A Model To Determine Local, Microscopic Flux Profiles within Composites from the Measured, Macroscopic Flux:  Application to Poly(4-vinylpyridine) Composites

Abstract

For any heterogeneous matrix with a parametrizable geometry, a model is provided to determine a microscopic, local flux profile within the matrix from the macroscopic, experimentally measured flux of a probe species. By analyzing the dependence of the measured flux on the geometric parameters that describe the matrix, a mathematical function describing a local flux profile within the matrix is obtained. Example applications of the model include composites of Nafion and neutron track-etched polycarbonate membranes. Composites of neutron track-etched polycarbonate membranes and 20% cross-linked poly(4-vinylpyridine) are also described. The measured flux of hydroquinone through the composites varies with r o -1 , where r o is the pore radius, a characteristic dimension of the matrix. The measured flux is characterized by a second-order polynomial in r o -1 . The local flux profile is found by application of the model and is shown to be higher toward the center of the pore. Poly(4-vinylpyridine) is cross-linked with various dibromoalkanes, with 2-8 and 12 carbons. A quadratic relationship is found between the measured flux and the chain length of the cross-linking agent, with a minimum for 1,6-dibromohexane. Effects of inter-and intrachain cross-linking are discussed.

Characterizing Flux through Micro- and Nanostructured Composites

“…The total flux is set by both J l , k and f l , k . If the local flux is high but sustained over a very small fraction of the matrix, then the total flux will be little affected …”

“…Example 2. A generic equation has been provided to characterize the local flux profile inside of micro- and nanostructures . In a system where only one characteristic dimension, x 0 , is important, and experiments are performed to measure the variation in flux with x 0 , then the local flux profile inside the structure ( J l ( x 0 , x )) for a datum at a given x 0 is related to the measured flux ( J ( x 0 )) as The geometric functions, g 1 ( x 0 ) and g 2 ( x 0 ) depend on the geometry of the system.…”

“…Equation 11 has been applied to composites formed by sorbing poly(4-vinylpyridine) into cylindrical pores . In this case, flux is higher at the center of the pore than at the edge.…”

“…In this case, flux is higher at the center of the pore than at the edge. Packing of monomers inside the pore leads to a response in contrast to the Nafion composites where the interface facilitates flux …”