Universal Scaling Formalism and Analytical Optimization Criterion for Multiscale Geometric Design of Thermoelectric Metamaterials

Abstract

Thermoelectric (TE) generators directly convert heat into electricity, yet their performance is often limited by small temperature gradients. Width-modulated metamaterials with constrictions and expansions (\textit{constricted} geometries) sustain larger temperature differences $\Delta T$ than constant-width counterparts due to reduced \textit{Transmissivity} ($Tr$)—the geometric ratio of constriction to expansion cross-sections. A scaling behavior of transport and key TE performance metrics with \textit{Transmissivity} is demonstrated from the nanoscale to the macroscale using analytical formalism and simulations across single- and multiple-constriction profiles. It is shown that $\Delta T$, electrical and thermal resistances, efficiency, and output power are governed by a single scaling function $g(Tr)$—the conductance of a constricted geometry relative to a uniform-width counterpart—independent of carrier type, material, or operating conditions. Universal scaling formalism leads to \textit{Performance Design Maps} and an analytical optimization criterion: maximum TE performance occurs at an optimal \textit{Transmissivity} $Tr_{opt}$, where $g(Tr_{opt})=Bi$, with $Bi = hL/k$ denoting the Biot number and $h$, $L$, and $k$ the convection coefficient, length, and thermal conductivity, respectively. Compared with the uniform geometry, the optimal constricted geometry produces a maximum output power reduced under fixed $\Delta T$, by a factor of $Bi/4$, and enhanced under identical convective operating conditions by a factor of $(1+Bi)^2/(4Bi)$. This work provides a theoretical framework for multiscale design and optimization of \textit{constricted} geometries, thereby enabling systematic exploration of design strategies for next-generation TE modules based on advanced thermoelectric metamaterials analogous to nature’s hierarchical structures for optimized functionality.