A fundamental study of the extent of meaningful application of Maxwell's and Wiener's equations to the permeability of binary composite materials. Part I: A numerical computation approach

Abstract

In studies of the permeability of composite materials, consisting of solid isometric particles A dispersed in a polymeric matrix B, Maxwell's classical equation, rigorously valid only for very low fractional volumes (vA) of spherical particles, is often used for data analysis up to much higher vA values. Theoretical justification for this practice can be provided, only up to vA≈0.5, by current analytical models based on cubic lattices of congruent spheres. Replacing spheres by cubes yields a model covering the full composition range vA=0−1, but lacking convenient general analytical tractability. Accordingly, to explore unrestrictedly the practical validity of the Maxwell equation, a simple cubic lattice-of-cubes model was combined with an appropriate numerical computation tool. The results establish satisfactory applicability of the Maxwell equation for isometric particles A, at all compositions and over a wide range of component permeability ratios (α=PA/PB=0−100).The practical applicability of the corresponding classical equation of Wiener (which extends Maxwell's treatment to anisometric particles via the value of a single geometrical parameter AW) was similarly explored, using appropriate model s.c. lattices of (a) unidimensionally anisometric particles (transverse square rods of varying length) or (b) bidimensionally anisometric particles (transverse square platelets of varying area). The computations covered the same vA and α ranges as above, as well as a range of aspect ratios in each of the cases (a) and (b). The results obtained enable determination of values of AW, independent of vA and α, linked directly to the aspect ratio of the embedded particles, and demonstrate very good practical applicability of the Wiener equation (with the proper AW value) under all conditions studied.