## Materials and Methods
### Nanocomposite Products  
The two nanocomposite formulations used in this study, hereafter referred to as WNC (water-based nanocomposite) and ANC (alcohol-based nanocomposite), were developed by companies (ChemSpec srl, Peschiera Borromeo, Italy; Colorobbia Consulting srl, Sovigliana, Italy) in the framework of the EU-funded NanoC Cathedral project [27]. They consist of TiO\({}_{2}\) nanoparticles dispersed in alkylalkoxysilane reactive sols. The details of their preparation are protected by a non-disclosure agreement, and only partial information on their components was supplied by the producers (Table 2).  
In order to characterize the rheological behaviour of the products, a CV0 120 Rheometer (Bohlin Instruments Vertrieb GmbH, now Malvern Panalytical Ltd., Malvern WR14 1 XZ, UK) was employed, using a cone-plate configuration (1\({}^{\circ}\) angle, 40-mm diameter) with a 0.03-mm gap. The measurements were conducted for 3 min under 0.01-3 Pa stress at 20 \({}^{\circ}\)C. Since the products exhibited a shear-thinning behaviour, the value of viscosity measured in the low shear rate region around 4 s\({}^{-1}\) (Table 2) was assumed to be representative of the rheological behaviour of the products in a capillary flow regime.  
Particle size was measured on a 90 Plus Dynamic Laser Light Scattering instrument (Brookhaven Instruments Corporation, Holtsville, NY, USA) equipped with a 35-mW laser and an Avalanche photodiode detector collecting the scattered light at 90\({}^{\circ}\).  
The protective effectiveness and the general properties of these products have already been assessed in previous studies [29; 30]. The alkyl groups provide the coatings with hydrophobic properties, while TiO\({}_{2}\) nanoparticles at low concentration are introduced to impart photocatalytic and self-cleaning properties. Upon solvent evaporation, the sol-gel condensation turns the alkylalkoxysilane precursors into cross-linked organosilica networks, their adhesion onto stone substrates being the result of both noncovalent (hydrogen bonding and dipolar interactions of the mineral surface with unreacted silanol  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stone Type** & **Open Porosity** & **Average Pore Diameter** & **RMS Roughness** \\  & **(vol\%)** & **(\(\upmu\)m)** & **(nm)** \\ \hline Apuan Marble & 0.7a & 0.08a & 6 \({}^{1}\) \\ Balegem & 9.9 \(\pm\) 0.8 & 0.28 \(\pm\) 0.06 & 38 \(\pm\) 8 \\ Obernkirchen & 24.1 \(\pm\) 0.1 & 0.8 \(\pm\) 0.3 & 10 \(\pm\) 9 \\ \hline \hline \end{tabular} \({}^{1}\) In the case of Apuan marble, measurements were only carried out on one specimen.  
\end{table}
Table 1: Open porosity, average pore diameter and surface roughness (Root Mean Square (RMS)) of the three lithotypes.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Main Features of Formulation** & **WNC (Water-Based Nanocomite)** & **ANC (Alcohol-Based Nanocomposite)** \\ \hline Solvent & water & 2-propanol \\ Main component \({}^{1}\) & aminoalkyl-trialkovysilane (oligomers) & monomeric alkylalkoxysilane \\ Concentration (wt/wt) \({}^{1}\) & 15\% & 40\% \\ Density (g/cm\({}^{3}\)) & 1.03 & 0.84 \\ Viscosity (mPa-s) & 12 & 6 \\ pH & 4.5 & n.a. \\ TiO\({}_{2}\) (ppm) \({}^{1}\) & 9600 & 1200 \\ TiO\({}_{2}\) particle size (nm) & 106 & 25 \\ \hline \hline \end{tabular} \({}^{1}\) Data supplied by the producers.  
\end{table}
Table 2: Main chemical properties of the two nanocomposites.  
groups, with additional ionic interactions involving protonated amine in WNC) and covalent (sol-gel condensation with surface silanols of silicate stones) interactions.