Charge Regulation of Colloidal Particles: Theory and Simulations

Coupling of Charge Regulation and Geometry in Soft Ionizable Molecular Assemblies

The size, shape, and charge of structures, such as proteins and amphiphile assemblies, respond in an interconnected manner to solution ionic conditions. We analyze assemblies of an amphiphile (C16K2), with two ionizable amino acids [lysine (K)] coupled to a 16-carbon alkyl tail, via small-angle X-ray scattering (SAXS), nonlinear Poisson-Boltzmann theory (nl-PB), and hybrid Monte Carlo-molecular dynamics (MC-MD) simulations. SAXS revealed structural transitions from spherical micelles to cylindrical micelles to bilayers with increasing pH. By combining SAXS-determined structural information and nl-PB, we derived the molecular degree of ionization as a function of pH. The back-calculated titration curves matched the experimental data over an extended pH range, without adjustable parameters. Similarly, the SAXS data on the evolution of spherical micelle structure with ionic strength were combined with nl-PB and MC-MD to derive the bare and effective charges. MC-MD, which considered finite ion sizes, showed that bare and effective charges saturate quickly with increasing salt concentration. Furthermore, the calculated effective charges closely matched results from Zeta-potential measurements. The presented approach has advantages over customary methods for charge regulation, such as the Henderson-Hasselbalch (HH) or Hill models, where molecular ionization/deionization in assemblies is described by effective pKs that are distinct from the pK for isolated molecules. However, these models lack a physical explanation for these pK shifts. By contrast, our approach of combining structural details with an electrostatic model and simulations provides a more intuitive understanding of structure-charge coupling and a framework for understanding charge regulation in many synthetic and biological systems.

Charge regulation of nanoparticles in the presence of multivalent electrolytes

We explore the charge regulation (CR) of spherical nanoparticles immersed in an asymmetric electrolyte of a specified pH. Using a recently developed reactive canonical Monte Carlo (MC) simulation method, titration isotherms are obtained for suspensions containing monovalent, divalent, and trivalent coions. A theory based on the modified Poisson-Boltzmann approximation, which incorporates the electrostatic ion solvation free energy and discrete surface charge effects, is used to compare with the simulation results. A remarkably good agreement is found without any fitting parameters, both for the ion distributions and titration curves, suggesting that ionic correlations between coions and hydronium ions at the nanoparticle surface play only a minor role in determining the association equilibrium between hydroniums and the functional sites on the nanoparticle surface. On the other hand, if suspension contains multivalent counterions, we observe a large deviation between theory and simulations, showing that the electrostatic correlations between counterions and hydronium ions at the nanoparticle surface are very significant and must be properly taken into account to correctly describe CR for such solutions.

Tracking surface charge dynamics on single nanoparticles

Surface charges play a fundamental role in physics and chemistry, in particular in shaping the catalytic properties of nanomaterials. However, tracking nanoscale surface charge dynamics remains challenging due to the involved length and time scales. Here, we demonstrate time-resolved access to the nanoscale charge dynamics on dielectric nanoparticles using reaction nanoscopy. We present a four-dimensional visualization of the spatiotemporal evolution of the charge density on individual SiO2 nanoparticles under strong-field irradiation with femtosecond-nanometer resolution. The initially localized surface charges exhibit a biexponential redistribution over time. Our findings reveal the influence of surface charges on surface molecular bonding through quantum dynamical simulations. We performed semi-classical simulations to uncover the roles of diffusion and charge loss in the surface charge redistribution process. Understanding nanoscale surface charge dynamics and its influence on chemical bonding on a single-nanoparticle level unlocks an increased ability to address global needs in renewable energy and advanced health care.

Disparate External Electric Field Effect on the Adsorption and Shear Behavior of Monovalent and Trivalent Ions in Electrolyte Solution

Reducing friction is of great interest, and an external potential applied to the friction pair can regulate lubricity. Electrochemical atomic force microscopy (EC-AFM) is used to study the tribological and adsorption behavior of monovalent and trivalent ionic solutions between charged surfaces. An opposite trend of coefficient of friction (COF) and normal force that varies with the applied electric potential is witnessed. Direct force measurements and theoretical models have disclosed that, for the NaCl solution, the negative electric field reduces the COF by increasing cation adsorption. As for LaCl3 solution, the positive electric field promotes the primary adsorption of anions on HOPG, resulting in the disappearance of the attractive ion-ion correlation between the trivalent ions, thereby reducing the COF. The shear behavior of adsorbed ions in electrolyte solution is sensitive to their valence, because of their different surface force contribution. The study further provides a framework to optimize the design of hydration lubrication.

