  Vapor adsorption experiments are widely used to assess pore size
distributions, but the large hysteresis sometimes observed between sorption and
desorption isotherms remains difficult to interpret. Such hysteresis is
influenced pore network connectivity, which has previously been modeled by
percolation on infinite lattices. Our hypothesis is that percolation occurs
instead through finite networks of micropores connecting accessible macropores,
always exposed to the outside environment. We derive a general formula for
sorption/desorption isotherms that introduces a simple measure of hierarchical
pore connectivity -- the fraction $f$ of always exposed pores. The model thus
accounts for "small world" connections in finite-size percolation, while also
incorporating other hysteresis mechanisms, in single-pore filling, liquid
insertion into the solid matrix, and cavitation. Our formula is able to fit and
interpret both primary and scanning sorption/desorption isotherms for a variety
of adsorbates (noble gases, water, and organics) and porous materials (cement
pastes, dental enamels, porous glasses, carbon black and nanotubes), including
cases with broad pore-size distributions and large hysteresis. It allows
quantification of the prevalence of percolating macropores in the material,
even though these pores are never filled during the sorption experiments. A
distinct bump in sorption isotherms is explained as a lowering of the barrier
to nucleation of the vapor phase with a universal temperature scaling.
