## 3 Experimental Approach  
The AFM was operated in force spectroscopy mode to perform the nanoindentation measurements. In AFM force measurements, the sample is mounted on a holder attached to the piezoelectric element of the instrument. By applying a voltage to the piezoelectric translator, the sample is moved up and down, and the resulting cantilever deflection is measured by the optical lever method as an electrical signal called DFL and recorded as a function of the piezo displacement [35]. The cantilever deflection, \(\delta_{c}\), can be then obtained dividing the DFL signal by the photodetector sensitivity, which is derived from the slope of the curve in the hard repulsive contact region obtained using cleaved muscovite mica as a hard material [36]. Knowing the cantilever spring constant \(K_{c}\), the interaction force \(F\) acting between the probe and the sample can be simply obtained from Hooke's law as:  
\[F=K_{c}\cdot\delta_{c} \tag{1}\]  
Considering that the deflection of the cantilever is always very small, it can be considered as an ideal spring. The cantilever spring constant was calculated from the geometrical beam theory by measuring the resonant frequency and neglecting the diamond coating [37; 38]. For the tips used in this study, \(K_{c}\) was measured to be equal to 95 \(\pm\) 5 N/m. The force-distance curves are the source of information related to the mechanical characteristics of the sample [31; 39]. A typical interactive force-distance curve measured on a PEN film is reported in Figure 1. The curve is divided into an approach and a retraction part. Initially, the distance between tip and surface is large enough to have no measurable interaction, the cantilever is not deflected and only the physical separation changes as the probe is moved (point A in Figure 1). As the sample approaches the tip, an instability called \"Jump-to-contact\" occurs and the cantilever undergoes a rapid deflection when the force gradient exceeds the spring constant (point B in Figure 1). The straight line after this point is referred to as the contact line. If the cantilever stiffness exceeds the sample stiffness, as the translator moves further the tip acts as an indenter and causes a deformation (point C in Figure 1). Hence, information about the mechanical properties of the sample can be deduced. Subsequently, the \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Sample \#1** & **Sample \#2** \\ \hline C/O Ratio & 0.67 & 0.85 \\ Equivalent Ratio \(\Phi\) & 2.01 & 2.55 \\ Cold gas velocity, cm/s & 9.8 & 10 \\ Sampling position, mm & 14 & 14 \\ Insertions number & 100 & 25 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Investigated flame reactor parameters. sample surface is retracted from the probe (point D in Figure 1), the adhesive forces hold the probe to the sample until the spring constant overcomes the force gradient, and the \"Pull-off\" instability occurs (point E in Figure 1). The adhesive force, \(F_{adh}\), can be measured from the cantilever deflection at the \"Pull-off\" as shown in Figure 1.  
The hardness of the sample was calculated from the maximum applied load \(F_{max}\), derived from the force-distance curve, and the projected area of the intend \(A_{p}\), derived from AFM topographic images, according to Equation (2):  
\[H=\frac{F_{max}}{A_{p}} \tag{2}\]  
A continuum elastic theory can then be used to further derive mechanical properties of the sample. The most popular models are based on the Hertz theory, which assumes the tip to be a sphere or a paraboloid indenting a semi-infinite plane and neglects adhesion forces [40; 41]. In this study, the Derjaguin-Muller-Toporov (DMT) model has been used to fit the elastic part of the contact line in the retraction force curves [42; 43].