Probability Distributions for Interfacial Reactivity Are Not New, Argues Surface Chemist

A quiet but pointed dispute has broken out in the pages of Nature Communications over how scientists should represent the messy, uneven chemistry of mineral surfaces in contact with water. In a Matters Arising comment published on 28 September 2026, geochemist Louise J. Criscenti, formerly of Sandia National Laboratories, challenges a 2024 Perspective by A.G. Ilgen and colleagues that proposed using probability distribution functions to capture the heterogeneous reactivity of oxide-water interfaces as a novel conceptual shift. Criscenti’s rebuttal is blunt in its central claim: the idea is not new at all. Probability distributions, she argues, were baked into the very first adsorption models for oxide-water interfaces more than four decades ago, and the field’s apparent forgetting of that history risks sending researchers in circles rather than forward.

The original Perspective had suggested that continuum-scale models, including surface complexation models and reactive transport models, are not designed to incorporate spatially differing reactivities of surfaces, and that introducing probability distributions to describe surface properties would transform how these models are built. Criscenti counters with a detailed historical record. As early as 1978, Davis, James and Leckie reported that the adsorption of metal ions at the oxide-water interface could not be described by assuming a homogeneous solid surface. In 1981, Benjamin and Leckie showed that the adsorption of cadmium, copper, zinc and lead onto amorphous iron oxyhydroxide could only be reproduced by assuming a large dispersion of site affinities across the surface. In other words, the energetic heterogeneity of oxide surfaces, and the statistical machinery needed to describe it, was foundational to the discipline from its earliest days.

The technical literature Criscenti marshals is extensive. Review chapters by van Riemsdijk and Koopal in 1992, Koopal in 1996, and Rudzinski and colleagues in 1996 all document early surface complexation models formulated explicitly with probability distributions describing surface heterogeneity. Koopal and van Riemsdijk went further in 1989, developing electrical double layer theory for three distinct types of spatial heterogeneity: random heterogeneity, isolated patches, and interacting patches. That work directly contradicts the Perspective’s assertion that continuum-scale models cannot handle spatially varying reactivity. Rudzinski and co-workers later fitted electrolyte ion adsorption data at the silica-sodium chloride interface using a single-site triple layer model in which heterogeneity parameters were proportional to the variance of a Gaussian-like function describing the dispersion of adsorption energies, a function originally grounded in studies of argon adsorption on silica. Charmas and colleagues extended this framework in 2002 to cadmium adsorption, fitting both surface charge titrations and electrophoretic mobility measurements across a range of pH values.

Criscenti is careful to acknowledge the parallel track that developed alongside the statistical one. Rather than describing heterogeneity with continuous distributions, some modelers chose to represent it explicitly, assigning distinct reactive groups each with its own affinity. This discrete approach underpins the MUSIC model developed by Hiemstra and colleagues in 1989, in which proton binding constants for different surface site types were estimated a priori from crystallographic considerations. The two strategies, continuous probability distributions and discrete multisite descriptions, have coexisted for decades, and Criscenti’s point is that neither represents the conceptual revolution the Perspective claimed. Both are mature tools with long track records, and choosing between them is a matter of data availability and purpose, not paradigm-shifting novelty.

The rebuttal also highlights a second, often overlooked use of probability distributions in geochemistry: quantifying uncertainty. Normal distributions have long been used to describe error bars in aqueous solution measurements, aqueous speciation equilibrium constants, and adsorption equilibrium constants. Monte Carlo methods have propagated these uncertainties through geochemical and reactive transport calculations, work Criscenti herself contributed to in the 1990s and early 2000s when assessing contaminant migration from uranium mill tailings, hazardous landfills, and potential nuclear waste repository sites. That tradition continues today. A community data-driven database and workflow for raw sorption data, developed by Zavarin and colleagues in 2022, propagates error bars from raw potentiometric titration data through to fitted acidity constants and representative average values for the community database. Even newer hybrid machine learning approaches, such as the random forest method of Chang and colleagues in 2023 for predicting radionuclide sorption, incorporate error propagation based on means and standard deviations of normal probability distributions.

Beyond the question of novelty, Criscenti identifies what she sees as misunderstandings of surface complexation model theory embedded in the Perspective. One claim held that chemical adsorption energies and Coulombic terms cannot be treated as constants because of intrinsic surface heterogeneity. Criscenti responds that these energies are indeed treated as constants for a given set of conditions, but that they properly vary with surface site type, surface coverage, solution composition and pH, and that models account for this through multiple site types and reaction-specific electrostatic terms. Another claim suggested that conventional models use only a single acidity constant and complexation constant. Criscenti points to the widely used diffuse double layer model of Dzombak and Morel, incorporated into the geochemical code PHREEQC, which requires two site types for cation sorption on hydrous ferric oxide, including a low-density, high-affinity site attributed to defects and impurities. The charge distribution MUSIC model, established enough to appear in readily available geochemical codes, allows for many more site types still.

