Understanding Gas Transport in Polymer-Grafted Nanoparticle Assemblies

We rationalize the unusual gas transport behavior of polymer-grafted nanoparticle (GNP) membranes. While gas permeabilities depend specifically on the chemistry of the polymers considered, we focus here on permeabilities relative to the corresponding pure polymer which show interesting, "universal" behavior. For a given NP radius, R_c, and for large enough areal grafting densities, σ, to be in the dense brush regime we find that gas permeability enhancements display a maximum as a function of the graft chain molecular weight, M_n. Based on a recently proposed theory for the structure of a spherical brush in a melt of GNPs, we conjecture that this peak permeability occurs when the densely grafted polymer brush has the highest, packing-induced extension free energy per chain. The corresponding brush thickness is predicted to be $$ , independent of chain chemistry and σ, i.e., at an apparently universal value of the NP volume fraction (or loading), ϕ_NP, ϕ_NP,max = [R_c/(R_c + h_max)]³ ≈ 0.049. Motivated by this conclusion, we measured CO_-2 and CH₄ permeability enhancements across a variety of R_c, M_n and σ, and find that they behave in a similar manner when considered as a function of ϕ_NP, with a peak in the near vicinity of the predicted ϕ_NP,max. Thus, the chain length dependent extension free energy appears to be the critical variable in determining the gas permeability for these hybrid materials. The emerging picture is that these curved polymer brushes, at high enough σ behave akin to a two-layer transport medium - the region in the near vicinity of the NP surface is comprised of extended polymer chains which speed-up gas transport relative to the unperturbed melt. The chain extension free energy increases with increasing chain length, up to a maximum, and apparently leads to an increasing gas permeability. For long enough grafts, there is an outer region of chain segments that is akin to an unperturbed melt with slow gas transport. The permeability maximum and decreasing permeability with increasing chain length then follow naturally.