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Introduction
Abstract
Corrosion in heritage buildings is critically influenced by the materials used, prevailing weather conditions, and the presence of atmospheric pollutants, as demonstrated by Sorrentino et al. (2024) [1] in their study on cultural heritage degradation under current and projected climate scenarios. In Macau, which was Portuguese colonial, many heritage sites feature construction materials such as limestone and cement, consistent with traditional masonry and later conservation practices [2].
Heritage buildings located in industrial or highly polluted areas experience enhanced deterioration, as atmospheric pollutants interact with building materials. These activities compromise the integrity and lifespan of cultural heritage[3]. To address this issue, we designed a prediction model which is used to estimate the percentage of corrosion mass loss in cultural heritage sites.
Background
Current models such as the[1] Dose-Response Functions (DRFs) used in the International Co-operative Programme on Effects on Materials and the standard rate-of-penetration models for civil infrastructure, have not been able to factor in the set of various acidity in real-life conditions. Also, the models still need to be refined by including additional material classes, particularly non-metallic and non-stone substrates such as cement-based materials. Furthermore, the programmes should be extended to Asia, Africa and Latin America so that the functions can be validated and, where necessary, recalibrated for global application. As a result, such models are not especially good at predicting real corrosion processes, especially in more complex heritage site situations where real-time pH plays a role in the corrosion rates. Our model fills this gap by including this parameter, enabling more realistic corrosion mass loss predictions for heritage buildings.
Objectives
Our main objective is to develop a predictive corrosion mass loss model under different pH over time for heritage sites. The Generalized Logistic Function provides flexibility to model processes with nonlinear, S-shaped growth of percentage mass loss over time. It captures rapid acceleration, maximum value, and smooth saturation. Key parameters (k, t₀, v) are interpretable, offering insights into rate, timing, and shape dynamics. This model is able to predict nonlinear transitions, which outperforms simple linear models in predicting corrosion mass loss.
Assumptions
- Corrosion occurs uniformly across the limestone surface, representing homogeneous corrosion rather than localized forms like pitting.
- Environmental variables such as pollutant concentrations, relative humidity, temperature, and rainfall acidity are treated as steady averages over time. Short-term fluctuations are neglected.
- Assumes the inflection time and shape parameters are pH-independent.
References
[1]B. Sorrentino, A. Screpanti, and A. De Marco, “Corrosion on cultural heritage buildings in Jordan in current situation and in future climate scenarios,” Scientific Reports, 2024.
[2]Ip, Kin Hong. “Sustaining Traditional Practice and Utilising Local Materials in Heritage Conservation.” Australia ICOMOS, 2019.
[3]Bai, Z., & Yan, Y. (2025). Dose–Response Functions for Assessing
[4]https://www.mdpi.com/3378862
Description
This mass loss model is a sigmoid-shaped generalized logistic function.
Observing from paper[1], the key characteristics of the corrosion process:
Initial stage: At the onset of corrosion, the rate of mass loss is relatively slow, as the corrosion reaction requires some time to initiate.
Acceleration stage: Over time, the corrosion reaction gradually accelerates, resulting in a significant increase in the rate of mass loss.
Saturation stage: As corrosion approaches the maximum possible loss, the rate gradually slows down and eventually stabilises, reaching a saturated state.
We propose a sigmoid-shaped generalized logistic function to depict the corrosion behavior of limestone. The S-shaped generalised logistic function effectively captures this dynamic process from initial to acceleration to saturation. Additionally, corrosion processes are typically non-linear, especially in real-world environments, where corrosion rates are influenced by various factors such as environmental acidity, humidity, and temperature[2]. The S-shaped logistic function can effectively capture this non-linear relationship, whereas simple linear models cannot. Thus, the plot of mass loss is anticipated to begin at zero, accelerate rapidly, and decelerate smoothly as it approaches an upper limit (L), typically set as 1 (100% loss). This function modeling time dependence of degradation is employed because it is capable of representing the initial slow phase, rapid acceleration, and final saturation phase approaching the upper limit.
Initially, to test our model, we collected data from a paper[1] demonstrating mass loss of limestone under solutions of pH=4.3 and pH=6. The percentage mass loss is then calculated using the data obtained, shown in Table 1.