On the Validity of Constant pH Simulations

Constant pH (cpH) simulations are now a standard tool for investigating charge regulation in coarse-grained models of polyelectrolytes and colloidal systems. Originally developed for studying solutions with implicit ions, extending this method to systems with explicit ions or solvents presents several challenges. Ensuring proper charge neutrality within the simulation cell requires performing titration moves in sync with the insertion or deletion of ions, a crucial aspect often overlooked in the literature. Contrary to the prevailing views, cpH simulations are inherently grand-canonical, meaning that the controlled pH is that of the reservoir. The presence of the Donnan potential between the implicit reservoir and the simulation cell introduces significant differences between titration curves calculated for open and closed systems; the pH of an isolated (closed) system is different from the pH of the reservoir for the same protonation state of the polyelectrolyte. To underscore this point, in this paper, we will compare the titration curves calculated using the usual cpH algorithm with those from the exact canonical simulation algorithm. In the latter case, titration moves adhere to the correct detailed balance condition, and pH is calculated using the recently introduced surface Widom insertion algorithm. Our findings reveal a very significant difference between the titration isotherms obtained using the standard cpH algorithm and the canonical titration algorithm, emphasizing the importance of using the correct simulation approach when studying charge regulation of polyelectrolytes, proteins, and colloidal particles.

Ion Concentration-Dependent Surface Charge Density Inside a Nanopore

Surface charges shape the electrical double layer (EDL) structure at solid-liquid interfaces, critically influencing the performance of energy storage and micro/nanofluidic devices. However, accurately measuring surface charge density in nanoconfined spaces continues to be a challenge. Here, we introduce a methodology via solid-state nanopores that can investigate the dependence of surface charge density on salt concentrations and nanopore diameters. Measurements, complemented by a theoretical model, reveal that the surface charge density decreases as both the salt concentration in bulk solutions and the nanopore sizes are reduced. Notably, when the salt concentration in the bulk solution drops below 10-3 M, protons dominate ion conductance in a nanopore, resulting in a constant surface charge density. This study introduces an effective approach to surface charge characterization and may serve in the design of electrokinetically driven nanofluidic systems.

Induced phase transformation in ionizable colloidal nanoparticles

Acid-base equilibria directly influence the functionality and behavior of particles in a system. Due to the ionizing effects of acid-base functional groups, particles will undergo charge exchange. The degree of ionization and their intermolecular and electrostatic interactions are controlled by varying the pH and salt concentration of the solution in a system. Although the pH can be tuned in experiments, it is hard to model this effect using simulations or theoretical approaches. This is due to the difficulty in treating charge regulation and capturing the cooperative effects in a colloidal suspension with Coulombic interaction. In this work, we analyze a suspension of ionizable colloidal particles via molecular dynamics (MD) simulations, along with Monte Carlo simulations for charge regulation (MC-CR) and derive a phase diagram of the system as a function of pH. It is observed that as pH increases, particles functionalized with acid groups change their arrangement from face-centered cubic (FCC) packing to a disordered state. We attribute these transitions to an increase in the degree of charge polydispersity arising from an increase in pH. Our work shows that charge regulation leads to amorphous solids in colloids when the mean nanoparticle charge is sufficiently high.

Canonical titration simulations

We present a Monte Carlo approach for performing titration simulations in the canonical ensemble. The standard constant pH (cpH) simulation methods are intrinsically grand canonical, allowing us to study the protonation state of molecules only as a function of pH in the reservoir. Due to the Donnan potential between a system and an (implicit) reservoir of a semi-grand canonical simulation, the pH of the reservoir can be significantly different from that of an isolated system, for an identical protonation state. The new titration method avoids this difficulty by using the canonical reactive Monte Carlo algorithm to calculate the protonation state of macromolecules as a function of the total number of protons present inside the simulation cell. The pH of an equilibrated system is then calculated using a new surface insertion Widom algorithm, which bypasses the difficulties associated with the bulk Widom particle insertion for intermediate and high pH values. To properly treat the long range Coulomb force, we use the Ewald summation method, showing the importance of the Bethe potential for calculating the pH of canonical systems.

