the Grahame equation and the non-linear Poisson-Boltzmann theory, we show here that this prescription is certainly exact for the mean electrostatic potential at the electrode's surface and for the capacitive compactness. Contrastingly, analytical and numerical solutions of the non-linear Poisson-Boltzmann equation show that, in the limit of an infinite surface charge density of the planar electrode, it is possible to observe finite differences between the local mean electrostatic potentials and electric fields associated to a 1:1 and a 1:z electrolyte at places near the electrode's surface. Thus, we prove here that even in the absence of ion correlations and ionic excluded volume effects, the counterions do not fully dominate the structural properties in the entire electrical double layer in the non-linear Poisson-Boltzmann picture, which is confirmed through comparisons with new Monte Carlo simulations.

Electrical double layer interactions between soil colloidal particles: Polarization of water molecule and counterion

The electrical double layer (EDL) interactions strongly affect aggregation and dispersion of soil colloidal particles. In this research a modified model taking the polarization of water molecule and counterion into account was proposed based on nonlinear Poisson-Boltzmann theory to describe electrostatic interaction. The polarizations of both the counterion and the water molecule can increase the interaction between ion/water and soil particle surface, and then decrease the surface potential. Particle surface potential also depends on the surface separation when the EDL overlaps. The midpoint potential between colloidal particles decreases with increase of ion polarization but increases in the presence of water polarization through the change of the double layer thickness. Both the electrostatic repulsive pressure between soil particles and its force range increase markedly in the presence of the polarization of water molecules, but decrease with increase of counterion polarization. The water molecule polarization, rather than the counterion polarization, contributes more to EDL interactions. The counterion polarization is the origin of Hofmeister effects of the EDL interactions, which is affected by water polarization. The theoretical predicted force curve between clay particles by the modified model was verified by the experimental determination of surface force. In addition, the hydration force, fitted to a double exponential decay function, can be estimated by subtraction of the electrostatic interaction force based on DLVO theory from that based on the present model.

Electrostatic pair-interaction of nearby metal or metal-coated colloids at fluid interfaces.

In this paper, we theoretically study the electrostatic interaction between a pair of identical colloids with constant surface potentials sitting in close vicinity next to each other at the fluid interface. By employing a simplified yet reasonable model system, the problem is solved within the framework of classical density functional theory and linearized as well as nonlinear Poisson-Boltzmann (PB) theory. Apart from providing a sound theoretical framework generally applicable to any such problem, our novel findings, all of which contradict common beliefs, include the following: first, quantitative and qualitative differences between the interactions obtained within the linear and the nonlinear PB theories; second, the importance of the electrostatic interaction between the omnipresent three-phase contact lines in interfacial systems; and, third, the occurrence of an attractive electrostatic interaction between a pair of identical metal colloids. The unusual attraction we report largely stems from an attractive line interaction, which although scales linearly with the size of the particle can compete with the surface interactions and can be strong enough to alter the nature of the total electrostatic interaction. Our results should find applications in metal or metal-coated particle-stabilized emulsions where densely packed particle arrays are not only frequently observed but also sometimes required.

Expansion and shrinkage of the electrical double layer in charge-asymmetric electrolytes: A non-linear Poisson-Boltzmann description

The spatial extension or thickness of the electrical double layer determines many thermodynamic and electrokinetic properties of charged colloids in solution. In the classical Debye-Hückel formalism of point ions, the thickness of the electrical double layer around a spherical macroion or next to an infinite planar electrode can be characterized by the bulk Debye length of the supporting ionic fluid. As a result, that approach neglects, at least, the influence of the colloidal charge on the spatial extension of the ionic cloud. Given that the Debye-Hückel formalism of point ions is valid only in the limit of very weak colloidal charges, in this work we use the non-linear Poisson-Boltzmann equation to study the thickness of the electrical double layer near a positively charged electrode in spherical and planar geometries, in the presence of several binary charge-asymmetric −1:z+ point-ions electrolytes with monovalent counterions and multivalent coions. The properties of counterions are maintained fixed, whereas the properties of coions (such as their valence and concentration) are varied fulfilling the bulk electroneutrality condition. The thickness of the ionic cloud is quantified here via the recently introduced capacitive compactness idea (Phys. Chem. Chem. Phys. 20 (2018) 262). Physically, this length represents essentially the separation distance between two electrodes associated to the corresponding effective electrical double layer capacitor, in both planar and spherical geometries. Our numerical calculations show that the capacitive compactness obtained via the non-linear Poisson-Boltzmann equation reduces to the Debye length of the supporting bulk electrolyte at the point of zero charge. In the presence of a charge symmetric −1:+1 electrolyte, the capacitive compactness always decreases as a function of the colloidal charge. Contrastingly, the electrical double layer may expand or shrink as a function of the surface charge density, in the presence of multivalent coions, which is confirmed here by primitive model Monte Carlo simulations. This last non-monotonic behaviour of the capacitive compactness for multivalent coions, depending on the colloidal charge, is related to the microscopic behaviour of the local electric field, the mean electrostatic potential, and the net ionic charge per unit volume close to the colloidal surface. At very large colloidal charges, the capacitive compactness of all −1:z+ electrolytes collapses onto a single curve illustrating the dominance of counterions in the non-linear Poisson-Boltzmann theory.

