Multicomponent Maxwell-Stefan Diffusivities at Infinite Dilution

Abstract

Diffusion plays an important role in (bio)chemical processes. It is usually difficult to obtain Maxwell−Stefan diffusivities from experiments as well as molecular simulation. Therefore, predictive models based on easily measurable quantities are highly desired. The Vignes equation is commonly used to describe the concentration dependence of Maxwell−Stefan diffusivities. In mixtures containing at least three components, the generalized Vignes equation requires the value of the quantity Đijxk→1, which describes the friction between components i and j when both are infinitely diluted in component k. Over the past decades, several empirical models were proposed for estimating Đijxk→1, and all of these are lacking a sound theoretical basis. In this study, we show that Đijxk→1 actually exists (i.e., its value does not depend on the molar ratio xi/xj), and we derive an analytical expression for Đijxk→1 that is based on the linear response theory and the Onsager relations. We find that Đijxk→1 can be expressed in terms of binary and pure-component self-diffusivities and integrals over velocity cross-correlation functions. By neglecting the latter terms, we obtain a convenient predictive model for Đijxk→1. Molecular dynamics simulations are used to validate the assumptions made in this model. The following test systems are considered: a ternary system consisting of particles interacting using Weeks−Chandler−Andersen (WCA) interactions and the ternary systems n-hexane−cyclohexane−toluene and ethanol−methanol−water. Our results show that (1) for the WCA system, as well as the n-hexane−cyclohexane−toluene system, neglecting the integrals over velocity cross-correlation functions results in accurate predictions for Đijxk→1; (2) for the WCA system, our model prediction is superior, compared to the existing models for Đijxk→1; (3) in the ethanol−methanol−water system, the integrals over velocity cross-correlation functions cannot be neglected, because of the presence of hydrogen bonds (models for predicting Đijxk→1 in this system will require detailed information on the collective motion of molecules); and (4) our model may provide an explanation why the Maxwell−Stefan diffusivity describing the friction between adsorbed components in a porous material is usually very large.