After fitting the logistic function with experimental data, k, t0, are found. Using k as the function of pH, A and β, constants of the exponential functions are then found. k is constructed as an exponential function of pH to capture the exponential effect of acidity on corrosion rate. For instance, at low pH, the exponent −βpH is less negative, so k is larger, demonstrating a faster corrosion, whereas at high pH, the exponent becomes more negative, so k is smaller, depicting a slower corrosion. These functions effectively capture dynamic systems with gradual onset, rapid change, and eventual saturation, predicting corrosion mass loss behaviours at different pH values.
Logistic function of mass loss over time
Interpretation of Parameters:
- L : Upper limit of corrosion(100%)
- k : Intrinsic rate constant
- t : Duration of immersion
- t0 : Inflection time
- v : Shape (steepness) parameter
k as function of pH
Interpretation of Parameters:
- k: Intrinsic rate constant
- A: Constant (scaling factor)
- β: Constant controlling the sensitivity of k to changes in pH
- t0 : Inflection time
- pH: Acidity of the solution
Graph plotted from the model
--- Individual fits: Richards logistic ---
pH=4.3: k=0.02958, t0=120.00, nu=0.359, R²=0.9995
pH=6.0: k=0.00596, t0=109.56, nu=0.129, R²=0.9907
Results
The figure 3 illustrated the relationship between pH and corrosion mass loss with time and reveal the strong influence environmental alkalinity or acidity has upon degradation reactions. Time (up to 10 years) is plotted along the X-axis and mass loss along the Y-axis in terms of fraction of original material. Prolongedly very acidic conditions (pH 3–5) result in loss of mass with extremely high rates, nearing almost complete breakdown within the first one to two years, while neutral to weakly acidic conditions (pH 6–7) have more even, slower rates of decomposition. Alkaline conditions (pH 8–10), however, have fairly level curves, and little more than a fraction of material is lost in a decade, indicating significantly greater stability. The experimental pH 4.3 and pH 6 values track almost perfectly the fitted model, as evidenced by comparative R² values of 0.9995 and 0.9907, respectively, as a testament to the robustness of the model in capturing real corrosion behavior. The table presents raw data for initial periods of degradation (10–60 days), which reveal that the material at pH 4.3 degraded in weight approximately four times faster than at pH 6 under equivalent time frames. Combined, these results display the trend unequivocally: more acidic conditions dramatically accelerate deterioration, while raising pH holds material structure intact.
In Figure 4, the exact interpolation (residual = 0) reflects the fact that only two pH levels are available. With only two observations, the regression residual is exactly zero, so the blue line passes precisely through both calibrated k values. it confirms the internal consistency of the rate-constant calibration within the measured range.
Reference
[1]https://pmc.ncbi.nlm.nih.gov/articles/PMC10912733/
[2]https://www.scitepress.org/Papers/2018/75586/75586.pdf
Experiment Design
Description
To validate the accuracy of the pH-dependent mass-loss predictions generated by our (pH-logarithmic corrosion) model, a controlled laboratory experiment was conducted. The primary objective was to observe and quantify the effect of varying pH levels on the corrosion behaviour of common construction materials, thereby confirming whether the theoretical curves align with empirical data.
The experiment was carried out in two versions:
Verison 1
Twelve red bricks were selected as test specimens. These bricks were divided into four groups, each consisting of three samples. Each group was assigned to an immersion solution with a specific pH value: pH 1, pH 3, pH 5, and pH 7 (water). Before immersion, the initial mass of each brick was recorded. The bricks were then fully submerged in their respective solutions for a period of one week. After removal, surface moisture was gently blotted using a towel, and the mass was measured again.
However, the results showed a consistent increase in mass across all samples, contrary to the expected mass loss due to corrosion. Upon consultation, it was determined that the bricks had absorbed significant amounts of water, and their initial mass measurements were not taken under fully dry conditions. This introduced a systematic error, rendering the data unreliable. Consequently, the first phase was discontinued, and the experimental protocol was revised.
Version 2
The materials were changed to limestones and cement cubes, both of which are more representative of heritage building materials currently in use. The cubes were oven-dried at 130 °C for three hours to ensure complete dehydration and then allowed to cool in a desiccator before initial mass measurement. The same grouping and pH conditions were applied: four groups per material, each exposed to solutions of pH 1, 3, 5, and 7.
