Counterion-only electrical double layer: A constrained entropy approach

Colloidal Stability of PFSA-Ionomer Dispersions. Part I. Single-Ion Electrostatic Interaction Potential Energies

Charged colloidal particles neutralized by a single counterion are increasingly important for many emerging technologies. Attention here is paid specifically to hydrogen fuel cells and water electrolyzers whose catalyst layers are manufactured from a perfluorinated sulfonic acid polymer (PFSA) suspended in aqueous/alcohol solutions. Partially dissolved PFSA aggregates, known collectively as ionomers, are stabilized by the electrostatic repulsion of overlapping diffuse double layers consisting of only protons dissociated from the suspended polymer. We denote such double layers containing no added electrolyte as "single ion". Size-distribution predictions build upon interparticle interaction potential energies from the Derjaguin-Landau-Verwey-Overbeek (DLVO) formalism. However, when only a single counterion is present in solution, classical DLVO electrostatic potential energies no longer apply. Accordingly, here a new formulation is proposed to describe how single-counterion diffuse double layers interact in colloidal suspensions. Part II (Srivastav, H.; Weber, A. Z.; Radke, C. J. Langmuir 2024 DOI: 10.1021/acs.langmuir.3c03904) of this contribution uses the new single-ion interaction energies to predict aggregated size distributions and the resulting solution pH of PFSA in mixtures of n-propanol and water. A single-counterion diffuse layer cannot reach an electrically neutral concentration far from a charged particle. Consequently, nowhere in the dispersion is the solvent neutral, and the diffuse layer emanating from one particle always experiences the presence of other particles (or walls). Thus, in addition to an intervening interparticle repulsive force, a backside osmotic force is always present. With this new construction, we establish that single-ion repulsive pair interaction energies are much larger than those of classical DLVO electrostatic potentials. The proposed single-ion electrostatic pair potential governs dramatic new dispersion behavior, including dispersions that are stable at a low volume fraction but unstable at a high volume fraction and finite volume-fraction dispersions that are unstable with fine particles but stable with coarse particles. The proposed single-counterion electrostatic pair potential provides a general expression for predicting colloidal behavior for any charged particle dispersion in ionizing solvents with no added electrolyte.

Mathematical model of electromigration allowing the deviation from electroneutrality

The structure of the double layer on the boundary between solid and liquid phases is described by various models, of which the Stern-Gouy-Chapman model is still commonly accepted. Generally, the solid phase is charged, which also causes the distribution of the electric charge in the adjacent diffuse layer in the liquid phase. We propose a new mathematical model of electromigration considering the high deviation from electroneutrality in the diffuse layer of the double layer when the liquid phase is composed of solution of weak multivalent electrolytes of any valence and of any complexity. The mathematical model joins together the Poisson equation, the continuity equation for electric charge, the mass continuity equations, and the modified G-function. The model is able to calculate the volume charge density, electric potential, and concentration profiles of all ionic forms of all electrolytes in the diffuse part of the double layer, which consequently enables to calculate conductivity, pH, and deviation from electroneutrality. The model can easily be implemented into the numerical simulation software such as Comsol. Its outcome is demonstrated by the numerical simulation of the double layer composed of a charged silica surface and an adjacent liquid solution composed of weak multivalent electrolytes. The validity of the model is not limited only to the diffuse part of the double layer but is valid for electromigration of electrolytes in general.

Experimental Evidence for Algebraic Double-Layer Forces

According to conventional wisdom, electric double-layer forces normally decay exponentially with separation distance. Here, we present experimental evidence of algebraically decaying double-layer interactions. We show that algebraic interactions arise in both strongly overlapping as well as counterion-only regimes, albeit the evidence is less clear for the former regime. In both of these cases, the disjoining pressure profile assumes an inverse square distance dependence. At small separation distances, another algebraic regime is recovered. In this regime, the pressure decays as the inverse of separation distance.

Interactions between silica particles in the presence of multivalent coions

Forces between charged silica particles in solutions of multivalent coions are measured with colloidal probe technique based on atomic force microscopy. The concentration of 1 : z electrolytes is systematically varied to understand the behavior of electrostatic interactions and double-layer properties in these systems. Although the coions are multivalent the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory perfectly describes the measured force profiles. The diffuse-layer potentials and regulation properties are extracted from the forces profiles by using the DLVO theory. The dependencies of the diffuse-layer potential and regulation parameter shift to lower concentration with increasing coion valence when plotted as a function of concentration of 1 : z salt. Interestingly, these profiles collapse to a master curve if plotted as a function of monovalent counterion concentration.

