Thermal characterization of suspended fine wires across continuum to free-molecular gas regimes using the 3$\omega$ method
The 3$\omega$ method is widely used to measure the thermal conductivity and the specific heat of wires and thin films. These measurements are typically performed under high vacuum conditions, which justify the use of heat transfer models that exclude thermal losses to a surrounding fluid. Here, we study the effect of thermal conduction from a joule-heated wire to a surrounding gas on pressure-dependent 3$\omega$ measurements, and show how a one-dimensional (1D) heat-transfer model may be used to reliably determine the wire's thermal properties. We derive a full analytical solution of the 1D heat-transfer equation with finite heat-transfer coefficient $h$ and validate it experimentally using a 16-$\mu$m diameter platinum wire in air across pressures from $10^{-5}$ to $10^3$ mbar. We introduce a model for heat transfer between the wire and the surrounding gas based on kinetic gas theory that accurately describes the data across continuum to free-molecular gas regimes, with $h$ varying from near-zero in high vacuum to approximately 700 W/(m$^2\cdot$K) at atmospheric pressure. We show that use of a validated $h(p)$ model allows extracting both thermal conductivity $\kappa$ and volumetric heat capacity $\rho c_p$, whereas volumetric heat capacity can be extracted even without invoking a specific $h(p)$ model. Our approach facilitates the characterization of fine wires with moderate to low thermal conductivities and may enable accurate thermal measurements of suspended wires with diameters on the nanometer scale.