(a) Cross-sectional FCOM image of a zeolite membrane reconstructed from (b) FCOM focal planes, depicting both (1) vertical and (2) horizontal grain boundaries in parallel and series, respectively, with crystal grains (dark regions) constituting the membrane layer. Membrane schematics (c) are also shown to depict the single-crystal and polycrystalline model membranes motivated by the single crystal and type 2 polycrystalline regions of (a).
Flux calculated via KMC simulations with dynamic boundary conditions (open symbols) compared to that predicted via the Darken approximations (comparable closed symbols) as a function of inverse membrane thickness, , over a range of temperature under a gradient in pressure of approximately (low-pressure reservoir held at ).
Temperature dependence of the flux through a -thick membrane under gradients of (a) and (b) for dynamic (diamonds) boundary conditions, compared to the flux resulting from enforcing local equilibrium (LE) loadings consistent with bulk isotherms (circles) and symmetric membrane isotherms (squares). Darken approximations are shown for bulk isotherm LE (solid circles) and LE concomitant with the dynamic boundary loading (solid triangles). The dashed lines in panel (b) depict Arrhenius behavior based on linear extrapolation from the low-temperature data.
Local equilibrium (LE) ratio as a function of inverse membrane thickness, , over a range of temperatures for the high-pressure side of the membranes studied in Fig. 2.
Sensitivity of coarse-grained axial loadings at two temperatures to the enforcement of local equilibrium (LE) loading at the membrane boundaries, consistent with either the symmetric membrane isotherm (thick line) or the bulk isotherm (squares).
Flux through the polycrystalline membrane of grains (of total crystal thickness ) separated by grain boundaries (GB) relative to the flux through the single-crystal membrane of thickness under a gradient in pressure. Grain boundaries are each in width, but the finite size is replaced for clarity with a vertical dotted line denoting each grain boundary. Error bars reflect the standard deviation of the flux through the membrane at steady state, and are smaller than the symbols at and for the single grain-boundary case at . Insets are schematics of the polycrystalline membrane model where single-crystal grains (hashed region) are separated by grain boundaries (shaded spacing between hashed region).
Temporally-averaged loading of weak (, circles) and strong (, squares) binding sites through a polycrystalline membrane consisting of grains (as shown in schematic inset) under a gradient in pressure at (a) and (b) . The average pressure in each of the seven grain boundaries (positions denoted by vertical dotted lines) is also shown (triangles).
Probability distribution function (pdf) of the time-average concentration of molecules in the first grain boundary (nearest the high-pressure boundary denoted as GB1 in the inset schematic) normalized to the benzene/ saturated lattice concentration for the polycrystalline membrane discussed in Fig. 6. Magnification of the higher-concentration region is shown in the inset.
Relative contribution of the flux pathway to the overall flux at (squares) and (circles) corresponding to the symmetric membrane enforced local equilibrium simulations of Fig. 5.
Loading dependence of the relative contribution of the and flux pathways to increases in the benzene mean-square displacement during canonical ensemble simulations of benzene in a fully periodic lattice.
Summary of apparent activation energies for benzene in zeolite under equilibrium and nonequilibrium conditions.
Article metrics loading...
Full text loading...