Depletion and double layer forces acting between charged particles in solutions of like-charged polyelectrolytes and monovalent salts

Interaction forces between silica particles were measured in aqueous solutions of the sodium salt of poly(styrene sulphonate) (PSS) and NaCl using the colloidal probe technique based on an atomic force microscope (AFM). The observed forces can be rationalized through a superposition of damped oscillatory forces and double layer forces quantitatively. The double layer forces are modeled using Poisson-Boltzmann (PB) theory for a mixture of a monovalent symmetric electrolyte and a highly asymmetric electrolyte, whereby the multivalent coions represent the polyelectrolyte chains. The effective charge of the polyelectrolyte is found to be smaller than the bare number of charged groups residing on one polyelectrolyte molecule. This effect can be explained by counterion condensation. The interplay between depletion and double layer forces can be further used to predict the phase of the depletion force oscillations. However, this picture holds only at not too elevated concentrations of the polyelectrolyte and salt. At higher salt concentrations, attractive van der Waals forces become important, while at higher polyelectrolyte concentrations, the macromolecules adsorb onto the like-charged silica interface.

Colloidal Stability in Asymmetric Electrolytes: Modifications of the Schulze-Hardy Rule

The Schulze-Hardy rule suggests a strong dependence of the critical coagulation concentration (CCC) on the ionic valence. This rule is addressed theoretically and confronted with recent experimental results. The commonly presented derivation of this rule assumes symmetric electrolytes and highly charged particles. Both assumptions are incorrect. Symmetric electrolytes containing multivalent ions are hardly soluble, and experiments are normally carried out with the well-soluble salts of asymmetric electrolytes containing monovalent and multivalent ions. In this situation, however, the behavior is completely different whether the multivalent ions represent the counterions or co-ions. When these ions represent the counterions, meaning that the multivalent ions have the opposite sign than the charge of the particle, they adsorb strongly to the particles. Thereby, they progressively reduce the magnitude of the surface charge with increasing valence. In fact, this dependence of the charge density on the counterion valence is mainly responsible for the decrease of the CCC with the valence. In the co-ion case, where the multivalent ions have the same sign as the charge of the particle, the multivalent ions are repelled from the particles, and the surfaces remain highly charged. In this case, the inverse Schulze-Hardy rule normally applies, whereby the CCC varies inversely proportional to the co-ion valence.

Derivation of the inverse Schulze-Hardy rule

The inverse Schulze-Hardy rule was recently proposed based on experimental observations. This rule describes an interesting situation of the aggregation of charged colloidal particles in the presence of the multivalent coions. Specifically, it can be shown that the critical coagulation concentration is inversely proportional to the coion valence. Here the derivation of the inverse Schulze-Hardy rule based on purely theoretical grounds is presented. This derivation complements the classical Schulze-Hardy rule, which describes the multivalent counterion systems.

Counterion-only electrical double layers: An application of density functional theory

Within the framework of density functional theory, a self-consistent approach of weighted correlation approximation is developed to give an accurate account of the cross correlations between the Coulombic interaction and the hard-sphere exclusion in the counterion-only electrical double layers. Application of the approach to the cases of practical interest, against the Monte Carlo simulations, shows that it is excellent in describing the structural properties and the pressures of the confined solutions involving both mono- and divalent counterions between two planar charged walls. In particular, the study suggests that the relative importance of electrostatic correlations in comparison to the effects of ionic excluded volume and direct Coulomb interactions depends on the valency of the counterions and the surface charge density. In a clay system with mixed counterions, the competition between the mono- and divalent ions results in a large swelling when the fraction of surface charge compensated by monovalent counterions is greater than 30%. In the opposite situation involving mostly divalent counterions, a limited swelling is found and the attraction between the clay particles favors the formation of stacks incorporating a water layer of about 1.0 nm. These findings are consistent with experimental observations, giving insight into some mechanisms governing the stability of colloidal clay in salt-free or dilute solutions.

