Generalized processing-structural functions for porous materials

An equation connecting a generalized overall structural property Y with a generalized processing variable X was proposed to monitor the structural evolution of porous materials. The equation PVp = const (P—mechanical strength, Vp—pore volume) suggested previously for granulated catalysts (Titelman 1989) was transformed into a generalized function for any porous material Y = f(X). Y is considered to be internal energy (work) establishing a conjugate pair of force/displacement, both are generalized. This was obtained in the form of a pore-to-wall volume ratio. For ordered mesoporous materials, this ratio was reduced to a ratio of pore to wall cross-sectional areas and further to a ratio of pore diameter to wall thickness. An additional type of Y is the ratio of the cross-sectional area of the pore shell to that of struts. X is any processing variable of synthesis and post-synthesis treatment of the material (quantitative, non-quantitative and a set of single variables). For X, a method of ranking X by Y was used. Large amounts of published data were recalculated, and typical sections of Y = f(X) (constant, linear, etc.) in different X ranges were found. Obtained dependencies allow for the monitoring of both collective and individual effects of independent variables on Y, discovering energetic-wise equal and similar states of material, controlling material stability, selecting optimal sample, correcting some published data and notably expanding on the information obtained from regular adsorption measurements. In the graphical abstract, the geometrical characteristics of the ordered mesoporous material unit cells with square (a) and hexagonal (b) pore arrangements are shown: shell and strut areas (a, b) and equivalent wall thickness (c); A- cross-sectional surface area;···A u.c. = a 0 2 (a), A u.c. = a 0 2 / (√3/2) (b), Ap = πD p 2 /4; Aw = Au.c.−Ap; Ash = π(a 0 2 −D p 2 )/4 , Astr = Au.c.−Ash and t eq = Aw/(πDp/2).