Controlling wetting with electrolytic solutions: Phase-field simulations of a droplet-conductor system.

The wetting properties of immiscible two-phase systems are crucial in applications ranging from laboratory-on-a-chip devices to field-scale oil recovery. It has long been known that effective wetting properties can be altered by the application of an electric field; a phenomenon coined as electrowetting. Here, we consider theoretically and numerically a single droplet sitting on an (insulated) conductor, i.e., within a capacitor. The droplet consists of a pure phase without solutes, while the surrounding fluid contains a symmetric monovalent electrolyte, and the interface between them is impermeable. Using nonlinear Poisson-Boltzmann theory, we present a theoretical prediction of the dependency of the apparent contact angle on the applied electric potential. We then present well-resolved dynamic simulations of electrowetting using a phase-field model, where the entire two-phase electrokinetic problem, including the electric double layers (EDLs), is resolved. The simulations show that, while the contact angle on scales smaller than the EDL is unaffected by the application of an electric field, an apparent contact angle forms on scales beyond the EDL. This contact angle relaxes in time towards a saturated apparent contact angle. The dependency of the contact angle upon applied electric potential is in good agreement with the theoretical prediction. The only phenomenological parameter in the prediction is shown to depend on the permeability ratio between the two phases. Based on the resulting unified description, we obtain an effective expression of the contact angle which can be used in more macroscopic numerical simulations, i.e. where the electrokinetic problem is not fully resolved.

Planar screening by charge polydisperse counterions

We study how a neutralising cloud of counterions screens the electric field of a uniformly charged planar membrane (plate), when the counterions are characterised by a distribution of charges (or valence), . We work out analytically the one-plate and two-plate cases, at the level of non-linear Poisson–Boltzmann theory. The (essentially asymptotic) predictions are successfully compared to numerical solutions of the full Poisson–Boltzmann theory, but also to Monte Carlo simulations. The counterions with smallest valence control the long-distance features of interactions, and may qualitatively change the results pertaining to the classic monodisperse case where all counterions have the same charge. Emphasis is put on continuous distributions , for which new power-laws can be evidenced, be it for the ionic density or the pressure, in the one- and two-plates situations respectively. We show that for discrete distributions, more relevant for experiments, these scaling laws persist in an intermediate but yet observable range. Furthermore, it appears that from a practical point of view, hallmarks of the continuous behaviour are already featured by discrete mixtures with a relatively small number of constituents.

Structure of primitive electrolytes in semi-permeable shells

We have theoretically studied the competing adsorption of the two- and three-component electrolytes with different sizes and ionic valences near a semi-permeable shell. The present theory provides interesting insights about the steric charge separation occurring at the shell, which cannot be explained by the linearized and nonlinear Poisson-Boltzmann theories. The smaller and divalent ions interact more strongly with the shell than the larger and monovalent ions because the smaller ions can approach more closely to the shell surface and the divalent ions interact more strongly with the charged shell than the monovalent ions. The large steric charge separation occurs at a highly asymmetric electrolyte and a high bulk concentration. The competition between the cations and anions gives rise to the charge reversal and charge inversion even for an uncharged shell. A positively charged surface enhances the charge surface amplification because the shell is impermeable for the anion, whereas a negatively charged surface enhances the charge reversal. The excess osmotic pressure, which is based on a nonlinear Poisson-Boltzmann theory, depends only on the permeated ion density at the shell, and increases with increasing the shell size and surface charge density.