Effective charges and virial pressure of concentrated macroion solutions

The stability of colloidal suspensions is crucial in a wide variety of processes, including the fabrication of photonic materials and scaffolds for biological assemblies. The ionic strength of the electrolyte that suspends charged colloids is widely used to control the physical properties of colloidal suspensions. The extensively used two-body Derjaguin-Landau-Verwey-Overbeek (DLVO) approach allows for a quantitative analysis of the effective electrostatic forces between colloidal particles. DLVO relates the ionic double layers, which enclose the particles, to their effective electrostatic repulsion. Nevertheless, the double layer is distorted at high macroion volume fractions. Therefore, DLVO cannot describe the many-body effects that arise in concentrated suspensions. We show that this problem can be largely resolved by identifying effective point charges for the macroions using cell theory. This extrapolated point charge (EPC) method assigns effective point charges in a consistent way, taking into account the excluded volume of highly charged macroions at any concentration, and thereby naturally accounting for high volume fractions in both salt-free and added-salt conditions. We provide an analytical expression for the effective pair potential and validate the EPC method by comparing molecular dynamics simulations of macroions and monovalent microions that interact via Coulombic potentials to simulations of macroions interacting via the derived EPC effective potential. The simulations reproduce the macroion-macroion spatial correlation and the virial pressure obtained with the EPC model. Our findings provide a route to relate the physical properties such as pressure in systems of screened Coulomb particles to experimental measurements.

Investigation of Changes in Tetracycline Repressor Binding upon Mutations in the Tetracycline Operator

The tetracycline operon is an important gene network component, commonly used in synthetic biology applications because of its switch-like character. At the heart of this system is the highly specific interaction of the tet repressor protein (TetR) with its cognate DNA sequence (tetO). TetR binding on tetO practically stops expression of genes downstream of tetO by excluding RNA polymerase from binding the promoter and initiating transcription. Mutating the tetO sequence alters the strength of TetR-tetO binding and thus provides a tool to synthetic biologists to manipulate gene expression levels. We employ molecular dynamics (MD) simulations coupled with the free energy perturbation method to investigate the binding affinity of TetR to different tetO mutants. We also carry out in vivo tests in Escherichia coli for a series of promoters based on these mutants. We obtain reasonable agreement between experimental green fluorescent protein (GFP) repression levels and binding free energy differences computed from molecular simulations. In all cases, the wild-type tetO sequence yields the strongest TetR binding, which is observed both experimentally, in terms of GFP levels, and in simulation, in terms of free energy changes. Two of the four tetO mutants we tested yield relatively strong binding, whereas the other two mutants tend to be significantly weaker. The clustering and relative ranking of this subset of tetO mutants is generally consistent between our own experimental data, previous experiments with different systems and the free energy changes computed from our simulations. Overall, this work offers insights into an important synthetic biological system and demonstrates the potential, as well as limitations of molecular simulations to quantitatively explain biologically relevant behavior.

Nanofluids mediating surface forces

Fluids containing nanostructures, known as nanofluids, are increasingly found in a wide array of applications due to their unique physical properties as compared with their base fluids and larger colloidal suspensions. With several tuneable parameters such as the size, shape and surface chemistry of nanostructures, as well as numerous base fluids available, nanofluids also offer a new paradigm for mediating surface forces. Other properties such as local surface plasmon resonance and size dependent magnetism of nanostructures also present novel mechanisms for imparting tuneable surface interactions. However, our fundamental understanding, experimentally and theoretically, of how these parameters might affect surface forces remains incomplete. Here we review recent results on equilibrium and dynamic surface forces between macroscopic surfaces in nanofluids, highlighting the overriding trends in the correlation between the physical parameters that characterise nanofluids and the surface forces they mediate. We also discuss the challenges that confront existing surface force knowledge as a result of this new paradigm.

Particle charging and charge screening in nonpolar dispersions with nonionic surfactants

The electrostatic stabilization of colloidal dispersions is usually considered the domain of polar media only because of the high energetic cost associated with introducing electric charge in nonpolar environments. Nevertheless, some surfactants referred to as "charge control agents" are known to raise the conductivity of liquids with low electric permittivity and to mediate charge stabilization of nonpolar dispersions. Here we study an example of the particularly counterintuitive charging and electrostatic interaction of colloidal particles in a nonpolar solvent caused by nonionic surfactants. PMMA particles in hexane solutions of nonionic sorbitan oleate (Span) surfactants are found to exhibit a field-dependent electrophoretic mobility. Extrapolation to zero field strength yields evidence for large electrostatic surface potentials that decay with increasing surfactant concentration in a fashion reminiscent of electrostatic screening caused by salt in aqueous solutions. The amount of surface charge and screening ions in the nonpolar bulk is further characterized via measurements of the particles' pair interaction energy. The latter is obtained by liquid structure analysis of quasi-2-dimensional equilibrium particle configurations studied with digital video microscopy. In contrast to the behavior reported for systems with ionic surfactants, we observe particle charging and a screened Coulomb type interaction both above and below the surfactant's critical micelle concentration.

