Adsorption isotherms of fractal surfaces

Adsorption of inert molecules on fractal surfaces by Van der Waals (VDW) interactions has long been proposed as a method to probe the fractal dimension D but convincing evidence is still missing after more than a decade. We have used small angle scattering (SANS) to determine the pore surface dimensions for three shale samples and carried out nitrogen adsorption studies. We found that analyzing the adsorption isotherm according to the available theoretical predictions always resulted in D values that are lower than those obtained by SANS. There are also systematic deviations from the theoretical curve common to all three samples. To understand the origin of these problems, we used a power law distribution of uncoupled spherical pores as a representation of a fractal surface and carried out numerical simulations of the adsorption isotherm. We found that the simulated isotherms exhibit the same general features observed in the experiment. A crossover between the VDW effect and capillary condensation always occurs and it precludes the use of isotherms to determine D in real systems.