Titration in Canonical and Grand-Canonical Ensembles

We discuss problems associated with the notion of pH in heterogeneous systems. For homogeneous systems, standardization protocols lead to a well-defined quantity, which, although different from Sørensen's original idea of pH, is well reproducible and has become accepted as the measure of the "hydrogen potential". On the other hand, for heterogeneous systems, pH defined in terms of the chemical part of the electrochemical activity is thermodynamically inconsistent and runs afoul of the Gibbs-Guggenheim principle that forbids splitting of the electrochemical potential into separate chemical and electrostatic parts, since only the sum of two has any thermodynamic meaning. The problem is particularly relevant for modern simulation methods which involve charge regulation of proteins, polyelectrolytes, nanoparticles, colloidal suspensions, and so forth. In this paper, we show that titration isotherms calculated using semigrand canonical simulations can be very different from the ones obtained using canonical reactive Monte Carlo simulations.

Charge fluctuations in charge-regulated systems: dependence on statistical ensemble

We investigate charge regulation of nanoparticles in concentrated suspensions, focusing on the effect of different statistical ensembles. We find that the choice of ensemble does not affect the mean charge of nanoparticles, but significantly alters the magnitude of its fluctuation. Specifically, we compared the behaviors of colloidal charge fluctuations in the semi-grand canonical and canonical ensembles and identified significant differences between the two. The choice of ensemble-whether the system is isolated or is in contact with a reservoir of acid and salt-will, therefore, affect the Kirkwood-Shumaker fluctuation-induced force inside concentrated suspensions. Our results emphasize the importance of selecting an appropriate ensemble that accurately reflects the experimental conditions when studying fluctuation-induced forces between polyelectrolytes, proteins, and colloidal particles in concentrated suspensions.

Theory of Charge Regulation of Colloidal Particles in Electrolyte Solutions

We present a theory that enables us to (i) calculate the effective surface charge of colloidal particles and (ii) efficiently obtain titration curves for different salt concentrations. The theory accounts for the shift of pH of solution due to the presence of 1:1 electrolyte. It also accounts self-consistently for the electrostatic potential produced by the deprotonated surface groups. To examine the accuracy of the theory, we have performed extensive reactive Monte Carlo simulations, which show excellent agreement between theory and simulations without any adjustable parameters.

Structural correlations in highly asymmetric binary charged colloidal mixtures

We explore structural correlations of strongly asymmetric mixtures of binary charged colloids within the primitive model of electrolytes considering large charge and size ratios of 10 and higher. Using computer simulations with explicit microions, we obtain the partial pair correlation functions between the like-charged colloidal macroions. Interestingly the big-small correlation peak amplitude is smaller than that of the big-big and small-small macroion correlation peaks, which is unfamiliar for additive repulsive interactions. Extracting optimal effective microion-averaged pair interactions between the macroions, we find that on top of non-additive Yukawa-like repulsions an additional shifted Gaussian attractive potential between the small macroions is needed to accurately reproduce their correct pair correlations. For small Coulomb couplings, the behavior is reproduced in a coarse-grained theory with microion-averaged effective interactions between the macroions. However, the accuracy of the theory deteriorates with increasing Coulomb coupling. We emphasize the relevance of entropic interactions exerted by the microions on the macroions. Our results are experimentally verifiable in binary mixtures of micron-sized colloids and like-charge nanoparticles.

Reactive Monte Carlo simulations for charge regulation of colloidal particles

We use a reactive Monte Carlo simulation method and the primitive model of electrolyte to study acid-base equilibrium that controls charge regulation in colloidal systems. The simulations are performed in a semi-grand canonical ensemble in which colloidal suspension is in contact with a reservoir of salt and strong acid. The interior of colloidal particles is modeled as a low dielectric medium, different from the surrounding water. The effective colloidal charge is calculated for different numbers of surface acidic groups, pH, salt concentrations, and types of electrolyte. In the case of potassium chloride, the titration curves are compared with the experimental measurements obtained using potentiometric titration. A good agreement is found between simulations and experiments. In the case of lithium chloride, the specific ionic adsorption is taken into account through the partial dehydration of lithium ion.

