Maximal rectification ratios for bi-segment thermal rectifiers

We study bi-segment thermal rectifiers whose forward heat fluxes are greater than reverse counterparts. Presently, a shortcoming of thermal rectifiers is that the rectification ratio, namely the forward flux divided by the reverse flux, remains too small for practical applications. In this study, we have managed to discover and theoretically derive the ultimate limit of such ratios, which are validated by numerical simulations, experiments, and micro-scale Hamiltonian-oscillator analyses. For rectifiers whose thermal conductivities are linear with the temperature, this limit is simply a numerical value of 3. For those whose conductivities are nonlinear with temperatures, the maxima equal $\kappa_{max}/\kappa_{min}$, where the two extremes denote values of the solid segment materials that can be possibly found or fabricated within a reasonable temperature range on earth. Recommendations for manufacturing high-ratio rectifiers are also given with examples.