Thermal characterization of suspended fine wires across continuum to free-molecular gas regimes using the 3$ω$ method

TL;DR

This work addresses the limitation of the 3ω method in non-vacuum environments by developing a finite-gas-transfer model for suspended wires. It derives a full analytical solution to the 1D heat-transfer equation with a finite heat-transfer coefficient $h$ and validates it via experiments on a $d=16\,\mu$m Pt wire in air across $p$ spanning $10^{-5}$ to $10^{3}$ mbar. A kinetic-gas-theory–based model for $h(p)$ is introduced, describing transport from continuum to free-molecular regimes and yielding $h$ values from near-zero to about $700$ W/(m$^2$ K) at ambient pressure; this enables simultaneous extraction of $\kappa$ and $\rho c_p$, while $\rho c_p$ can be retrieved even without a specific $h(p)$ model. The results broaden the applicability of the 3ω technique to suspended wires in gas environments, offering enhanced signal strength and enabling $in\,situ$ thermal characterization of low-κ and nanoscale wires across a wide pressure range.

Abstract

The 3$ω$ 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$ω$ 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-$μ$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 $κ$ and volumetric heat capacity $ρc_p$, 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.

Figures (6)

• Figure 1: (a) Schematic four-terminal setup for $3\omega$ measurements of a fine wire. (b) Calculated temperature rise along the suspended wire section of length $L$ with (red) and without (blue) heat loss to a surrounding gas as quantified by the heat-transfer coefficient $h$. Solid lines illustrate the dc temperature rise, and shaded areas show the extent of ac temperature fluctuations based on Equations (\ref{['eq:temp rise with gas']}) and (\ref{['eq:temp rise with gasAC']}). Open circles represent numerical calculations of the maximum (ac+dc) temperature rise at select positions along the wire (see Appendix B). The current used for the calculation was $I_\mathrm{rms}=7.58$ mA.
• Figure 2: $3\omega$ measurements of the 16-µm diameter Pt wire. (a) Root-mean-squared amplitude and (b) tangent of the phase $\phi'$ of the 3$\omega$ voltage measured at different gas pressures. (c) Electrical resistance $R$ of the free-standing Pt wire section measured as a function of temperature. Solid line is a linear fit of the data used to determine the temperature coefficient $R'=\mathrm{d}R/\mathrm{d}T$ of the Pt wire. (d) $V_{3\omega}$ vs $I_\mathrm{rms}$ on a log-log scale at two different pressures. All pressure-dependent data is measured at $T = 297$ K.
• Figure 3: Gas pressure-dependent 3$\omega$ measurements. (a) The $3\omega$ voltage at low frequency (0.25-0.57 Hz data averaged) is plotted as a function of gas pressure. (b) The extracted $h$ as a function of gas pressure. The dash-dotted and dotted lines are best fits of Eq. (\ref{['eq:hmodel-new']}) and (\ref{['eq:h']}), respectively. Green, yellow, and red shaded regions indicate the free-molecular, transitional (slip and transition), and continuum regimes, respectively. The inset shows a schematic cross section of the model used to derive Eq. (\ref{['eq:hmodel-new']}), subdividing space around the wire into continuum and noncontinuum ("nonc.") regions. (c) $\omega_{\mathrm{inflection}}$ as a function of $h$. The data is obtained at 297 K.
• Figure 4: Volumetric heat capacity of the Pt wire determined under different gas pressures. Gray dashed line is $\rho c_p=2.85\times10^6~$J/(m$^3$K) expected for Pt at 297 K Furukawa1974.
• Figure 5: Benefit of finite $h$ for the thermal characterization of suspended wires. (a) Calculated $V_{3\omega,\mathrm{rms}}$ and (b) calculated $\omega_\mathrm{inflection}$ for our Pt wire. Black dashed lines in (a) and (b) are trajectories for a maximum dc temperature rise of 2 K. (c) Calculated frequency dependence of $V_\mathrm{3\omega,rms}$ of a carbon nanotube with 20-nm outer diameter.