Relationship between Pore Connectivity and Mean Pore Size in Modulated Mesoporous Vanado-Phosphoro-Aluminates and Some Similarities with the Branching of Trees

Sixteen mesoporous vanado-phosphoro-aluminate solids, of the general formula Al 100 P X V Y ( X, Y = 0, 5, 10, 20), have been prepared and characterized by their N 2 adsorption/desorption isotherms. The materials were truly mesoporous, and the addition of P and/or V affects in a very precise and profound way the corresponding specific surface areas S p (m 2 g - 1 ), specific pore volumes (cm 3 g - 1 ), the mean hydraulic pore diameter d p (4 V p / S p ) (nm), the maximum of the pore size distribution (psd) d max (nm), and the full width at half-maximum (fwhm) (nm) of the psd. The pore connectivity c was also determined and the dimensionality D cc of adsorption was estimated in the pressure range where capillary condensation of N 2 in the pores takes place. The dimensionality D cc was found to be related to the d max, reduced over the fwhm, via the relation D cc ≈ 3.7( d max /fwhm). The two parameters, dimensionality D cc and connectivity c, are also related to each other via the simple relation log c ∝ −1.83 log D cc, which in turn results in the relation log c ∝ −1.83 log( d max /fwhm). A parallelism is drawn between the connectivity of pores in solids and its relation to the pore size, on one hand, and the degree of branching b of trees and its relation with the branch diameter d k, on the other. Using allometric relations already established for such hierarchically developed natural systems, that is, trees, we reach a relation between b and d k of the form log b ∝ −2.34 log( d k,max /2 d mean ) where d k,max is the average diameter of the higher order (thinner) branches in a typical tree and d mean is the arithmetic mean diameter of all branches of that tree. The reasons for these similarities are discussed.