Dendritic polyelectrolytes with monovalent and divalent counterions: the charge regulation effect and counterion release

The charge regulation and the release of counterions are extremely important and substantial in determining the charge state of polyelectrolytes and the interaction between polyelectrolytes and proteins. Going beyond monovalent to multivalent cations, it is well-known that the effects of ions are qualitatively different. Therefore, the well-accepted descriptions of the charge regulation and the counterion release based on monovalent ions do not immediately apply to systems with multivalent ions. Here, we study the key structural and electrostatic features of charged dendrimers at hand of the pharmaceutically important dendritic polyglycerol sulfate (dPGS) macromolecule equilibrated with monovalent and divalent salts by molecular dynamics (MD) simulations. Following a simple but accurate scheme to determine its effective radius, the counterion condensed layer of the dPGS is determined with high accuracy and we observe the sequential replacement of condensed monovalent cations (MCs) to divalent cations (DCs) rendering a smaller dPGS effective charge versus the DC concentration. We resolve and track the release of counterions on the dPGS along its binding pathway with the plasma protein Human Serum Albumin (HSA). We find that the release of MCs remains favorable for the complexation leading to a considerable amount of release entropy as the driving force for complexation. The release of DCs only occurs above a certain DC concentration with a comparably smaller number of released ions than MCs. Its contribution to the binding free energy is small indicating a subtle cancellation between the entropy gain in releasing DCs and the enthalpy penalty from dissociating DCs from the dendrimer.

Charge regulated solid-liquid interfaces interacting on the nanoscale: Benchmarking of a generalized speciation code (SINFONIA)

Surface chemistry of mineral phases in aqueous environments generates the electrostatic forces involved in particle-particle interactions. However, few models directly take into account the influence of surface speciation and changes in solution speciation when the diffuse layer potential profiles of approaching particles overlap and affect each other. These electrostatic interactions can be quantified, ideally, through charge regulation, considering solution and surface speciation changes upon particle approach by coupling state-of-the-art surface complexation models for the two particle surfaces with a Poisson-Boltzmann type distribution of electrostatic potential and ions in the inter-particle space. These models greatly improve the accuracy of inter-particle force calculations at small inter-particle separations compared to constant charge and constant potential approaches. This work aims at advancing charge regulation calculations by including full chemical speciation and advanced surface complexation models (Basic Stern-, three-, or four plane models and charge distribution concepts), for cases of similar and dissimilar surfaces involving the numerical solution of the Poisson-Boltzmann equation for arbitrary electrolytes. The concept was implemented as a Python-based code and in COMSOL. The flexibility and precision of both, concept and implementations are demonstrated in several benchmark calculations testing the new codes against published results or simulations using established speciation codes, including aqueous speciation, surface complexation and various interaction force examples. Due to the flexibility in terms of aqueous chemistry and surface complexation models for various geometries, a large variety of potential applications can be tackled with the developed codes including industrial, biological, and environmental systems, from colloidal suspensions to gas bubbles, emulsions, slurries like cement paste, as well as new possibilities to assess the chemistry in nano-confined systems.

Charge Regulation Effects in Nanoparticle Self-Assembly

Nanoparticles in solution acquire charge through the dissociation or association of surface groups. Thus, a proper description of their electrostatic interactions requires the use of charge-regulating boundary conditions rather than the commonly employed constant-charge approximation. We implement a hybrid Monte Carlo/molecular dynamics scheme that dynamically adjusts the charges of individual surface groups of objects while evolving their trajectories. Charge regulation effects are shown to qualitatively change self-assembled structures due to global charge redistribution, stabilizing asymmetric constructs. We delineate under which conditions the conventional constant-charge approximation may be employed and clarify the interplay between charge regulation and dielectric polarization.

Lattice-gas model of a charge regulated planar surface

In this work, we consider a lattice-gas model of charge regulation with electrostatic interactions within the Debye-Hückel level of approximation. In addition to long-range electrostatic interactions, the model incorporates the nearest-neighbor interactions for representing non-electrostatic forces between adsorbed ions. The Frumkin-Fowler-Guggenheim isotherm obtained from the mean-field analysis accurately reproduces the simulation data points.

Charge regulation of colloidal particles in aqueous solutions

We study the charge regulation of colloidal particles inside aqueous electrolyte solutions. To stabilize a colloidal suspension against precipitation, colloidal particles are synthesized with either acidic or basic groups on their surface. On contact with water, these surface groups undergo proton transfer reactions, resulting in colloidal surface charge. The charge is determined by the condition of local chemical equilibrium between hydronium ions inside the solution and at the colloidal surface. We use a model of Baxter sticky spheres to explicitly calculate the equilibrium dissociation constants and to construct a theory which is able to quantitatively predict the effective charge of colloidal particles with either acidic or basic surface groups. The predictions of the theory for the model are found to be in excellent agreement with the results of Monte Carlo simulations. This theory is further extended to treat colloidal particles with a mixture of both acidic and basic surface groups.

