The total and the differential mean pore anisotropy in porous solids and the ranking of pores according to Zipf's law

Abstract

In this work the property of total pore anisotropy in porous solids is introduced. Its calculation is based on a combination of the specific surface area S_p and the specific pore volume V_p estimated via typical nitrogen porosimetry data and tested in two kinds of porous materials: a group of spinels CoAl₂O₄ with differentiated random porosities and a second group of silicas SiO₂ with ordered porosity modulated by the addition of LaFeO₃ nanoparticles. Two basic complementary expressions of total pore anisotropy were estimated: (i) the specific total mean pore anisotropy b_mean,total = (N·b) ≈ [S_p³]/[V_p²] corresponding to the total anisotropy value of all N hypothetical similar pores in one gram of a solid with mean size D_mean = 4V_p/S_p and anisotropy b = L_total/D_mean. The b_mean,total takes a unique value for each particular porous material. (ii) The specific differential mean pore anisotropies b_mean,diff = (N_i·b_i) ≈ [S_pi³]/[V_pi²] corresponding to the spectrum of partial anisotropy values b_mean,diff = L_i/D_i of N_i pores with similar size D_i = 4V_i/S_i possessing differential pore volume V_pi and differential specific surface area S_pi. The b_mean diff takes different values at each particular partial pressure and exhibits a distribution as a function of pore radius b_mean,diff = f(r_i). It is shown that plots of log(b_mean,diff) = f(log(r_i)) lead to the ranking of pores according to the Zipf's law log(N_i) = A − B log(V_pi). This ranking is not obeyed by the pores exhibiting appreciable local pore anisotropy.