Density functional theory of adsorption in spherical cavities and pore size characterization of templated nanoporous silicas with cubic and three-dimensional hexagonal structures

Adsorption in spherical cavities is studied by the nonlocal density functional theory (NLDFT). Theoretical results are compared with experimental data on ordered nanoporous materials with cubic Pm3n (SBA-1, HMM-3), cubic Im3m (SBA-16), and 3D hexagonal P6₃/mmc (SBA-2, SBA-12) cagelike structures. Quantitative comparison shows that capillary condensation of N₂ at 77 K in sufficiently small cavities (pore diameters 3 < D < 6 nm) occurs reversibly; the equilibrium condensation pressure is determined by the cavity diameter. In the case of hysteretic isotherms on materials with cavity diameters of >ca. 6 nm, the capillary condensation step corresponds to the theoretical limit of stability of the metastable adsorption film. For pores wider than ca. 10 nm, this limit is approximated by the macroscopic Derjaguin-Broekhoff-de Boer equations. Desorption from cavities of >6 nm is controlled by the size of the windows that connect the cavity with the bulk fluid. If the diameter of the window is below ca. 4 nm, desorption occurs via spontaneous cavitation of condensed liquid. We developed a NLDFT method for calculating pore size distributions (PSD) of cavities, the amount of intrawall porosity, and, in combination with X-ray diffraction, the wall thickness in siliceous materials with cagelike pores. We demonstrate that the adsorption method allows one to differentiate between the materials of different morphological symmetry. For regular cagelike structures, the NLDFT results are in remarkably good agreement with the estimates derived from geometrical considerations. In contrast, the conventional Barrett-Joyner-Halenda method of PSD analysis, based on the Kelvin equation, underestimates the pore sizes in cagelike nanopores by up to 100%.