Perhaps the most technically charged disagreement concerns ion pairing at the interface. The Perspective argued that models should account for surface-promoted ion pairing, such as strontium adsorbing as a strontium chloride ion pair subject to reduced lateral Coulomb repulsion, and claimed they currently do not. Criscenti’s rebuttal walks through the evidence that present-day models do exactly this. James and Healy showed as early as 1972, using Born solvation theory, that aqueous metal hydroxide species should adsorb preferentially over bare metal ions. Since then, numerous studies have proposed surface complexes involving the metal cation together with hydroxide or electrolyte anions, a literature comprehensively reviewed by Criscenti and Sverjensky before 2000. In extended triple layer models, the Coulombic term in the equilibrium expression changes according to the reaction, so that complexes carrying different charges and occupying different planes carry different electrostatic corrections. The curvature of metal adsorption isotherms with increasing coverage has long been interpreted as arising from sequential adsorption onto different site types or from changes in surface charge and potential as the dominant complex type shifts.

So where should the field go, if not back to distributions it never really left? Criscenti argues for more comprehensive data collection on bulk systems, since surface complexation theory is constrained by fitting calorimetric, radiometric, titration and adsorption data, and complete datasets remain scarce. Molecular-scale studies were originally recruited precisely to reduce the number of unconstrained variables, for example when several different proposed surface complexes fit the same batch adsorption data and spectroscopy or molecular modeling is needed to narrow the field. Recent work shows how powerful this integration can be: an SCM built using intrinsic acidity constants calculated from first-principles molecular dynamics successfully predicts montmorillonite surface properties as a function of pH and ionic strength, and has been extended to arsenic and uranium adsorption on clay edges. Criscenti suggests this multiscale integration provides more mechanistic information than circling back to a purely statistical representation of heterogeneity, and points to particle-continuum hybrid methods and micro-continuum models, already used for dissolving interfaces, alteration layers and mineral precipitation, as promising but still untested avenues for full multicomponent geochemical systems.

She is equally candid about the weaknesses on every side. A major limitation of large-scale molecular simulations is their still-poor description of aqueous solutions, particularly water dissociation at surfaces and the formation of metal hydroxide and metal-electrolyte complexes. Even the simplest batch adsorption experiment, one solid, one electrolyte and one trace metal, cannot yet be fully described by molecular simulation because aqueous speciation changes with pH in ways the simulations struggle to capture, making direct comparison with macroscopic experiments difficult. Meanwhile, the acknowledged soft spot of surface complexation models is the Stern layer, the near-surface region whose behavior is exquisitely sensitive to surface topography, charge distribution, ion-ion correlations and interfacial hydrogen bonding. Any new molecular-level description of that layer, she stresses, must be coupled back into geochemical equilibrium and reactive transport codes to be useful.

The takeaway from this exchange reaches beyond one contested Perspective. Criscenti’s argument is a reminder that in mature fields, proposals framed as revolutions may be rediscoveries, and that the collective memory of a discipline, preserved in its older literature, is itself a scientific resource. Her prescription is pragmatic: molecular-scale experiments and computations should be designed from the outset to feed continuum-scale constructs, with macroscopic observations guiding the questions asked at the atomic scale. Whether the original authors respond in kind remains to be seen, but the episode has already performed a service, dusting off four decades of heterogeneity modeling and putting the question of how best to represent interfacial reactivity, probabilistically or mechanistically, squarely back on the table.

Subject of Research: Probability distributions and surface complexation modeling of adsorption at oxide-water interfaces

Article Title: Should we return to probability distributions to represent interfacial reactivity?

Article References: Criscenti, L. J. (2026). Should we return to probability distributions to represent interfacial reactivity?. Nature Communications, 17(1), Article 10216. https://doi.org/10.1038/s41467-026-78022-w

Image Credits: AI Generated

DOI: 10.1038/s41467-026-78022-w

Keywords: interfacial reactivity, surface complexation models, probability distributions, surface heterogeneity, oxide-water interface, adsorption, electrical double layer, geochemical modeling, reactive transport, Stern layer, molecular simulation, Monte Carlo uncertainty