A hybrid model of sulphation reactions: stochastic particles in a random continuum environment

TL;DR

This work develops a hybrid stochastic-continuum model for marble sulphation, coupling stochastic Itô SDEs for sulfuric acid particles with random ODEs for calcium carbonate and gypsum densities, linked through a mollified empirical concentration $u_N$. Particle interactions include Lennard-Jones-type repulsion and nonlocal environment forces that bias movement toward gypsum-rich regions, while reactions occur according to a Poisson process with intensity driven by local calcium carbonate density. The model is non-dimensionalized to a dimensionless system $(P')$ and simulated using Euler–Maruyama with finite-element-like spatial discretization, revealing heterogeneous corrosion patterns even from symmetric initial data. Numerical experiments explore various boundary and initial conditions, as well as parameter studies, demonstrating the model’s ability to capture key mechanistic features of sulphation and gypsum formation. The approach presents a flexible framework for discrete-continuum coupling in corrosion and related transport-reaction problems and points to future analytical homogenization avenues for macroscopic descriptions.

Abstract

We present a hybrid stochastic-continuum model to study the sulphation of calcium carbonate and the consequent formation of gypsum, a key phenomenon driving marble deterioration. While calcium carbonate and gypsum are continuous random fields evolving according to random ordinary differential equations, the dynamics of sulfuric acid particles follows Itô-type stochastic differential equations. The particle evolution incorporates both strong repulsion between particles via the Lennard-Jones potential, and non-local interactions with the continuum environment. The particle-continuum coupling is also achieved through the chemical reaction, modeled as a Poisson counting process. We simulate the spatiotemporal evolution of this corrosion process using the Euler-Maruyama algorithm with varying initial data combined with finite elements to take care of the spatial discretization. Despite symmetric initial data, our simulations highlight an uneven progression of corrosion due to the stochastic influences in the model.

Figures (8)

• Figure 1: (a) The standard Lennard-Jones potential \ref{['eq:LJ']} and force in dimension $d=3$, with well depth $\epsilon = 1$ and range parameter $\varsigma = 1$. (b) The smoothed potential \ref{['eq:LJpot_regularized']} and force, with $\delta = 0.5$. (c), (d) Interaction ranges of the Lennard-Jones potential \ref{['eq:LJ']}. Most of the interaction is concentrated around the equilibrium distance $r_0 = r_1 + r_2$, while for particles at distances greater than $2r_0$ the interaction is negligible.
• Figure 2: (a) In a heterogeneous environment, particles are attracted toward regions with higher gypsum density, \ref{['eq:force_environment']}, as the porosity of gypsum facilitates particle diffusion. (b) Dynamics in a homogeneous environment, where motion is primarily stochastic and resembles Brownian motion.
• Figure 3: Case 1. (Top) Evolution of the system at specific times: spatial gypsum (orange) and calcium (black) densities; active (green circle) and reacted (red cross) sulfuric acid particles locations. (Bottom) Temporal evolution of the total mass mass of gypsum (solid light blue) and calcium carbonate (orange).
• Figure 4: Case 1. (Top) Evolution of particles concentrations at specific times. (Bottom) Temporal evolution of the total mass of sulfuric acid particles.
• Figure 5: Case 2. (Top) Evolution of the system at specific times: spatial gypsum (orange) and calcium (black) densities; active (green circle) and reacted (red cross) sulfuric acid particles locations. (Bottom) Temporal evolution of the total mass mass of gypsum (solid light blue) and calcium carbonate (orange).

Theorems & Definitions (1)

• Remark 1