Theory of Charge Regulation of Colloidal Particles in Electrolyte Solutions

CO2-induced drastic decharging of dielectric surfaces in aqueous suspensions

We study the influence of airborne CO2 on the charge state of carboxylate stabilized polymer latex particles suspended in aqueous electrolytes. We combine conductometric experiments interpreted in terms of Hessinger's conductivity model with Poisson-Boltzmann cell (PBC) model calculations with charge regulation boundary conditions. Without CO2, a minority of the weakly acidic surface groups are dissociated and only a fraction of the total number of counter-ions actually contribute to conductivity. The remaining counter-ions exchange freely with added other ions like Na+, K+ or Cs+. From the PBC-calculations we infer a corresponding pKa of 4.26 as well as a renormalized charge in reasonably good agreement with the number of freely mobile counter-ions. Equilibration of salt- and CO2-free suspensions against ambient air leads to a drastic de-charging, which exceeds by far the expected effects of to dissolved CO2 and its dissociation products. Further, no counter-ion-exchange is observed. To reproduce the experimental findings, we have to assume an effective pKa of 6.48. This direct influence of CO2 on the state of surface group dissociation explains our recent finding of a CO2-induced decrease of the ζ-potential and supports the suggestion of an additional charge regulation caused by molecular CO2. Given the importance of charged surfaces in contact with aqueous electrolytes, we anticipate that our observations bear substantial theoretical challenges and important implications for applications ranging from desalination to bio-membranes.

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.

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.

On the nature of screening in charge-regulated macroion solutions

We present a derivation of the screening length for a solution containing a charge-regulated macroion, e.g. protein, with its counterions. We show that it can be obtained directly from the second derivatives of the total free energy by taking recourse to the "uncertainty relation" of the Legendre transform, which connects the Hessians or the local curvatures of the free energy as a function of density and its Legendre transform, i.e., osmotic pressure, as a function of chemical potentials. Based on the Fowler-Guggenheim-Frumkin model of charge regulation, we then analyze the "screening resonance" and the "overscreening" of the screening properties of the charge-regulated macroion solution.

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.

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.

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.

A new method for reactive constant pH simulations

We present a simulation method that allows us to calculate the titration curves for systems undergoing protonation/deprotonation reactions-such as charged colloidal suspensions with acidic/basic surface groups, polyelectrolytes, polyampholytes, and proteins. The new approach allows us to simultaneously obtain titration curves both for systems in contact with salt and acid reservoir (semi-grand canonical ensemble) and for isolated suspensions (canonical ensemble). To treat the electrostatic interactions, we present a new method based on Ewald summation-which accounts for the existence of both Bethe and Donnan potentials within the simulation cell. We show that the Donnan potential dramatically affects the pH of a suspension. Counterintuitively, we find that in suspensions with a large volume fraction of nanoparticles and low ionic strength, the number of deprotonated groups can be 100% larger in an isolated system, compared to a system connected to a reservoir by a semi-permeable membrane-both systems being at exactly the same pH.

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.