Theoretical and experimental studies of electrical conductivity for functionally graded, heterogeneous surfaces

A theoretical approach for estimating solutions to Maxwell's equations for structures with spatially-varying electromagnetic properties is presented for conductive media containing surfaces modified with functionally graded, heterogeneous electrical conductivity. The basis of the approach is an equivalent depth technique that replaces a graded conductivity region consisting of a phase mixture with a series of thin layers with uniform, multi-phase properties locally matching the effective mixture properties of the graded region. Radio frequency field propagation within each layer is determined as if it had existed within a constant conductivity medium but its depth is electromagnetically equivalent to the replaced graded region existing prior to the layer. The equivalent depth approach was applied to planar, thin foil, and cylindrical media to enable comparison with experimental results. Model predictions were compared with total transmission results for Pt-doped titanium thin foils and steady-state temperature rise in closed wire loops made from Sn-modified copper wire. For the thin foil case, the model-predicted total transmissivity shows good agreement with trends in the experimental results due to property changes in the modified surface layers. In the cylindrical wire case, similar agreement between the predicted effective conductivity values for the modified layers and experimental results was observed. Thus, the equivalent depth approach is an effective method for estimating solutions to Maxwell's equations in complex media and a useful tool for predicting the performance of tailored surface conductivity modifications.