🔬 materials science

Kinetic Cellular Model of Corrosion

This paper presents a kinetic cellular model for simulating aqueous metal corrosion that utilizes generalized rate equations to bridge the gap between atomistic simulations and macroscopic scales, effectively reproducing established electrochemical behaviors like Nernst-Planck and Butler-Volmer dynamics while reducing to phase-field or cellular automata methods in specific limits.

Original authors: Orkan Sezer, Andrew Horsfield

Published 2026-07-13

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Orkan Sezer, Andrew Horsfield

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the surface of a metal pipe as a bustling city made of tiny, square rooms called "cells." In this city, atoms are the citizens, and they are constantly trying to move around, swap places, or even change their identity. This is the world of corrosion, the slow, destructive process where metal rots away in water or air.

For a long time, scientists trying to simulate this rot had to choose between two very different tools. Some used atomistic simulations, which are like taking a super-magnified photo of just a few atoms. These photos are incredibly detailed, but they only last for a split second (femtoseconds) and cover a tiny patch of space. Others used continuum models, which are like looking at the city from a helicopter. You can see the whole neighborhood and how long it takes to rust, but you can't see the individual citizens or the specific rules they follow when they bump into each other.

The paper by Orkan Sezer and Andrew Horsfield introduces a new way to play this game called the Kinetic Cellular Model (KCM). Think of KCM as a hybrid video game engine. It divides the world into those same little square rooms (cells), but it gives each room a set of "kinetic rules" that dictate how citizens (particles) hop from one room to the next and how they rearrange themselves inside.

The Main Discovery: A Bridge Between Scales
The big finding here is that this new model successfully bridges the gap between the super-fast, tiny world and the slow, big world. The authors show that their equations can shrink down to look exactly like the famous Nernst-Planck and Butler-Volmer equations (the standard rules used by engineers to predict corrosion) when the conditions are right. But unlike those old rules, KCM can also zoom in to watch individual atoms rearrange themselves during chemical reactions.

In their simulations, they tested this "city" with four different scenarios:

Simple Diffusion: They watched particles spread out like a drop of ink in water. The simulation matched the perfect mathematical answer almost perfectly.
Electric Fields: They simulated ions (charged particles) settling between two charged plates. The result matched the Gouy-Chapman theory, which predicts how ions arrange themselves in a solution.
Hydrogen Evolution: They created a scenario where electrons in a metal met protons in water to form hydrogen gas. The model showed the electrons rushing to the interface to react, just as expected.
Magnesium Dissolving: This was the big test. They simulated solid magnesium metal dissolving into water to become magnesium ions. The model showed the metal atoms leaving the solid, turning into ions in the water, and leaving behind a layer of electrons in the metal. This created an electric "dipole" layer, exactly as physics predicts.

What the Paper Says "No" To
It is important to know what this model doesn't do yet. The authors explicitly state that they are not including mechanical stress fields (like the metal bending or cracking under pressure) in this version. They also assume that the chemical reactions happening inside the cells are simple: they are not catalytic (speeded up by a helper), not inhibited (slowed down by a blocker), and they don't change the temperature.

Furthermore, while the model handles the movement of charged particles well, it treats the electrons in a simplified way. The authors admit that a more complex treatment of electrons is needed for a fully realistic electrochemical model, but they have laid the groundwork for it.

How Sure Are We?
The results presented here are simulations, not physical experiments in a lab. The authors ran their code on computers to solve one-dimensional (1D) and two-dimensional (2D) problems.

In the 1D diffusion test, the simulation matched the analytic solution "essentially perfectly."
In the 2D diffusion test, the match was "very good," though there were tiny differences at the edges of the simulation box because the computer simulation had to use a finite box, while the math assumed an infinite one.
For the magnesium dissolution, the simulation ran for only 0.7 femtoseconds before it became unstable. The authors note that the dissolution appeared "restricted," likely because the charge built up at the interface wasn't removed. They suggest that in a real-world scenario, the formation of a solid corrosion product (like magnesium hydroxide) or the presence of other chemicals (like salt) would be needed to keep the reaction going, but simulating those extra steps is left for future work.

The Bottom Line
This paper doesn't claim to have solved corrosion forever. Instead, it proposes a new "universal" framework that can handle the messy, multi-scale nature of corrosion. It suggests that by treating the material as a grid of cells that exchange energy and particles, we can simulate everything from the slow drift of ions to the fast jump of electrons. The authors hope that in the future, this framework can be fed with even more detailed energy data from quantum mechanics (DFT) to create a truly accurate digital twin of a rusting bridge or a corroding pipeline. For now, it's a promising new tool that works well in simple tests, waiting to be upgraded for the complex, real world.

