replying to L. Criscenti et al. Nature Communications https://doi.org/10.1038/s41467-026-78022-w (2026)
The central argument of our Perspective1 article is that continuum-scale models should represent interfacial properties using probability distributions of surface descriptors rather than fixed average values. Traditional methods address surface heterogeneity by treating a surface as a collection of distinct reactive site types, with each type characterized by fixed equilibrium or reaction rate constants. For instance, in surface complexation modeling (SCM), multi-parameter approach assigns constant values to different site types, effectively representing heterogeneity as an assembly of discrete site classes (see, for example Table II in ref. 2 and Eq. (5) in ref. 3). Similarly, other studies4,5 discussed in the Comment propose that a surface may be treated as an assembly of different groups of sites, each group characterized by its own affinity constant and adsorption isotherm (see for example, Eq. (27) in ref. 4 or Table 1 in ref. 5). However, the probabilities in these models reflect the likelihood of occurrence of each site type, not variability in the chemical descriptors themselves.
Our proposed probabilistic framework differs fundamentally by suggesting that the chemical descriptors—such as equilibrium constants and rate constants—should themselves be modeled as probability distribution functions rather than fixed constants. This concept has not yet been incorporated into existing modeling codes and remains largely unexplored in the literature. By retaining the full probability distribution at the continuum scale, rather than condensing it into a single value, we aim to better capture the complexity of heterogeneous and dynamic interfacial systems.
Similar to SCMs, in reactive transport models (RTM), equilibrium and kinetic processes are described by equations involving equilibrium constants and rate constants, which are currently treated as fixed values. For example, equilibrium constants are used in mass action equations to calculate species concentrations, and rate constants are used in rate laws for mineral dissolution6. These constants are typically derived from laboratory measurements or thermodynamic relationships assuming well-mixed, homogeneous systems.
However, solid surfaces are inherently heterogeneous and locally variable, violating the fundamental thermodynamic assumption of a well-mixed state, where a chemical potential of a species can equalize across the system. Unlike gas or aqueous systems where such equilibration could occur, real surfaces can vary compositionally or structurally at the atomic scale. Consequently, their local environment and energetic contributions lead to fluctuations in chemical potentials and reaction parameters even among identical site types. In particular, surface charge distributions can be heterogeneous, as individual sites could dynamically fluctuate between neutral and protonated/deprotonated states7. Additionally, the observed distributions of calcite dissolution rates exhibit spatial and temporal variability8 even at small length scales, causing the assumption of well-mixed behavior to break down. If this assumption held, the distributions of dissolution rates would remain consistent over time and across different surface locations (assuming chemical conditions remain constant). In other words, when the distribution of activation energies is narrow, the rate constant derived from it closely approximates the system’s behavior. However, if the distribution is broad, multi-modal, or if spatial heterogeneity creates distinct domains with different local distributions, a single rate constant cannot capture the complexity. This variability arises because even nominally identical surface sites can have different activation energies due to atomic-scale defects, strain, impurities, or differences in the molecular environment, leading to a distribution of rate constants rather than a single value.
To address this, we propose representing equilibrium constants and rate constants as time-dependent probability distribution functions that capture spatial and temporal variability. A promising mathematical tool for this is the Fokker-Planck equation9, which can describe the evolution of the probability density function of a chemical descriptor, such as a surface site acidity constant (Ka), under the influence of deterministic drift and stochastic fluctuations. This approach accounts for both systematic changes and random noise affecting surface properties.
Recent studies provide initial support for this probabilistic perspective. For example, spatially resolved measurements of calcite dissolution rates reveal Gaussian-like distributions of rates across a single crystal surface, indicating variability beyond a single constant value8,10,11. Models incorporating surface morphology descriptors, such as surface slope related to kink site density12, have improved the accuracy of reactive transport simulations by linking local structural variability to activation energy distributions. Similarly, in heterogeneous catalysis, probabilistic treatments based on Fokker-Planck theory have been proposed to describe variations in binding energies between catalysts and reactants13.
Our vision is that complex interfacial systems characterized by compositional and structural variability, including defects, can be better modeled probabilistically. This approach advances beyond the limitations of well-mixed system assumptions, which were developed for near-ideal, chemically homogeneous surfaces. While probability distributions may not be necessary for surfaces known to have a single chemically active site, they provide a valuable framework for representing uncertainty and heterogeneity in more complex systems.
We emphasize that collaboration across disciplines, including chemistry, statistics, applied mathematics, computer science, artificial intelligence (AI)-enhanced data analysis, and engineering, is essential to develop innovative tools that accurately capture the chemistry of solid-water interfaces in continuum models.
Regarding specific critiques of our discussion on SCMs, we find no major contradictions or inaccuracies between the Comment article and our writing. While it is true that chemical adsorption energies and Coulombic energies are often treated as constants in SCMs, these values vary with surface site types, coverage, solution composition, and pH. Our argument is that these variations should be represented as distributions rather than fixed constants, even for identical site types, due to local energetic fluctuations.
The term “conventional SCMs” in our article refers to commonly used models such as the constant capacitance model, diffuse layer model, and triple layer model, shown in Fig. 1 in the Perspective article (reproduced from ref. 14). These models assume that the total free energy of ion adsorption is the sum of chemical adsorption and Coulombic energies. More advanced SCMs incorporate multiple site types with distinct acidity or complexation constants, but these constants remain fixed values in current practice.
We acknowledge that SCMs can incorporate multiple surface sites, including high-affinity binding sites attributed to defects or impurities, as in models such as the Double Diffuse Layer Model15 and CD-MUSIC16. However, the properties associated with these sites are still represented by fixed constants rather than probability distributions, which is the key distinction of our proposed approach.
Regarding ion pairing at surfaces, while current SCMs can consider adsorption products involving ion pairs, accurately predicting changes in ion pairing due to surface effects remains challenging. Surface-induced shifts in solvation properties affect equilibrium constants for ion pairs, and these effects are not yet fully captured by existing models. This limitation supports our view that probabilistic frameworks are needed to better represent such complex interfacial phenomena.
In summary, our Perspective advocates for a paradigm shift in continuum-scale modeling of solid-water interfaces, moving from fixed constants to probability distributions for chemical descriptors. This shift acknowledges the inherent heterogeneity and dynamic nature of real surfaces and aims to bridge molecular-scale interfacial science with continuum-scale models in novel ways. We believe this approach will enhance understanding and predictive capabilities across geochemistry, catalysis, chemical separations, corrosion science, energy storage, and related fields.
Author contributions
The Reply article was led by A.G.I., with input provided by E.B., F.M.G., J.M.G., V.H.G., Y.-S.J., N.K., and J.D.K.
Peer review
Peer review information
Nature Communications thanks Andreas Luttge and the other anonymous reviewer(s) for their contribution to the peer review of this work.
Funding
A.G.I. was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division under Field Work Proposal Number 23-015452. E.B. gratefully acknowledges the support of the National Science Foundation (CHE 2102557). F.M.G. was supported by the US National Science Foundation (CHE- 2153191 and CHE-2602170) and the Department of Energy (DE-SC0023342 and DE-SC0026327). V.H.G. research was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-23-1-0181. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. Y.S.J. is grateful for the support received from the American Chemical Society’s Petroleum Research Fund (62756-ND5) and the U.S. Department of Energy Office of Science (DE-SC0023390). This article has been authored by an employee of National Technology & Engineering Solutions of Sandia, LLC under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all right, title and interest in and to the article and is solely responsible for its contents. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes. The DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan https://www.energy.gov/downloads/doe-public-access-plan. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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