## 2 Materials and Methods
### Fractal Model  
A phenomenological model based on fractal geometry was developed with the aim to reproduce the porous structural features (pore size distribution, pore size range, pore fraction) and predict the specific surface area and Young's modulus of NP Al [25]. In particular, the proposed fractal model, called IFU, was obtained by using Sierpinski carpet, a well-known example of fractal geometry as a base unit. This fractal figure was constructed by starting from a square in which each side (\(l\)) is divided by a numerical factor \(F\) generating equal sub-squares (Equation (1)). Some of these sub-squares can be removed by creating a structure formed by solids and voids with the same characteristic length [25]. \[L_{e}^{(max)}\ =\frac{l}{F} \tag{1}\]  
The obtained objects were characterized by the non-integer dimension calculated by  
\[D_{f}=\frac{\log\bigl{(}N_{RSQ}\bigr{)}}{\log(F)} \tag{2}\]  
where \(N_{RSQ}\) denotes the number of sub-squares not removed at the first iteration. The described step was repeated on all sub-squares, generating a self-similar pattern in which at the \(n\)-th iteration, the size of the voids was:  
\[L_{e}^{(n)}\ =\frac{L_{e}^{(max)}}{F^{(n-1)}} \tag{3}\]  
Iteration by iteration, it was possible to note that new classes of voids were generated, and monotonic pore size distribution could be obtained. However, real materials in general have different pore size distribution curves (non-monotonic) in which one or more maxima can be found in correspondence with any pore class. These structures could be approximated by a modeling procedure based on IFU. The number of unit base type was related to the complexity of the cumulative curve of porous structures, one unit for every inflection point. For the investigated systems, two structural types of units' bases were used: 1-unit Sierpinski carp \(A\) and \(n_{B}\) Sierpinski carpet B (for simplicity unit A and unit B), with different characteristic lengths and \(D_{f}\). The pore fraction resulting from intermingling was given by:  
\[\varepsilon_{mod}=\bigl{(}A_{Ap}+n_{B}\cdot A_{Bp}\bigr{)}/\bigl{(}A_{A}+n_{B} \cdot A_{B}\bigr{)}\]  
where \(A_{A}\), \(A_{B}\), \(A_{Ap}\), and \(A_{Bp}\) are the total areas and total pore areas of A and B units, respectively. After porosimetric tests or image analysis, it was possible to use the experimental measurement value for calculating \(n_{B}\)  
\[n_{B}=\bigl{(}A_{Ap}-\varepsilon_{exp}\cdot A_{A}\bigr{)}/\bigl{(}\varepsilon _{exp}\cdot A_{B}-A_{Bp}\bigr{)}, \tag{4}\]  
where \(\varepsilon_{exp}\) is the experimental pore fraction. In order to improve the versatility of the model, different settings could be used, such as: (a) maintain solid some sub-squares of unit A and units B forever; (b) introduce non-porous surfaces to control the pore fraction of the entire model; (c) make the pore profile tortuous in a longitudinal or transverse direction for reproducing specific surface and morphological characteristics. In particular, concerning the tortuous pore profile, it can be characterized by rough path \(L_{t}^{(n)}\) which was calculated by  
\[L_{t}^{(n)}\ =L_{e}^{(n)}\,\delta^{-\mathrm{i}^{1-D_{t}}} \tag{5}\]  
where \(D_{t}\) is tortuous fractal dimension and  
\[\delta=F^{-n} \tag{6}\]  
A tortuous pore profile was constructed around the square edge of a void to leave the solid and void fraction unaffected. Consequently, the pore cross-sectional area was considered constant and independent of the specific profile. Since the IFU is the smallest representative part of a microstructure, a method to compare data with IA (referred to the examined surface) had to be used.