Should we return to probability distributions to represent interfacial reactivity?

arising from A.G. Ilgen et al. Nature Communications https://doi.org/10.1038/s41467-024-49598-y (2024)

The Ilgen et al. Perspective suggests that using probability distribution functions (PDFs) to represent surface properties in adsorption models is a novel idea. In fact, PDFs were used to describe the heterogeneous nature of metal oxide surfaces in the very first adsorption models for the oxide-water interface. In addition, PDFs have long been used to evaluate the effects of parameter uncertainty in geochemical and reaction-transport models. Finally, much of the discussion on Surface Complexation Models (SCMs) suggests some misunderstanding of these models, their structure, and their application.

Surface heterogeneity and probabilistic distributions for adsorption

The main thesis of the Perspective is to propose the use of probability distributions to represent the properties of oxide-water interfaces and suggest that this is a new concept that will lead to a conceptual shift in continuum-scale modeling. In fact, the very first continuum-scale models for adsorption at oxide-water interfaces used probability distributions to describe the heterogeneous nature of oxide surfaces. These models were developed from those for oxide-gas interfaces which recognized that oxide surfaces are energetically heterogeneous. Davis et al. (1978)¹ and Benjamin and Leckie (1981)² were the first to report that the adsorption of Me²⁺ ions onto ferrihydrite from aqueous solution could not be described by ion adsorption onto homogeneous solid surfaces but only by assuming a large dispersion of site affinities.

The first SCMs were formulated with the consideration of surface heterogeneity and the use of probability distributions to describe this heterogeneity. Much of this early development is included in review articles/chapters by van Riemsdijk and Koopal (1992)³, Koopal (1996)⁴, and Rudzinski et al. (1996)⁵. The Perspective states “continuum-scale models, such as SCM and RTMs, are not designed to incorporate spatially differing reactivities of surfaces.” In fact, Koopal et al. (1989)⁶ developed electric double layer (EDL) theory for three different types of surface spatial heterogeneity, including random, isolated patches, and interacting patches. Rudzinski et al. (1999)⁷ fitted data for electrolyte ion adsorption at the silica/NaCl aqueous solution interface with a single-site triple layer model (TLM) with heterogeneity parameters proportional to the variance of a Gaussian-like, fully symmetrical function describing the dispersion of adsorption energies on different complexes across the surface. The Gaussian-like function to represent the energetic heterogeneity of silica surface oxygens was based on studies of argon adsorption to silica. Charmas et al. (2002)⁸ used this model to investigate Cd adsorption at the aqueous silica/NaClO₄ interface, fitting data from both surface charge (titration) and electrophoretic mobility measurements, and monitoring both  electrolyte cation and Cd ion adsorption at different pH values.

Concurrently with the use of probability distributions to describe heterogeneity, the option to describe this heterogeneity by considering distinctly different reactive groups, each with its specific affinity developed. This approach (explicit, discrete heterogeneity with a priori estimated proton binding constants) was used by Hiemstra et al. (1989)⁹ in the development of the MUSIC model.

Probability distributions used to describe parameter uncertainty

Normal probability distributions have also been used to describe uncertainty (i.e., error bars) in aqueous solution measurements, aqueous speciation equilibrium constants, and adsorption equilibrium constants. Monte-Carlo generated uncertainties in these input parameters have been propagated through geochemical and reactive-transport calculations (e.g., Criscenti et al., 1996, 2002)^10,11 to determine the range of potential downstream outcomes for contaminant migration from uranium mill tailings, hazardous and municipal landfills, and potential nuclear waste repository sites. Developing new approaches to address this uncertainty for applied applications has continued to the present day. Most recently, a new community data-driven database and workflow has been developed for raw sorption data and SCM database development (Zavarin et al., 2022)¹². This database was used to simulate potentiometric titrations of mineral surfaces (Han et al., 2023)¹³. Error bars for the raw data were propagated through the workflow and reported with the fitted pK_a1 and pK_a2 for each individual dataset as well as for the representative average pK_a values for the community database. Another approach to calculate adsorption at oxide-water interfaces has been to use a hybrid random forest (RF) machine-learning (ML) approach and bypass SCM modeling constraints entirely (Chang et al., 2023)¹⁴. Again, error propagation based on means and standard deviations (for a normal bell curve of probability) is included in this approach.

