Some pitfalls in the use of the Knudsen equation in modelling diffusion in nanoporous materials

Abstract

The Knudsen model of diffusion in small pores, originally verified in macropores, is widely applied at the mesopore scale with adsorption effects neglected, largely based on linearity of the √ T/M correlation. Here, we show that this approach is misleading, and that the correlation masks inconsistencies arising from neglect of van der Waals forces in the Knudsen model. We examine the tortuosity for diffusion of light gases in nanoporous carbons using the Oscillator model of low pressure transport developed in the first author's laboratory, which incorporates van der Waals interactions. Pore network effects are considered through a hybrid correlated random walk-effective medium theory approach. It is shown that in the presence of a pore size distribution the apparent tortuosity is not a porous medium property alone, but depends also on the temperature and on the diffusing molecule, because of the temperature and the gas-dependent short circuiting effects associated with pores that have high conductance. This short circuiting effect leads to a complex and rich variety of behaviour with respect to pore size, temperature and diffusing gas, which is consistent with experimental evidence, but is absent when the Knudsen model is used with adsorption effects neglected. It is shown that when effects of adsorption on equilibrium and transport are overlooked the commonly used correlation of diffusivity with √ T/M is deceptive, as the product of the adsorption equilibrium constant and diffusivity (or pore conductance) also approximately scales linearly with the Knudsen diffusivity (i.e. with √ T/M ). Such behaviour is found for diffusion in mesoporous carbons and silica as well as in silicon. Consequently, claims of validity of the Knudsen model based on such a correlation may be misconceived.