The experiment ran for several weeks, with weekly mass measurements. After the first few weeks, it was observed that the mass changes plateaued.
Result
--- (Cement Data) Individual fits: Richards logistic ---
pH=1.0: k=0.95871, t0=49.70, nu=0.399, R²=0.0528
pH=3.0: k=0.10700, t0=71.49, nu=0.700, R²=0.3361
pH=5.0: k=0.00239, t0=0.00, nu=0.700, R²=0.1333
pH=7.0: k=0.00474, t0=120.00, nu=0.478, R²=0.9820
--- (Limestone Data) Individual fits: Richards logistic ---
pH=1.0: k=0.00088, t0=0.02, nu=5.000, R²=-0.5823
pH=3.0: k=0.00082, t0=0.00, nu=5.000, R²=-0.1027
pH=5.0: k=0.00109, t0=0.00, nu=5.000, R²=-0.2578
pH=7.0: k=0.00143, t0=0.00, nu=5.000, R²=-0.2883
Investigation revealed that the pH of the immersion solutions had drifted over time due to neutralization reactions between the added acids and bases, leading to the formation of carbonate salts. These salts either altered the solution chemistry or deposited onto the cement cubes' surfaces, potentially affecting mass readings. Notably, under sulfuric acid conditions, the correlation between model predictions and measured data was poor, with a low R² value (eg. R² = 0.0528 at pH 1), indicating significant deviation. This is likely due to the formation of salts such as calcium sulfate (CaSO₄), which may have remained as solid deposits on the sample surface or within its pores, and were not fully removed, resulting in a net increase in sample weight. In contrast, the water group exhibited an r value very close to 1, indicating excellent agreement between model and data. Since water does not react with limestone nor produce solid byproducts, the water group's result strongly supports the hypothesis that salt deposition is the true cause of the observed mass deviation.
For limestone samples, the model showed extremely poor fits across all pH conditions, with R² values ranging from -0.58 to -0.10. These negative R² values indicate that the model performed worse than a horizontal line, suggesting no meaningful correlation between predicted and observed mass changes. This is primarily attributed to the insufficient experimental duration, which was too short to induce detectable mass loss in limestone, a material known for its low reactivity under the tested pH range.
References:
[1]https://www.mdpi.com/2075-5309/15/13/2271
Future Improvements
In our corrosion experiments, the formation of sulfate-based residues—such as CaSO₄—was found to interfere with accurate mass loss quantification. These residues tend to remain on the sample surface after corrosion, leading to underestimation of material degradation. To mitigate surface salt formation without interfering with acid-induced corrosion, we adopted the pre-treatment strategy described by Liu et al. [2], applying CF-S2 densifier to limestone and cementitious samples prior to immersion
CF-S2 is a carbon-fiber-based inorganic densifier that modifies the pore structure of the substrate. It is applied prior to corrosion testing and cured to ensure stable adhesion. CF-S2 exhibits strong bonding to both limestone and cement surfaces and remains attached throughout immersion. It does not interfere with the corrosion process and does not contribute to mass loss. The sample mass is recorded after CF-S2 application and used as the baseline for corrosion-related mass loss calculations.
A 2025 study published in Frontiers in Materials[1] reported that CF-S2 treatment reduced sulfate ion accumulation in cementitious samples subjected to wet-dry cycling. These findings support its use in improving the reliability of mass loss measurements in corrosion modelling.
References:
[1]J. Liu, P. Huang, H. Gan, X. Wang, and Z. Liu, “Novel mechanism of sulfate erosion mitigation in cement mortar using CF-based densifier: microstructural and durability perspectives,” Frontiers in Materials, vol. 12, Jun. 2025. DOI: 10.3389/fmats.2025.1619529.
Limitations
This model cannot detect multifactor interactions, such as the combined effects of fluctuating pollutant levels, varying temperature, and humidity. Localized factors like pitting and uneven surface corrosion are also neglected. This results in an underestimation of localized damage that accelerates structural degradation. Since the model is based on full immersion of limestone in a corrosive solution, it cannot precisely predict corrosion behaviour of limestone in exposed environments where surfaces undergo partial wetting and drying cycles. Furthermore, we are unable to simulate all heritage sites, as not all of them are constructed with limestone. The diversity of materials, structural designs, and environmental exposures across different sites introduces additional variables that fall outside the scope of this model.