Surface complexation models

There are numerous statements in the Perspective about SCMs that suggest some misunderstanding of SCM theory.

The Perspective states “…neither ΔG_chem nor ΔG_coul can be considered constants in any given interfacial system because of the variability of surface structures that define local surface charge or the reactivity of isolated surface groups, which should lead to variability in the surface potential across the same surface caused by intrinsic surface heterogeneity.” The chemical adsorption energy (ΔG_chem) and Coulomb static energy (ΔG_coul) are each typically described as a constant, not a probability distribution (i.e., a distribution of values), for a specific P, T, etc. However, these energies do vary as a function of surface site types, specific surface coverages, and different solution compositions and pH values.

The Perspective states “…conventional SCMs describe surface properties and reactivities with a single surface acidity constant and surface complexation constant for a given surface and adsorbate (the more advanced SCMs may go as far as to incorporate two- or three-site models with distinct acidity and/or complexation constants).” A “conventional SCM” is not defined in the Perspective; here I will define it as an SCM that has been in use for over thirty years. Single-site and multisite SCM theories developed in parallel. One SCM that is widely used and incorporated into geochemical codes like PHREEQC (Parkhurst and Appelo, 2013)¹⁵ is the Double Diffuse Layer Model (DDLM) developed by Dzombak and Morel (1990)¹⁶ for ion adsorption to hydrous ferric oxide (HFO). The DDLM requires two site types to describe cation sorption on HFO. The density for one site-type is the total reactive sites available for protons, cations, and anions. The second site-type has a much lower density and is a high-affinity binding site attributed to surface defects and impurities. Another SCM, the CD-MUSIC (van Riemsdijk and Hiemstra (1996)¹⁷ models which allows for many different types of surface sites is also sufficiently established to be incorporated into readily available geochemical codes.

The Perspective states “Sr²⁺ then absorbs as the [SrCl]⁺ ion pair, which is subject to reduced lateral Coulomb repulsion… SCMs should take this effect into account but currently do not…. The surface-promoted ion-pairing processes need to be incorporated into new descriptions of Stern layer for ion speciation.” Present-day SCMs do account for surface-promoted ion-pairing at the interface. James and Healy¹⁸ showed that the aqueous metal hydroxide species (MOH⁺) should adsorb preferentially to the bare metal ion (M²⁺) based on Born solvation theory. Since then, numerous studies have proposed the formation of surface complexes that involve both the metal cation and either the hydroxide or electrolyte anion. A comprehensive review of this literature before 2000 is found in Criscenti and Sverjensky^19,20. Adsorption isotherm shapes and surface coverages vary as a function of surface complex type and the associated electrostatic term²⁰. SCMs can account for different types of surface complexes. The Coulombic term in the equilibrium expressions changes according to the reaction. For example, in the extended TLM, equilibrium constants that account for the formation of surface complexes SOH₂⁺ and SOHCd²⁺-NO₃^- include coulombic terms ${\mathrm{\Psi }}_o$ and ${2\mathrm{\Psi }}_0-{\mathrm{\Psi }}_{\beta }$ respectively²⁰. The curvature of metal adsorption isotherms with increasing surface coverage has been interpreted to be due to either sequential adsorption onto different site types or changes in surface charge and potential based on the predominant type of metal surface complex²⁰. Single-site multispecies models have been successfully applied with the CCM, DDLM, and TLM.

Although surface protonation schemes for different surface site types have been developed using different theoretical approaches and incorporated into SCMs (e.g., CD-MUSIC model¹⁷), the adsorption of metal ions can often be fit using fewer site types. This is consistent with the outcome for adsorption models that use probability distributions to describe surface heterogeneity. Evidence that there is no direct correspondence between aqueous and surface metal speciation was found for transition and heavy metal adsorption over a range of ionic strengths in both NaCl and NaClO₄ solutions¹⁹. This difference in metal speciation schemes is not surprising given that the physical properties of the mineral–solution interface are quite distinct from those of the bulk aqueous solution.

