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

Highlights

• Surface electrostatic interactions are considered via charge regulation (CR)

• CR allows precise assessment of inter particle forces on the nanoscale

• CR allows precise assessment of surface and aqueous speciation in nanopores

• A multipurpose CR code (SINFONIA) is implemented and benchmarked

• The python implementation of SINFONIA is for free and open source available

Abstract

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.

Introduction

Surface chemistry and colloidal stability are topics of broad interest concerning technological applications and environmental studies. Several fields of engineering rely on understanding the behaviour of colloidal systems for future development and societal needs. To name a few examples: understanding the side effects of nano-silica in pharmaceutics and cosmetics [1], applications in remediation techniques for recovering nano-titania from wastewaters [2], potential colloid-mediated radionuclide migration from radioactive waste disposal areas [3], or workability of cementitious materials, with or without addition of superplasticizers in cement pastes for concrete technologies [[4], [5], [6]]. Furthermore, material and environmental sciences require comprehensive knowledge on the chemical speciation and the behaviour of solid-liquid interfaces under nano-confined conditions. For instance, such knowledge helps in further understanding the governing factors in interfacial phenomena such as aggregation of particles or bacterial attachment to particles in aquatic systems [7]. Other examples include processes involved in trace element mobility and fate in aquatic/soil systems [8], as well as the potential anthropogenic impacts of nanoparticles in natural ecosystems [9]. All these examples ultimately require the understanding of chemical properties and specific surface behaviour (i.e., charges, potentials and forces) of the involved mineral phases such as silica (e.g., [10,11]), mica (e.g., [12]), metal oxides (e.g., [13]), and calcite (e.g., [14,15]), or cement hydration products such as CSH and ettringite (e.g., [[16], [17], [18]]). This understanding can be achieved by direct measurements of surface properties (i.e., electrophoretic mobility, acid-base titrations, force-distance measurements) and through appropriate, more or less complex models.

Coupling electrostatic surface complexation models (SCMs, including the corresponding electric double layer – EDL description) and van der Waals (vdW) forces, in order to account for charge regulation effects as addition to the classical DLVO theory [[19], [20], [21]], which originally involves assumptions of constant potential, results in a powerful tool to account for the effects of the individual surface chemistry (i.e., adsorption equilibria) on colloidal stability. Vice versa, this approach allows to assess the effect of close proximity of surfaces on surface chemistry, where charge regulation becomes most relevant. This is a recurring issue for systems under nano-confinement or involving narrow pores with overlapping EDLs. Besides the surface chemistry, aqueous chemistry in narrow pores is potentially affected by electrostatic effects. Our model concept, often addressed as charge regulation (CR) or full regulation, implies that:

• The electrostatic forces developing upon approach of the diffuse layers are self-consistent with respect to the surface chemistries, such that surface chemistry can be independently studied and parametrized.

• When the diffuse layers of two approaching surfaces overlap, their surfaces need to be allowed to adjust surface speciation, interfacial charges and potentials in accordance with the distance between the surfaces. As a result, the individual surface chemistries affect each other.

The aim of this study is to present and validate a general-purpose code capable of modelling the speciation of solid-liquid interfaces interacting down to nanoscale separation and the surrounding aqueous solution on the Poisson-Boltzmann (PB) level. Furthermore, this study will help to overcome restrictions of previous codes (i.e., restricted to diffuse layer – DLM, Basic Stern – BSM, or a triple layer model – TLM, or to monovalent electrolytes) and their non-availability in order to broaden the applicability and facilitate future applications.

Compared to previous implementations (e.g., [22]), our work particularly aims at advancing charge regulation calculations for overlapping EDLs from opposing surfaces by including full scale chemical speciation and advanced surface complexation models (Basic Stern-, three-, or four layer models), considering interactions in both, symmetric and asymmetric systems (interactions between equal and unequal surfaces), and a numerical solution for the Poisson Boltzmann equation for arbitrary electrolytes. Ionic strength may be fixed (constant ionic medium approach) or adjusted, depending on the aqueous speciation (e.g., often required for systems of low salt content). The possibility to address such complex cases is particularly beneficial for systems such as cement suspensions, where particle interactions are of key importance to predict rheological properties.

Activity corrections are so far based on the Davies equation [23] but could easily be adapted to other models. We implemented the concept in two ways, (1) based on the open source programming language Python, in order to ensure general availability and accessibility, and (2) within the commercial software COMSOL, in order to enable coupling of SCMs and charge regulation problems to systems readily described with COMSOL (e.g., [[24], [25], [26], [27], [28]]). The implementations are adapted to simulate force-distance curves for the common simple particle interaction geometries: sphere-sphere and sphere-plane, where the latter corresponds to crossed cylinders of equal radii as usually applied in surface force apparatus approaches (e.g., [29]). Some shortcomings of the present concept include the difficulty to account for the Stern layer thickness for quantifying precise surface separations, solvent effects, which may play a dominant role in the local structuring of the charged interface for very short separation distances (ca. < 1 nm), as well as the lack of finite ion sizes in the diffuse part of the double layer and ion correlations. These are specific shortcomings related to the application of PB theory (e.g., [29,30]). We note that alternative methods are available which circumvent these shortcomings for specific systems and may also provide information on inter-particle interactions, like density functional theory - DFT, molecular dynamics - MD or Monte Carlo - MC calculations, at different scales and resolutions [31,32]. However, these are to our knowledge not currently directly applicable in the context of general aqueous and surface speciation calculations and cannot yet readily be implemented in the approach proposed in this work, which in turn builds on earlier work [22,[33], [34], [35], [36]].

To validate the two implementations, we present a series of benchmarking exercises:

• Results of the developed code are compared to a published DLM-based CR code simulating force-distance data [37,38] for surfaces involving simple surface chemical reactions (i.e., protonation and deprotonation here for silica and hydrous ferric oxide-HFO, also known as ferrihydrite, with and without ion sorption) in simple electrolytes. The main goal of this exercise is to verify the accurate solution of the PB-equation for overlapping diffuse layers, i.e., the differential equation-solver for this specific boundary value problem.

• We benchmark the codes against PHREEQC [39], a wide-spread geochemical speciation software including surface complexation, on a surface- and aqueous speciation problem in an electrolyte solution (NaCl) involving specifically adsorbing ions (Eu³⁺) and quartz. This benchmark includes the option to adjust ionic strength and activity coefficients according to aqueous speciation and demonstrates that such systems can be solved robustly with high precision. Another purpose of this exercise is to demonstrate the precision of the implementations with respect to the solution of advanced electrostatic SCMs for surfaces that do not interact, including the PB-equation for arbitrary electrolytes (H₂O, Na⁺, Cl⁻, Eu³⁺, and derived species). This example also addresses difficulties related to multidentate sorption models [40].

• We extend the code to a four layer model (FLM), applicable for the simulation of interactions of air bubbles in water, and for predicting colloidal stability of approaching asymmetric surfaces from defined systems (i.e., goethite or silica in contact with TiO₂). With this exercise we describe published force distance data (calculated or measured) using available surface complexation models, in order to demonstrate the wide field of potential applications of the codes.