Charged plate in asymmetric electrolytes: One-loop renormalization of surface charge density and Debye length due to ionic correlations.

Self-consistent field theory (SCFT) is used to study the mean potential near a charged plate inside a m:-n electrolyte. A perturbation series is developed in terms of g=4πκb, where band1/κ are Bjerrum length and bare Debye length, respectively. To the zeroth order, we obtain the nonlinear Poisson-Boltzmann theory. For asymmetric electrolytes (m≠n), the first order (one-loop) correction to mean potential contains a secular term, which indicates the breakdown of the regular perturbation method. Using a renormalizaton group transformation, we remove the secular term and obtain a globally well-behaved one-loop approximation with a renormalized Debye length and a renormalized surface charge density. Furthermore, we find that if the counterions are multivalent, the surface charge density is renormalized substantially downwards and may undergo a change of sign, if the bare surface charge density is sufficiently large. Our results agrees with large MC simulation even when the density of electrolytes is relatively high.

Poisson-Boltzmann study of the effective electrostatic interaction between colloids at an electrolyte interface

The effective electrostatic interaction between a pair of colloids, both of them located close to each other at an electrolyte interface, is studied by employing the full, nonlinear Poisson-Boltzmann (PB) theory within classical density functional theory. Using a simplified yet appropriate model, all contributions to the effective interaction are obtained exactly, albeit numerically. The comparison between our results and those obtained within linearized PB theory reveals that the latter overestimates these contributions significantly at short inter-particle separations. Whereas the surface contributions to the linear and the nonlinear PB results differ only quantitatively, the line contributions show qualitative differences at short separations. Moreover, a dependence of the line contribution on the solvation properties of the two adjacent fluids is found, which is absent within the linear theory. Our results are expected to enrich the understanding of effective interfacial interactions between colloids.

Reduction in Tension and Stiffening of Lipid Membranes in an Electric Field Revealed by X-Ray Scattering.

The effect of ac electric fields on the elasticity of supported lipid bilayers is investigated at the microscopic level using grazing incidence synchrotron x-ray scattering. A strong decrease in the membrane tension up to 1 mN/m and a dramatic increase of its effective rigidity up to 300 k_{B}T are observed for local electric potentials seen by the membrane ≲1 V. The experimental results are analyzed using detailed electrokinetic modeling and nonlinear Poisson-Boltzmann theory. Based on a modeling of the electromagnetic stress, which provides an accurate description of the bilayer separation versus pressure curves, we show that the decrease in tension results from the amplification of charge fluctuations on the membrane surface whereas the increase in bending rigidity results from the direct interaction between charges in the electric double layer. These effects eventually lead to a destabilization of the bilayer and vesicle formation. Similar effects are expected at the tens of nanometers length scale in cell membranes with lower tension, and could explain a number of electrically driven processes.

Potential of mean force between like-charged nanoparticles: Many-body effect.

Ion-mediated interaction is important for the properties of polyelectrolytes such as colloids and nucleic acids. The effective pair interactions between two polyelectrolytes have been investigated extensively, but the many-body effect for multiple polyelectrolytes still remains elusive. In this work, the many-body effect in potential of mean force (PMF) between like-charged nanoparticles in various salt solutions has been comprehensively examined by Monte Carlo simulation and the nonlinear Poisson-Boltzmann theory. Our calculations show that, at high 1:1 salt, the PMF is weakly repulsive and appears additive, while at low 1:1 salt, the additive assumption overestimates the repulsive many-body PMF. At low 2:2 salt, the pair PMF appears weakly repulsive while the many-body PMF can become attractive. In contrast, at high 2:2 salt, the pair PMF is apparently attractive while the many-body effect can cause a weaker attractive PMF than that from the additive assumption. Our microscopic analyses suggest that the elusive many-body effect is attributed to ion-binding which is sensitive to ion concentration, ion valence, number of nanoparticles and charges on nanoparticles.

Poisson-Boltzmann thermodynamics of counterions confined by curved hard walls.

