## 4 Theoretical Considerations  
### PTR Method  
The Modulated Photothermal Radiometry basics are presented in detail in the literature [36; 37]. This method involves the illumination of the sample by an intensity-modulated beam of laser light. When the material is illuminated by a light beam, it absorbs energy,  
Figure 3: The BDS experimental setup. and its temperature increases. The temperature of the material is directly related to its thermal and optical properties, such as thermal conductivity, diffusivity, capacity, and optical absorption. As a result of the optical absorption of this light, periodical temperature waves arise in the sample, too. By measuring the thermal emission from the material using a sensitive detector, information about its internal structure, composition, and properties can be obtained. A simplified schematic of the investigated samples consisting of two layers, i.e., the PBMA-siloxane coating deposited on the substrate, is presented in Figure 4.  
In our previous studies, we dealt with sol-gel coatings that were covered with a thin layer of gold [38,39]. The presence of this layer resulted from the preparation of samples for SEM studies. Since this layer also played the role of an absorber, the PTR signal was proportional to the temperature of the surface, and the following model (Equation (1)) for the PTR signal could be used:  
\[PTR\approx T=\frac{I}{\lambda_{c}\cdot\sigma_{c}}\cdot\frac{\cosh(d_{c}\cdot \sigma_{c}+d_{s}\cdot\sigma_{s})+R_{cs}\cdot\cosh(d_{c}\cdot\sigma_{s}-d_{c} \cdot\sigma_{s})}{\sinh(d_{c}\cdot\sigma_{c}+d_{s}\cdot\sigma_{s})+R_{cs}\cdot \sinh(d_{c}\cdot\sigma_{c}-d_{s}\cdot\sigma_{s})} \tag{1}\]  
where \(I\)--intensity of the laser beam; \(d_{c}\) and \(d_{s}\)--thickness of the coating and the substrate, respectively; \(\lambda_{c}\) and \(\lambda_{s}\)--coating and substrate thermal conductivities, respectively; \(\alpha_{c}\) and \(\alpha_{s}\)--coating and substrate thermal diffusivities; and \(R_{cs}\)--thermal reflection coefficient between the coating and the substrate. It is described by Equation (2):  
\[R_{cs}=\frac{b_{cs}-1}{b_{cs}+1} \tag{2}\]  
In Equation (2), \(b_{cs}=e_{c}/e_{s}\), where \(e_{c}\) and \(e_{s}\) are the thermal effusivities of the coating and the substrate, respectively. The complex thermal wave number \(\sigma_{i}\) (subscript \(i\) can be \(s\) or \(c\), which refers to the coating and the substrate, respectively) is described by Equation (3):  
\[\sigma_{i}=(1+i)\cdot(2\alpha_{i}/\omega)^{1/2} \tag{3}\]  
where \(\omega\) is the angular modulation frequency. In the case of PBMA samples, the coating was optically transparent to the beam of light. The illuminating light was absorbed in the steel substrate, which was optically opaque and thermally thick. Figure 4: Schematic cross-section of the investigated two-layer sample. Therefore, the model presented above was found to be insufficient to reproduce the PTR characteristics measured for PBMA samples. An extended model was proposed that assumes that the PTR signal comes from the full thickness of the coating. In this case, as described by Equation (4), the PTR signal is proportional to the definite integral of the spatial temperature distribution \(T(x)\) with respect to the spatial variable \(x\) in the interval [0, \(d_{\rm c}\)], i.e., over the thickness of the layer. \[PTR\approx\int_{0}^{d_{\rm c}}\frac{1-R_{cs}}{\lambda_{s}\cdot\sigma_{\rm s}} \cdot\frac{\exp[-(d_{c}-x)\cdot\sigma_{c}]+\exp[-(d_{c}+x)\cdot\sigma_{c}]}{1- R_{cs}\cdot\exp(-2\cdot\sigma_{c}\cdot d_{c})}dx \tag{4}\]  
The integrand of the integral is the spatial temperature distribution \(T(x)\) of the periodical component of the temperature in the coating. The model assumes that the light was absorbed by the metal substrate, as the coating was optically transparent. The resulting thermal wave in the metal substrate was then transmitted to the coating and propagated, reflecting multiple times at the air-coating and metal-coating interfaces.