Dense fluid transport through nanoporous graphene membranes in the limit of steric exclusion

We investigate transport of dense fluid flow through nanoporous membranes in the limit of steric exclusion using molecular dynamics (MD) and finite element simulations. Simulation results suggest that, for simple fluids, deviations from Sampson flow are a consequence of the competition between slip and finite atomic size effects. The latter manifest themselves by introducing an effective pore size, as well as an effective membrane thickness. We propose an analytical model for the membrane permeance that accounts for all these factors. We also show how this model can be modified to describe transport of low molecular weight aromatic hydrocarbons across these membranes in the steric limit. Extensive validation of this model is conducted through MD simulations of Lennard-Jones fluids permeating single- and multilayer graphene membranes, as well as low molecular weight organic liquids permeating single-layer graphene membranes.

Figure 1

Membrane geometry and FE calculations for the permeance across a single-layer membrane. (a) Model geometry for single-layer and multilayer membranes. Left: Single-layer membrane; right: multilayer membrane. Fluid atoms are shown in blue, atoms at the edge of the pore (functionalization) in red and atoms on the membrane in green. The blue line represents the boundary that fluid atoms can reach. $L$ represents the respective membrane width (pore length) for use in model (9). (b) Computational domains for FE calculations. The membrane surface is denoted by a blue line. (c) FE results for the reduced permeance as a function of reduced radius. Blue line corresponds to Sampson result for a pore of radius $R-\delta$. Red dots and hollow squares denote FE results for the full-slip case with domain I and domain II, respectively, while the solid curved line denotes the approximate model given by Eq. (5). Black dots and hollow squares denote FE results for the no slip case with domain I and domain II, respectively, while the solid line denotes the approximate model given by Eq. (3).

Figure 2

MD simulation results. (a) MD simulation geometry. Carbon atoms on graphene membrane shown in green, hydrogen atoms in red, and LJ fluid shown in blue dots. (b) Reduced permeance in MD simulations as a function of reduced radius with $\delta$ chosen as discussed in the text. Circles denote MD results. The red solid line denotes the full-slip model given by Eq. (5). The shaded regions show the range of values of ${\tilde{R}}_{\delta }$ for each fluid over which the permeance reduces to zero due to steric effects. (c) Lennard-Jones fluid permeance as a function of nominal radius. Circles denote the MD results. Solid lines denote the full-slip model given by Eq. (5) with corresponding $\delta$. Inset: Details of permeance in the small pores. (d) Normalized axial flow velocity as a function of radius for $R=5.9$. Circles represent MD simulation results with error bars determined based on the standard error. Red solid lines correspond to additional FE simulation results with $R=5.9$ and $\delta$ as determined for the comparisons in (b) and (c) (see text for details). Blue solid lines correspond to Sampson's velocity profile for a pore of radius $R$ [Eq. (8)]. Black solid lines correspond to Sampson's velocity profile for a pore of radius $R-\delta$. The vertical gray dashed line represents $R-\delta$.

Figure 3

Simulation results using the shifted LJ potential (13). (a) Reduced permeance as a function of ${\tilde{R}}_{{\delta }^{\prime }}$ from MD simulations using the shifted potential. (b) Density distribution near the pore for varied shift parameter $\mathrm{\Delta }$.

Figure 4

Flow resistance in LJ units as a function of membrane thickness obtained from MD simulations for three different pore sizes. Solid lines denote model given by Eq. (9) with $l_{\mathrm{s}}=0.14$. Inset: Change in resistance due to stronger fluid-solid interactions in a medium-sized pore; dashed line denotes model given by Eq. (9) with $l_{\mathrm{s}}=0$.

Figure 5

(a) Molecular structure [47] and (b) permeance of $\mathit{o}$-xylene, $\mathit{p}$-xylene, benzene, $\mathit{n}$-octane, and $\mathit{i}$-octane. Circles denote MD results. Solid lines denote model given by Eq. (5). Inset in (b) shows detailed comparison for small pores.

Figure 6

Effective radius and extended permeance model for aromatic liquids. (a) Definition of characteristic width $d_{\mathrm{eff}}\left(\theta \right)$ and (b) its variation with $\theta$ for benzene. (c) Schematic of effective radius $R_{\mathrm{eff}}$ for aromatic liquids, where red and blue dots represent the hydrogen atoms at the edge of the pore and end of the aromatic molecules, respectively. (d) $R_{\mathrm{eff}}$ for benzene. (e) Permeance of $\mathit{o}$-xylene, $\mathit{p}$-xylene, and benzene; circles denote MD results; solid lines denote extended model given by Eq. (10) with $R_{\mathrm{eff}}={\langle R_{\mathrm{eff}}\rangle }_{\theta }$ as given by Eq. (11).