Morphometry of Porous Solids:  Lacunarity, Fractal Dimensions, Connectivity, and Some Topological Similarities with Neurons

The topological morphometry of 16 mesoporous phosphoro−vanado−aluminate solids possessing random porous network and 5 Al-modified MCM 48 materials, possessing ordered porosity, were investigated using the parameters lacunarity (L), fractal dimension (D v) and connectivity (c) of their porous structure. The phosphoro−vanado−aluminate group of materials employed had the general formula Al100P X V Y , with X, Y = 0, 5, 10, 20, the balance being oxygen. The Al−MCM 48 solids contained Al in 0, 5, 10, 15, and 20%. The porosities of the solids were determined using standard N2 adsorption−desorption measurements. The BJH methodology was employed for the determination of the pore size distribution (psd) curves for the Al100P X V Y solids while for the Al−MCM 48 materials the psd was estimated using the Howarth−Kawazoe method. From the psd curves, the L of the solids was determined using the formula L = M(2)/[M(1)]2, suggested previously by Allain and Cloitre, where M(1) and M(2) are the first- and second-order momenta of distribution. The D v of the solids was estimated from plots of the form ln V p = f[ln(ln(P o/P))] suggested originally by Avnir and Jaroniec, where V p (cm3 g-1) corresponds to the specific pore volume of the porous materials. Finally the connectivity c of the porous network was determined according to the method of Seaton. The quantities L and c for all materials were found to be interrelated via the relationship: ln L = 0.24 − 1.62 ln c, while the quantities D v and L are interrelated via the equation D v = 2.47 − 1.41L. Of interest is the fact that a relationship between D and L, similar to that described above for porous networks, has also been observed previously by Smith and Lange for neurons. The physical meaning underlying this kind of heterosimilarity is discussed from the point of view of natural necessity imposed on the development of such seemingly dissimilar systems that is pores and neurons.