Modeling Surface Charge Regulation of Colloidal Particles in Aqueous Solutions

Colloidal particles are mostly charged in an aqueous solution because of the protonation or deprotonation of ionizable groups on the surface. The surface charge density reflects a complex interplay of ion distributions within the electric double layer and the surface reaction equilibrium. In this work, we present a coarse-grained model to describe the charge regulation of various colloidal systems by an explicit consideration of the inhomogeneous ion distributions and surface reactions. With the primitive model for aqueous solutions and equilibrium constants for surface reactions as the inputs, the theoretical model is able to make quantitative predictions of the surface-charge densities and zeta potentials for diverse colloidal particles over a wide range of pH and ionic conditions. By accounting for the ionic size effects and electrostatic correlations, our model is applicable to systems with multivalent ions that exhibit charge inversion and provides a faithful description of the interfacial properties without evoking the empirical Stern capacitance or specific ion adsorptions.

Correlation function approach for diffusion in confined geometries

This paper describes a formalism for extracting spatially varying transport coefficients from simulations of a molecular fluid in a nanochannel. This approach is applied to self-diffusion of a Lennard-Jones fluid confined between two parallel surfaces. A numerical grid is laid over the domain confining the fluid, and fluid properties are projected onto the grid cells. The time correlation functions between properties in different grid cells are calculated and can be used as the basis for a fitting procedure for extracting spatially varying diffusion coefficients from the simulation. Results for the Lennard-Jones system show that transport behavior varies sharply near the liquid-solid boundary and that the changes depend on the details of the liquid-solid interaction. A quantitative difference between the reduced and detailed models is discussed. It is found that the difference could be associated with assumptions about the form of the transport equations at molecular scales in lieu of problems with the method itself. The study suggests that this approach to fitting molecular simulations to continuum equations may guide the development of appropriate coarse-grained equations to model transport phenomena at nanometer scales.

Critical behavior of charge-regulated macro-ions

Based on a collective description of electrolytes composed of charge-regulated macro-ions and simple salt ions, we analyze their equilibrium charge state in the bulk and their behavior in the vicinity of an external electrified surface. The mean-field formulation of mobile macro-ions in an electrolyte bathing solution is extended to include interactions between association and dissociation sites. We demonstrate that above a critical concentration of salt, similar to the critical micelle concentration, a non-trivial distribution of charge states sets in. Such a charge state can eventually lead to a liquid-liquid phase separation based on charge regulation.

Interfacial solvation can explain attraction between like-charged objects in aqueous solution

Over the past few decades, the experimental literature has consistently reported observations of attraction between like-charged colloidal particles and macromolecules in aqueous solution. Examples include nucleic acids and colloidal particles in the bulk solution and under confinement, and biological liquid-liquid phase separation. This observation is at odds with the intuitive expectation of an interparticle repulsion that decays monotonically with distance. Although attraction between like-charged particles can be rationalized theoretically in the strong-coupling regime, e.g., in the presence of multivalent counterions, recurring accounts of long-range attraction in aqueous solution containing monovalent ions at low ionic strength have posed an open conundrum. Here, we show that the behavior of molecular water at an interface-traditionally disregarded in the continuum electrostatics picture-provides a mechanism to explain the attraction between like-charged objects in a broad spectrum of experiments. This basic principle will have important ramifications in the ongoing quest to better understand intermolecular interactions in solution.

Osmotic stress and pore nucleation in charged biological nanoshells and capsids

A model system is proposed to investigate the chemical equilibrium and mechanical stability of biological spherical-like nanoshells in contact with an aqueous solution with added dissociated electrolyte of a given concentration. The ionic chemical equilibrium across the permeable shell is investigated in the framework of an accurate Density Functional Theory (DFT) that incorporates electrostatic and hardcore correlations beyond the traditional mean-field (e.g., Poisson-Boltzmann) limit. The accuracy of the theory is tested by a direct comparison with Monte Carlo (MC) simulations. A simple analytical expression is then deduced which clearly highlights the entropic, electrostatic, and self-energy contributions to the osmotic stress over the shell in terms of the calculated ionic profiles. By invoking a continuum mean-field elastic approach to account for the shell surface stress upon osmotic stretching, the mechanical equilibrium properties of the shell under a wide variety of ionic strengths and surface charges are investigated. The model is further coupled to a continuum mechanical approach similar in structure to a Classical Nucleation Theory (CNT) to address the question of mechanical stability of the shells against a pore nucleation. This allows us to construct a phase diagram which delimits the mechanical stability of capsids for different ionic strengths and shell surface charges.