Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Integro-differential Model of Cadmium Yellow Photodegradation

Maurizio Ceseri, Roberto Natalini, Mario Pezzella

TL;DR

This work formulates a non-local integro-differential model for the photodegradation of cadmium yellow by coupling CdS chemistry to environmental factors through Beer-Lambert attenuation and a nonlocal light-penetration operator. It introduces a second-order, positivity-preserving Predictor-Corrector scheme based on an exponential reformulation of a Volterra integral equation, ensuring stable, monotone, and accurate simulations. Numerical experiments reveal surface-dominated degradation with a CdSO$_4$-driven passivation effect, and show that UV exposure accelerates damage, while sensitivity analysis demonstrates model robustness to parameter changes. The framework offers a practical tool for cultural heritage preservation and points to future enhancements via Kubelka-Munk light propagation and binder-yellowing dynamics, along with parameter calibration against experimental data.

Abstract

Many paintings from the 19th century have exhibited signs of fading and discoloration, often linked to cadmium yellow, a pigment widely used by artists during that time. In this work, we develop a mathematical model of the cadmium sulfide photocatalytic reaction responsible for these damages. By employing non-local integral operators, we capture the interplay between chemical processes and environmental factors, offering a detailed representation of the degradation mechanisms. Furthermore, we present a second order positivity-preserving numerical method designed to accurately simulate the phenomenon and ensure reliable predictions across different scenarios, along with a comprehensive sensitivity analysis of the model.

An Integro-differential Model of Cadmium Yellow Photodegradation

TL;DR

This work formulates a non-local integro-differential model for the photodegradation of cadmium yellow by coupling CdS chemistry to environmental factors through Beer-Lambert attenuation and a nonlocal light-penetration operator. It introduces a second-order, positivity-preserving Predictor-Corrector scheme based on an exponential reformulation of a Volterra integral equation, ensuring stable, monotone, and accurate simulations. Numerical experiments reveal surface-dominated degradation with a CdSO-driven passivation effect, and show that UV exposure accelerates damage, while sensitivity analysis demonstrates model robustness to parameter changes. The framework offers a practical tool for cultural heritage preservation and points to future enhancements via Kubelka-Munk light propagation and binder-yellowing dynamics, along with parameter calibration against experimental data.

Abstract

Many paintings from the 19th century have exhibited signs of fading and discoloration, often linked to cadmium yellow, a pigment widely used by artists during that time. In this work, we develop a mathematical model of the cadmium sulfide photocatalytic reaction responsible for these damages. By employing non-local integral operators, we capture the interplay between chemical processes and environmental factors, offering a detailed representation of the degradation mechanisms. Furthermore, we present a second order positivity-preserving numerical method designed to accurately simulate the phenomenon and ensure reliable predictions across different scenarios, along with a comprehensive sensitivity analysis of the model.

Paper Structure

This paper contains 8 sections, 2 theorems, 28 equations, 26 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mathematical Modeling
  3. Dimensionless Formulation
  4. Unconditionally Positive and Monotonic Numerical Method
  5. Convergence and Performances
  6. Numerical Simulations
  7. Sensitivity Analysis
  8. Conclusions and Future Perspectives

Key Result

Theorem 3.1

Consider equation eq:PC method ADIM and assume that $I(\lambda)\geq 0,$ for all $\lambda\in\Omega.$ Then, independently of the positive values $\Delta x$, $\Delta t$, $\Delta \lambda,$ the solution $\{c^j_n\}_{n \in \mathbb{N}_0},$$j\geq 0$ is positive and non-increasing with respect to $n$.

Figures (26)

  • Figure 1: Experimental convergence and performances of the P and PC methods.
  • Figure 2: Absolute irradiance as a function of the wavelength for the UV-filtered xenon lamp.
  • Figure 3: Molar absorptivities as functions of the wavelength for the \ref{['defn: BCT']}$(\varepsilon_g(\lambda)=4\varepsilon_c(\lambda))$.
  • Figure 4: \ref{['defn: BCT']}: Space-time evolution of the cadmium sulfide concentration
  • Figure 5: \ref{['defn: BCT']}: Space evolution of the cadmium sulfide concentration at different times
  • ...and 21 more figures

Theorems & Definitions (4)

  • Remark
  • Theorem 3.1
  • proof
  • Theorem 3.2