A density functional theory of diffusion is developed for lattice fluids with molecular flux as a functional of the density distribution. The formalism coincides exactly with the generalized Ono-Kondo density functional theory when there is no gradient of chemical potential, i.e., at equilibrium. Away from equilibrium, it gives Fick's first law in the absence of a potential energy gradient, and it departs from Fickian behavior consistently with the Maxwell-Stefan formulation. The theory is applied to model a nanopore, predicting nonequilibrium phase transitions and the role of surface diffusion in the transport of capillary condensate.

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The dependence of the diffusivity on temperature, pressure, and composition is not understood well; consequently, data is preferred significantly over correlations in most practical situations. Even in dilute gases, the contributions of attractions and repulsions to the diffusivity are difficult to understand on a molecular level without performing simulations. We have developed a Lattice Density Functional Theory (LDFT) approach for modeling diffusion to supplement existing methods that are very rigorous but computationally demanding. The LDFT approach is analogous to the van der Waals equation in how it accounts for molecular interactions in that it has first-order corrections to ideal behavior; it is an extension of the Equilibrium LDFT for adsorption and phase behavior. In this work, the LDFT approach is presented and demonstrated by modeling the problem of color counterdiffusion in an externally-applied potential field. This potential field, in combination with the intermolecular potential function, creates a diffusion regime in which repulsions cause oscillations in the density profile. Using the LDFT approach, the oscillations were described and attributed to nearest-neighbor and next nearest-neighbor interactions. The LDFT approach gives qualitative and quantitative agreement with dual control-volume Grand Canonical Molecular Dynamics simulations.

Membrane permeability and flux of pure gases can exhibit maxima with respect to temperature. For zeolites, this has been explained as a competition between surface and nonsurface diffusion within pores and as a process that depends on the diffusive activation energy and the heat of adsorption. This behavior is reproduced for nanoscale pores by using the lattice density functional theory approach for modeling diffusion. The approach can give expressions for the permeability of noncondensable fluids through nanoscale pores in terms of bulk densities and intermolecular interactions; it also gives the following molecular explanation for the permeation maximum:  with decreasing temperature, (i) attractions at the pore's entrance increase permeation because molecules in the pore experience difficulty in back-diffusing to the feed, and (ii) at even lower temperatures, attractions at the pore's exit reduce permeation because molecules in the pore experience difficulty in escaping from the walls at the end of the pore. The approach shows that permeation maxima can occur without competition between surface and nonsurface diffusion. However, when this competition occurs, the maximum in the surface flux leads to the overall permeation maximum with respect to temperature.

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