Heat and Charge Conductivities in Superlattices: Two-Scale Measuring and Modeling

Conventional reasoning and established procedures for measurement of heat and charge conductivities at the continuum micrometer scale, or higher scales, results in a number of variables and physical entities being the subject of measurement. These variables themselves are not point values if to define them with the lower scale concepts. When the media overall properties are sought, their dependence on lower (smaller) scale physical phenomena and their mathematical descriptions need to be considered and incorporated into the higher (larger) scale description and mathematical modeling. This is not a new problem. How to treat or solve multi-scale problems is the issue. Effective scaled heat and charge conductivity are studied for a morphologically simple 1D layered heterostructure with the number of components being n ≥ 2, the effective scaled heat and charge conductivities. It is a two-scale media with the lower scale physics of energy and charge carriers being described by commonly used models. A continuum ↔ continuum description of ηm ↔ μm transport of electron-phonon energy fields, as well as the electromagnetic and temperature fields for ηm scale coupled with the microscale (μm) mathematical models are studied. The medium is heterogeneous because it has multiple phases, volumetric phases 1, 2, 3 .... and (n+m) phases that are the interfaces between volumetric phases. The fundamental peculiarities of interface transport and hierarchical mathematical coupling bring together issues that have never actually been addressed correctly. It is shown that accurate accounting for scale interactions and, as is inevitable in scaled problems, application of fundamental theorems to a scaled description of the Laplace and ▽ operators bring to the upper scales completely different mathematical governing equations and models. We have conducted and report some preliminary quantitative assessment of the differences between the static upper scale and transient nanoscale transport coefficients and show how the lattice morphology and its irregularities influence the effective conductivities.