A Lattice Model for the Simulation of Diffusion in Heterogeneous Polymer Systems. Simulation of Apparent Diffusion Constants as Determined by Pulse-Field-Gradient Nuclear Magnetic Resonance

Abstract

The diffusion of molecules through heterogeneous media is not simple Fickian diffusion on the length scale of the heterogeneities. Pulse-field-gradient (PFG) NMR measurements of diffusion reflect this by yielding apparentdiffusion constants that are dependent on the time over which diffusion is observed. Equations for simple cases of tortuous diffusion and restricted diffusion are available where some fraction of the medium is assumed to be permeable and some fraction is impermeable. In many polymer systems, the medium has domains that are more permeable and less permeable. In the limit of long times and above the percolation threshold, diffusion in such systems can be characterized with effective medium theory (EMT). To characterize diffusion as a function of time in heterogeneous media and at all compositions, a lattice model is presented based on an approach developed by Ediger. The lattice model gives results that match EMT at long times and produces tortuous and restricted diffusion above and below the percolation limit when one domain is made impenetrable. It also shows a wide range of behavior of the apparent diffusion constant, relative to observation time, which is intermediate between tortuous diffusion and restricted diffusion. The intermediate behavior arises for various ratios of the diffusion constant in the more- and less-permeable domains and for various ratios of solubility between the more- and less-permeable domains. Brief comparisons are made with PFG NMR data on a heterogeneous, high-permeability copolymer and on a polymer blend composed of more- and less-permeable constituents. The model demonstrates that apparent diffusion data from PFG NMR provide a new structural view of the heterogeneities in systems with more- and less-permeable domains.