Technical Summary: Kinetic Cellular Model of Corrosion

Problem Statement
Aqueous corrosion is a complex phenomenon involving interconnected processes operating across disparate length and time scales. These range from ultrafast electron transfer (femtoseconds) and atomic-scale ion dissolution (micrometers, microseconds) to slower ion diffusion and reaction (milliseconds, micrometers). Furthermore, corrosion involves long-range electrostatic interactions, environmental changes due to corrosion product formation, and microstructural influences like grain boundaries. Traditional computational approaches face limitations: atomistic simulations (e.g., Density Functional Theory) provide detailed interface insights but are restricted to short timescales, while continuum models (e.g., Finite Element Methods, Phase Field, Cellular Automata) handle larger scales but often lack explicit atomic-level detail or struggle to link geometry with phase evolution seamlessly. There is a need for a unified method that captures atomic-scale information while extending accessible length and time scales to reliably track corrosion propagation for infrastructure lifetime analysis.

Methodology: The Kinetic Cellular Model (KCM)
The authors propose the Kinetic Cellular Model (KCM), a method that divides a system into discrete cells. Each cell is treated as being locally in equilibrium, exchanging energy and particles with neighbors via generalised kinetic rate equations. The model distinguishes between two primary processes:

Particle Transfer Between Cells: Modeled as random hopping of species ( $\alpha$ ) between neighboring cells ( $i$ and $j$ ). The rate of transfer, $R_{\alpha, i \to j}$ , is governed by the difference in electrochemical potential ( $\mu$ ) between cells, following an Arrhenius-like expression (Eq. 2). This formulation links the KCM to the Butler-Volmer equation. The electrostatic potential is solved via Poisson's equation, accounting for charge density within each cell.
Local Rearrangement Within Cells: This includes chemical reactions and phase changes. If rearrangements are fast, they are treated as instantaneous equilibrium states; if slower, they are governed by internal rate equations. The model connects these internal changes to inter-cell transfer through the derivative of free energy with respect to particle number, ensuring consistency with the electrochemical potential.

The KCM is designed to reduce to established methods in specific limits: it recovers the Nernst-Planck equation for diffusion under small potential gradients and aligns with Phase Field (PF) and Cellular Automata (CA) methods under certain conditions. Unlike standard PF, the KCM treats the phase field parameter as a function of local concentration rather than an independent dynamic variable, implying rapid phase transformation.

Key Contributions and Results
The paper demonstrates the KCM's validity through four one-dimensional (1D) test cases and extensions to 2D:

Diffusion: The model reproduces the analytic solution for particle diffusion with near-perfect agreement.
Gouy-Chapman Distribution: The simulation of ion distribution between charged plates matches the analytic Gouy-Chapman solution, with minor deviations attributed to the KCM's lack of linear approximation and its treatment of electrodes as finite-density metals rather than fixed charge sheets.
Hydrogen Evolution: The model successfully simulates the reaction of protons in solution with electrons in a metal cathode. It captures the diffusion of species to the interface, the instantaneous reaction, and the resulting potential drop caused by the interfacial dipole layer.
Dissolution of Mg: A critical test involving the dissolution of solid Magnesium into water to form $Mg^{2+}$ ions. The model introduces a mechanism for cells to contain mixed phases (solid metal and water) based on a phase field parameter derived from effective local concentrations. The simulation shows Mg dissolving into the solution, leaving electrons in the metal, and forming an electric dipole layer. The authors note that the dissolution appears restricted due to charge buildup, suggesting a need for future mechanisms (e.g., corrosion product formation) to remove ions.
2D/3D Extension: The method was extended to 2D diffusion, showing good agreement with analytic Gaussian solutions, though boundary effects in finite simulation boxes caused slight deviations in peak heights at later times.

Significance and Claims
The authors claim the KCM provides a foundational framework for simulating corrosion that bridges the gap between atomistic and continuum descriptions. By focusing on cells near equilibrium, the method allows for the incorporation of accurate free energy models derived from Density Functional Theory (DFT), including defects and complex interfaces. The cell-based structure offers an intrinsic coordinate system to describe interfaces more naturally than traditional continuum methods.

The paper remains modest regarding its current scope. It acknowledges that the theory is in an early stage: mechanical stress fields are not yet included, chemical reactions are assumed non-catalytic and non-inhibited, and the treatment of electrons is simplified. The authors state that the current results are illustrative and that future work must address stable time integration (potentially using implicit schemes like Crank-Nicolson), 3D simulations of interface roughness and pitting, and the coupling of corrosion product formation with elastic fields to describe lattice mismatches. The method is presented as a step toward a "multi-field universal corrosion model" capable of informing lifetime analyses for critical infrastructure in corrosive environments.

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