## 2 Materials and Methods
### Characterization of the Applied Coatings  
Raman measurements were carried out in a backscattered geometry with a LabRAM micro-spectrometer (Jobin Yvon Horiba, Kyoto, Japan) equipped with an Olympus BX40 microscope (Olympus Corporation, Tokyo, Japan). The 632.8 nm line of a 15 mW He-Ne laser was used, and the spectra were collected, at room temperature, by a long working distance \(\times\)50 microscope objective lens. XRD patterns were collected using a Thermo ARL X'TRA X-ray diffractometer equipped with Si-Li detector (Thermo Fisher Scientific Inc., Waltham, MA, USA). Cu-K\(\alpha\) radiation (40 kV and 40 mA) at a 0.2\({}^{\circ}\) scan rate (in 2\(\theta\)) in the range of 10-60\({}^{\circ}\) was used. Powdered silicon was used for 2\(\theta\) calibration. Morphological observations before and after the treatments and elemental analysis to study the distribution of the coatings on the raw stone surface were performed by means of a SEM-EDS system: a scanning electron microscope Jeol 6400 (Jeol Ltd.; Tokyo, Japan) equipped with an Oxford Instruments (Abingdon-on-Thames, UK) Analytical Link Si(Li) Energy Dispersive System Detector, with INCA V7.2 software. The chromatic changes in the stone surface's appearance due to the TP-Hy and N-TiA treatments were evaluated according to UNI EN 15886:2010 rules [80] by colorimetric analysis performed by a Spectrodens colorimeter (Techkon GmbH, Konigstein, Germany) on nine small zones. The results were averaged on three samples for each case. The color difference (\(\Delta E^{*}\)) due to the coating was measured as  
\[\Delta E^{*}=\sqrt{\Delta L^{*2}+\Delta a^{*2}+\Delta b^{*2}} \tag{1}\]  
with \(\Delta L^{*}\) being the change in lightness and \(\Delta a^{*}\) and \(\Delta b^{*}\) being the changes of the colorimetric coordinates \(a^{*}\) (red/green) and \(b^{*}\) (yellow/blue) in the CIELAB space. The water absorption by capillarity was estimated on five samples before and after the treatments, according to UNI EN 15801:2010 [81]. The dried samples were placed on a Whatman multilayer paper (1 cm thick) saturated with deionized water. Samples were weighed at times t\({}_{\text{i}}\) =10 min, 20 min, 30 min, 60 min, 2 h, 4 h and 6 h and at following 24 h intervals until the difference between two successive weighings was \(<\)1% of the initial weight of the samples. The amount of absorbed water per unit area Q\({}_{\text{i}}\) (kg/m\({}^{2}\)) was reported as a function of the square root of time (t\({}_{\text{i}}\)\({}^{1/2}\)), as, according to a well-known capillarity absorption model [82], there should be a linear relationship between the adsorbed water and the square root of time in short times. The value of the capillary water absorption coefficient AC (kg/(m\({}^{2}\) s\({}^{1/2}\))) has been obtained from the slope of the linear fit in the first part (up to 60 min) of the capillarity graph (Q\({}_{\text{i}}\)). The water vapor permeability test was performed first on untreated stone and then after treatments, following the method proposed by [PERSON] et al. [16]: the 5 \(\times\) 5 \(\times\) 1 cm\({}^{3}\) samples were fixed on top of PVC containers, partially filled (one half) with water, and then the sealed containers were placed in a climatic chamber (RH 25% and T = 25 \(\pm\) 0.5 \({}^{\circ}\)C) and weighed every 24 h until constant vapor flow through the stone (i.e., the condition (\((\Delta M_{\text{i}}-\Delta M_{\text{i-1}})/\Delta M_{\text{i}}\)) \(\times\) 100 \(\leq\) 5 was reached). The vapor permeability reduction (\(VPR\)) was estimated as the mass of water vapor passing though the surface unit (cm\({}^{2}\)) in 24 h by \(VPR\%=((M_{uv}-M_{tv})/M_{uv})\) \(\times\) 100, where \(M_{uv}\) and \(M_{tv}\) are the mass of water vapor through the untreated and treated stone, respectively, measured at the steady state [83].