## 4 Results and Discussion  
The primary simulation results are provided by the software in terms of concentration profiles within the cells stack. An example of results is shown in Figure 9 for an overlapping FCC unit cell stack, where the concentration profile mainly develops from the bottom (higher concentration) towards the top (lower concentration). From these data, the total diffusive flux normal to the surface at the top and the bottom of the stack is calculated in post-processing, thus giving all the information required to evaluate the effective diffusivity and, from that, the tortuosity (Equation (7)). Besides the meaning of the 3D profiles, an important aspect to notice is that the concentration gradient is steeper in the voids where boundaries are closer. This indicates that the diffusion resistance is mainly due to such zones of the stack, it implies that tortuosity, as well as connectivity, is determined by the shape of these narrow regions. As voids become gradually narrower with decreasing porosity, the diffusion resistance increases; this causes a higher tortuosity and, thus, a lower connectivity. Therefore, in the limit case in which the volume of a narrow region becomes null, the diffusional path encounters a dead-end channel. If all channels become dead ends, the inner voids become completely disconnected islands (Figure 1b) and, thus, diffusional resistance and tortuosity become infinite (no diffusive flux possible). Since the considered structures are regular, the condition of infinite diffusional resistance is reached at a specific porosity value--here referred to as _Limit Porosity Value_\(\varepsilon_{Lim}\)--corresponding to the limit void volume that is not available for diffusion (\(V_{Lim}\), Equation (10)). Such a parameter can be considered a sort of _\"hidden\"_ morphological property, as it is not apparent when voids are interconnected. The numerically calculated \(\varepsilon_{Lim}\) for all structures are reported in Table 2, from where we can observe as a mere coincident the very similar values of SC and FCC. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mesh** & **\(\tau\) (-)** \\ \hline _Finer_ & 1.35535 \\ _Coarser_ & 1.35674 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of tortuosity sensitivity to mesh for the FCC. Figure 9: Example of concentration gradient obtained as a solution of the simulation performed for the overlapping FCC structure. To clearly show the geometry corresponding to the limit porosity, Figure 10 shows the completely disconnected island-shaped morphology of the structures. The results obtained in terms of tortuosity vs. porosity are shown in Figures 11 and 12. The lower bound for the structure is represented by the model developed by [PERSON] and [PERSON] (1973) ([PERSON] and [PERSON], 1973) for a homogeneous non-overlapping swarm of spheres, which is actually an ideal structure composed of an infinitely dispersed particles (Equation (9)). \[\tau=\frac{3-\varepsilon}{2} \tag{9}\]  
It is curious as well as interesting that this expression is the same as that obtained by [PERSON] (1881), which is valid in the limit of porosity approaching unity. In the plot, we prefer indicating the dashed line as [PERSON] and [PERSON]'s model instead of [PERSON]'s one since the former is virtually valid in the whole porosity range. Another fact to observe is that all the curves, which are reported on log-scale on the \(y\)-axis (here represented by the actual calculated values) show a more or less large convex portion (i.e., negative second derivative), which means that the tortuosity trend increases with decreasing porosity with a functionality less than exponential. A possible reason for such a behaviour is that, just after particles start overlapping because of decreasing inter-particle distance, the narrow channels present just before overlapping disappear because of the surface-surface merging, resulting in a certain limitation for tortuosity to increase.