The way forward

EDL/SCM theory is constrained by fitting several different types of data (calorimetric, radiometric, titration, metal adsorption edges and isotherms). For many systems, not all these sources of data are available, and there is heavy reliance on a limited set of complete datasets to evaluate and/or improve these theories. More comprehensive data collection on bulk systems is required to improve these models. The more data collected, the more model variables can be constrained. The initial reason that SCM developers turned to molecular-scale studies was to include additional data sources to reduce the number of unconstrained variables in the SCM models. For example, in some cases, batch adsorption data can be fit equally well with several different proposed surface complexes. Spectroscopic analysis and molecular modeling can narrow these possibilities by providing additional data. At first, the molecular-level data incorporated into SCMs was only the stoichiometry of the surface complexes. More recently, an SCM has been built using intrinsic acidity constants calculated from first-principles MD methods that successfully predicts montmorillonite surface properties as a function of pH and ionic strength²¹. This SCM was then expanded to include the adsorption of arsenic and uranium on montmorillonite clay edges²². This type of integrated multiscale approach to address the challenge of predicting heterogeneities in mineral surface reactivity provides more mechanistic information than circling back to a statistical approach incorporating molecular-scale information.

One approach to capture interfacial reactivity might be to couple atomistic/molecular and continuum simulations using a particle-continuum hybrid method such as that developed by Delgado-Bascalioni et al.²³. Many different hybrid numerical methods have been developed to simulate coupled processes such as fluid flow, transport, and reaction that occur over different spatial and temporal scales^24–26. However, to my knowledge, these methods have not been used to simulate complex geochemical systems with multicomponent aqueous solutions reacting with crystalline interfaces through adsorption, dissolution and/or precipitation. In geosciences, micro-continuum models have been used to investigate reaction fronts at dissolving interfaces, alteration layers along fractures, and mineral precipitation^27–29. A key advantage of the micro-continuum approach is that it allows for reaction between a solid interface and the complex multicomponent aqueous solutions found in natural systems.

To date, a major weakness of large-scale molecular simulations for investigating reactivity at solid-water interfaces is the inability to accurately describe aqueous solutions, including (1) the dissociation of water both in solutions and at solid surfaces, and (2) the formation of aqueous complexes like metal hydroxide and metal-electrolyte anion complexes. Even the simplest batch adsorption experiment (one solid, an electrolyte, and one trace metal), cannot be well described through molecular simulation because of limitations in describing aqueous solutions and changes in both aqueous speciation and surface protonation as a function of pH. Because aqueous speciation impacts the equilibrium between the solution and oxide surfaces, molecular simulations are difficult to compare with simple macroscopic adsorption experiments. Developing better molecular-scale simulation tools to describe multicomponent aqueous solutions is critical to improving comparisons between molecular models and continuum-scale observations.

SCMs were developed to provide a description of equilibrium adsorption within a thermodynamic framework consistent with that used to calculate multicomponent aqueous speciation. They use mass action laws analogous to aqueous solution reactions to describe adsorption, and the adsorption constants can be coupled with thermodynamic databases for aqueous speciation and solubilities³⁰. The major weaknesses of SCMs are due to our lack of understanding of the Stern layer because the behavior of water and ions in this layer is sensitive to surface topography, surface charge distribution, ion-ion correlations, and interfacial hydrogen bonding arrangement³¹. New molecular-level studies may better describe the Stern layer, but this new description must be coupled to geochemical equilibrium and reactive-transport models to be successfully implemented.

After several decades of developing molecular-scale tools to investigate solid-water interfaces, I believe molecular-scale analytical or computational experiments should be designed with the intent of incorporating data into or building new continuum-scale constructs of the EDL and SCMs. I would suggest that if the objective for molecular-level studies is to improve continuum-scale models, then examining the differences in molecular- and continuum-scale observations and theory is a good approach to improving our understanding of interfacial systems at both scales. Scientific questions regarding adsorption at mineral-water interfaces should be guided by macroscopic observations and questions. Experimental and computational tools should be used to address the propagation of atomistic reactivity into continuum observables, to provide information to incorporate into existing macroscopic (e.g., SCM) theories, or to revisit the assumptions and approximations that are inherent in present-day macroscopic interfacial concepts.

Author contributions

The sole author has made all contributions to this document.

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Competing interests

The author declares no competing interests.