We consider a set of identical mobile pointlike charges (counterions) confined to a domain with curved hard walls carrying a uniform fixed surface charge density, the system as a whole being electroneutral. Three domain geometries are considered: a pair of parallel plates, the cylinder, and the sphere. The particle system in thermal equilibrium is assumed to be described by the nonlinear Poisson-Boltzmann theory. While the effectively one-dimensional plates and the two-dimensional cylinder have already been solved, the three-dimensional sphere problem is not integrable. It is shown that the contact density of particles at the charged surface is determined by a first-order Abel differential equation of the second kind which is a counterpart of Enig's equation in the critical theory of gravitation and combustion or explosion. This equation enables us to construct the exact series solutions of the contact density in the regions of small and large surface charge densities. The formalism provides, within the mean-field Poisson-Boltzmann framework, the complete thermodynamics of counterions inside a charged sphere (salt-free system).

The non-dominance of counterions in charge-asymmetric electrolytes: non-monotonic precedence of electrostatic screening and local inversion of the electric field by multivalent coions.

The asymptotic convergence of the thermodynamic and structural properties of unequally-sized charge-symmetric ions in strong electric fields was postulated more than thirty years ago by Valleau and Torrie as the dominance of counterions via the non-linear Poisson-Boltzmann theory [Valleau and Torrie, J. Chem. Phys., 1982, 76, 4623]. According to this mean field prescription, the properties of the electrical double layer near a highly charged electrode immersed in a size-asymmetric binary electrolyte converge to those of a size-symmetric electrolyte if the properties of counterions are the same in both instances. On the other hand, some of the present authors have shown that, in fact, counterions do not dominate the electrical properties of a spherical macroion in the presence of unequally-sized ions, symmetric in valence, if ion correlations and ionic excluded volume effects are taken into account consistently. These ingredients are neglected in the classical Poisson-Boltzmann picture. In the present work, we show the occurrence of the non-dominance of counterions in the opposite scenario, that is, when ions are equally-sized but asymmetric in valence. This is performed in the presence of highly charged colloidal surfaces of spherical and planar geometries for different ionic volume fractions. In addition to the phenomenon of non-dominance of counterions, our simulations and theoretical data also exhibit a non-monotonic order or precedence in the mean electrostatic potential, or electrostatic screening, at the Helmholtz plane of a charged colloid. This interesting behaviour is analyzed as a function of the coion's valence, the ionic volume fraction, and the charge and size of the colloidal particle. All these phenomena are explained in terms of the decay of the electric field near the colloidal surface, and by the appearance of a local inversion of both the electric field and the integrated surface charge density of the colloidal particle in the presence of monovalent counterions and multivalent coions.

The apparent charge of nanoparticles trapped at a water interface.

Charged spherical nanoparticles trapped at the interface between water and air or water and oil exhibit repulsive electrostatic forces that contain a long-ranged dipolar and a short-ranged exponentially decaying component. The former are induced by the unscreened electrostatic field through the non-polar low-permittivity medium, and the latter result from the overlap of the diffuse ion clouds that form in the aqueous phase close to the nanoparticles. The magnitude of the long-ranged dipolar interaction is largely determined by the residual charges that remain attached to the air- (or oil-) exposed region of the nanoparticle. In the present work we address the question to what extent the charges on the water-immersed part of the nanoparticle provide an additional contribution to the dipolar interaction. To this end, we model the electrostatic properties of a spherical particle - a nanoparticle or a colloid - that partitions equatorially to the air-water interface, thereby employing nonlinear Poisson-Boltzmann theory in the aqueous solution and accounting for the propagation of the electric field through the interior of the particle. We demonstrate that the apparent charge density on the air-exposed region of the particle, which determines the dipole potential, is influenced by the electrostatic properties in the aqueous solution. We also show that this electrostatic coupling through the particle can be reproduced qualitatively by a simple analytic planar capacitor model. Our results help to rationalize the experimentally observed weak but non-vanishing salt dependence of the forces that stabilize ordered two-dimensional arrays of interface-trapped nanoparticles or colloids.

Effective charge of cylindrical and spherical colloids immersed in an electrolyte: the quasi-planar limit

We consider the nonlinear Poisson–Boltzmann theory for a single cylindrical or spherical macro-ion in symmetric 1:1, together with asymmetric 1:2 and 2:1 electrolytes. We focus on the regime where , the ratio of the macro-ion radius a over the inverse Debye length in the bulk electrolyte, is large. Analyzing the structure of the analytical expansion emerging from a multiple scale analysis, we uncover a hidden structure for the electrostatic potential. This structure, which appears after a heuristic resummation, suggests a new and convenient expansion scheme that we present and work out in detail. We show that novel exact results can thereby be obtained, in particular pertaining to effective charge properties, in complete agreement with the direct numerical solution to the problem.