# AFM Characterization of Track-Etched Membranes:
## 4 Characterization Parameters and Model Calculations  
In order to characterize TM, we suggest using the following parameters and their distributions:  
* pore diameters \(D\) and their distribution \(P_{\text{D}}\);
* the relative number of somehow defected pores \(\eta_{\text{defect}}\);
* distances between the nearest pores \(R_{\text{min}}\) and their distribution \(P_{\text{Rmin}}\);
* pore density \(\rho\) (one can also characterize the spatial inhomogeneity of pores distribution by finding the distribution \(P_{\rho}\) of pores densities in different regions of the sample, i.e., search for density fluctuations);
* presence/absence of \"anisotropy\" of pores distribution over a surface. The number of defective pores was calculated in the following way. The pore was considered defective if the approximate parameter \(D\) exceeded by more than \(2\sigma\) the average size <\(D\)> of all pores (here \(\sigma\) is the standard deviation of \(P_{\text{D}}\)). Two adjacent pores were considered defective pores also if the distance between their centers \(R_{\text{min}}\) was comparable to their sizes \(D\). To determine the presence of \"anisotropy\" of pores distribution over a surface, we investigated the distribution of the angles \(\varphi\) of the segments connecting the centers of the nearest pores with respect to the direction specified for the sample used. The histogram of the distribution \(P\varphi\) (some histogram show the orientations of the nearest pores pairs) was plotted in the polar coordinate system (see Figure 2g-i). Each petal in such a ring histogram represents the relative number of nearest pores pair, which are oriented with respect to each other with a certain angle bin. Figure 2: Model pores spatial distributions: fully ordered in a squared net (**a**), random homogeneous (**b**), random with a preferred orientation of the nearest pores pairs (**c**). Corresponding distributions of the minimum distances \(P_{\text{Rmin}}\) between the nearest pores (**d–f**) and orientation diagram \(P\varphi\) in polar coordinates—the distributions of the values of the inclination angles of the segments between the centers of the nearest pores (**g–i**). To investigate these parameters and distributions in porous films with known parameters, we performed some model calculations. The above-mentioned distributions were found for three different model cases (pore spatial distribution patterns):  
1. a strongly ordered spatial distribution, when the pores were located in the nodes of the square grid with a certain period \(R\);
2. a random homogeneous distribution with a certain surface density \(\rho\);
3. a distribution with the preferred orientation distribution of the pores, i.e., \"anisotropy\" (here, the nearest pores were oriented with respect to each other according to a preferred angle \(\varphi\)) and with three selected preferable minimal distances between nearest pores (\(\sim\)3, 4, and 7 a.u.). For all three cases, the size of the model sample surface was 1000 a.u. per 1000 a.u, and the number of generated pores was 10,000, with a diameter of 1 a.u. For each case, the histograms of the distances distribution \(P_{\text{Rmin}}\) between the nearest pairs of pores and the histogram of the orientations of the nearest pores pairs \(P\varphi\) were studied (see Figure 2). As we see in Figure 2, for a fully ordered case (pores in nodes of the squared net), the distribution of distances between nearest pores pairs is the delta function (Figure 2d), and the orientation diagram has only four petals at angles of 0, 90, 180 and 270\({}^{\circ}\) (Figure 2g). For the random homogeneous distribution, there was a (quasi) normal distribution \(P_{\text{Rmin}}\) (Figure 2e) and a ring histogram \(P\varphi\) without any preferred orientation (Figure 2h).