## 2 Materials and Methods
### Surface Analysis  
The stereometric parameters that were the basis for the thin film's morphology surface analysis, were in accordance with the ISO 25178-2:2012 standard, whose parameters have their physical meaning largely described in [33; 34; 35; 36]. To compute several parameters, such as height, feature, spatial, functional, hybrid, volume, and core Sk, MountainsMap 8.0 commercial software was used (Digital Surf, Mountains 8.0, (2020). [[https://www.digitalsurf.com](https://www.digitalsurf.com)]([https://www.digitalsurf.com](https://www.digitalsurf.com)) accessed on 5 December 2020). Furthermore, contour lines, furrows, and texture directions, obtained by Fourier transform on the height function, were obtained from surface microtexture evaluation of the films.  
The fractal characterization of rough surfaces is one of the many fields of science and engineering that makes use of Mandelbrot fractal mathematics, for example, Weierstrass-Mandelbrot (WM) function [37]. This is because a single mathematical expression (the WM function) contains characteristics that mimic the emergence of roughness. As an example, based on a method using image analysis, [PERSON] [38] investigated the link between fractal geometry and prime numbers. Moreover, as wavelets are powerful mathematical tools to analyze 1D/2D signal data in the time-frequency domain, many methods have been proposed based on wavelet and fractal theories, such as image classification using wavelet transform [39]. Thus, in addition to the stereometric parameters, and to have a complete evaluation of the sample surface, we also calculate advanced fractal parameters to study the surface microtexture. According to the method described by [PERSON] and [PERSON], fractal dimension (FD) was computed using a counting box [40] as well as the fractal lacunarity (FL) which was computed using a model described by [PERSON] et al. [41]. From the lacunarity curve, we estimated the lacunarity coefficient (\(\beta\)) using Equation (1), to obtain data regarding the surface texture homogeneity [42].  
\[L(r)=\alpha.r^{\beta} \tag{1}\]  
where \(L(r)\) is lacunarity, \(\alpha\) is a constant, and \(r\) is the box size.  
Using linearized graphs obtained from the mathematical theory by [PERSON] et al. [43], the average power spectrum density (PSD) of fractal regions of the spectra were calculated. Besides, we have estimated the Hurst coefficients of all spectra using Equation (2), from a linearized graph, where \(\gamma\) is the slope of the linearized curve, obtained using WSxM 5.0 software [44].  
\[Hc=\frac{\gamma-2}{2} \tag{2}\]Moreover, using the model described by [PERSON] and [PERSON] [45], fractal succolarity (FS) was calculated using Equation (3).  
\[FS(T(k),dir)=\frac{\sum_{k=1}^{n}P_{0}(T(k)).PR(T(k),p_{c})}{\sum_{k=1}^{n}PR(T(k),p_{c}} \tag{3}\]  
where \(dir\) is the liquid entry direction, \(P_{0}(T(k))\) is the occupation percentage, \(T(k)\) are boxes of equal size \(T(n)\), \(PR\) is the occupation pressure, and \(p_{c}\) is the centroid position \((x,y)\).  
Finally, the last parameter named surface entropy (E) was obtained from the information theory using the Shannon entropy equation [46; 47], Equation (4), where \(p_{ij}\) is assigned to be the probability that a height matrix term \(h_{ij}\) promotes a complete uniformity of the height distribution.  
\[E^{(2)}=-\sum_{i=1}^{N}\sum_{j=1}^{N}p_{ij}.log(p_{ij}) \tag{4}\]  
The obtained value was centralized and normalized according to Equation (5) to give us a normalized value of \(E\)[48].  
\[E=\frac{E^{(2)}-E^{(2)}_{min}}{E^{(2)}_{max}-E^{(2)}_{min}} \tag{5}\]  
All advanced fractal parameters were computed from the AFM topographical matrix extracted by WSxM software. The computational routines were programmed in R language (For FS and E) and Fortran 77 (for FL).