Adhesion forces controlled by chemical self-assembly and pH: application to robotic microhandling

Robotic microhandling is a promising way to assemble microcomponents in order to manufacture a new generation of hybrid microelectromechanical systems. However, at the scale of several micrometers, the adhesion phenomenon highly perturbs the micro-object release and positioning. This phenomenon is directly linked to both the object and the gripper surface chemical composition. We propose to control the adhesion by using a chemical self-assembled monolayer on both surfaces. Different types of chemical functionalization have been tested, and this paper focuses on the presentation of aminosilane-grafted 3-(ethoxydimethylsilyl)propylamine and (3-aminopropyl)triethoxysilane. We show that the liquid pH can be used to modify the adhesion and to switch from an attractive behavior to a repulsive behavior. The pH control can thus be used to increase the adhesion during handling and cancel the adhesion during release. Experiments have shown that the pH control is able to control the release of a micro-object. This paper shows the relevance of a new type of reliable submerged robotic microhandling principle, which is based on adjustment of the chemical properties of the liquid.

Electrostatic interactions of colloidal particles at vanishing ionic strength

Electrostatic interactions of colloidal particles are typically screened by mobile ions in the solvent. We measure the forces between isolated pairs of colloidal polymer microspheres as the density of bulk ions vanishes. The ionic strength is controlled by varying the concentration of surfactant (NaAOT) in a nonpolar solvent (hexadecane). While interactions are well-described by the familiar screened-Coulomb form at high surfactant concentrations, they are experimentally indistinguishable from bare Coulomb interactions at low surfactant concentration. Interactions are strongest just above the critical micelle concentration, where particles can obtain high surface potentials without significant screening, kappaa << 1. Exploiting the absence of significant charge renormalization, we are able to construct a simple thermodynamic model capturing the role of reverse micelles in charging the particle surface. These measurements provide novel access to electrostatic forces in the limit where the particle size is much less than the screening length, which is relevant not just to the nonpolar suspensions described here, but also to aqueous suspensions of nanoparticles.

Electrostatic interactions of colloidal particles in nonpolar solvents: role of surface chemistry and charge control agents

We study the electrostatic and hydrodynamic interactions of colloidal particles in nonpolar solvents. Using blinking optical tweezers, we can extract the screening length, kappa-1, the effective surface potential, |ezeta*|, and the hydrodynamic radius, ah, in a single measurement. We apply this technique to suspensions of polystyrene and poly(methyl methacrylate) particles in hexadecane with soluble charge control agents, aerosol sodium di-2-ethylhexylsulfosuccinate (AOT) and polyisobutylene succinimide (OLOA-1200). We find that the electrostatic interactions of these particles depend sensitively on surface composition as well as on the concentration and chemistry of the charge control agent.

Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces

We describe and implement a technique for extracting forces from the relaxation of an overdamped thermal system with normal modes. At sufficiently short time intervals, the evolution of a normal mode is well described by a one-dimensional Smoluchowski equation with constant drift velocity v, and diffusion coefficent D. By virtue of fluctuation dissipation, these transport coefficients are simply related to conservative forces, F, acting on the normal mode: F=kBTv/D. This relationship implicitly accounts for hydrodynamic interactions, requires no mechanical calibration, makes no assumptions about the form of conservative forces, and requires no prior knowledge of material properties. We apply this method to measure the electrostatic interactions of polymer microspheres suspended in nonpolar microemulsions.

Intermicellar interactions may induce anomalous size behavior in micelles carrying out bulky heads with multiple spatial arrangements

We report experimental and theoretical results on the concentration dependence of the micellar size of GM1 and GM1acetyl gangliosides, five-sugar-headed anionic glycolipids. Contrary to one of the mainstays of colloid science, that the aggregation number of amphiphile aggregates grows with concentration, an anomalous region is found at intermediate concentrations, where a sharp decrease of the aggregation number occurs. Experiments were performed by small-angle X-ray and neutron scattering (SAXS and SANS). Two models are discussed, reproducing the observed behavior of either GM1acetyl or GM1. The first one is a conventional picture of interacting micelles where a reduction in the molecular surface area, leading to an increase of the aggregate dimension, is paid to reduce intermicellar interactions: it foresees a monotonous increase of the aggregation number with concentration. The second one accounts for a conformational bistability of the bulky headgroups of GM1, modifying the amphiphilic molecular surface area and protrusion from the aggregate surface, and contributing to the inter- and intramicellar interaction balance. Energy minimization leads to a complex behavior of the aggregation number, which is consistent with the anomalous behavior of GM1.