Validation of the Wiedemann-Franz Law in Solid and Molten Tungsten above 2000 K through Thermal Conductivity Measurements via Steady-State Temperature Differential Radiometry

We measure the thermal conductivity of solid and molten tungsten using steady state temperature differential radiometry. We demonstrate that the thermal conductivity can be well described by application of Wiedemann-Franz law to electrical resistivity data, thus suggesting the validity of Wiedemann-Franz law to capture the electronic thermal conductivity of metals in their molten phase. We further support this conclusion using ab initio molecular dynamics simulations with a machine-learned potential. Our results show that at these high temperatures, the vibrational contribution to thermal conductivity is negligible compared to the electronic component.

Figure 1

(a) Schematic of experimental setup. A key in this approach is the ability to locally heat a region of the sample to its molten state while pyrometrically sensing radiance temperature in the middle of the heated region. This “self-crucible” approach then allows for thermal conductivity measurements by just sustaining melt only the middle of the sample during quasi-steady-state conditions. Note, we refer to this state as a “quasi” or “near” steady-state condition since the sample has not reached a true steady state due to the slow heating of the chamber from both conduction through the sample holder and radiation from the sample surfaces; however, this temperature variation is small during the time over which we collect data for thermal conductivity analysis, and we account for this slowly varying temperature in our thermal conductivity analysis. (b) Typical thermogram of the W sample heated to achieve quasi-steady-state conditions with a molten center, with the specific onset times of melting and resolidification (i.e., “freezing”) indicated in the plot. The incident laser power generating these radiance temperatures was 1.9 kW. The voltage response of our particular pyrometer was not linear with temperature until above $\sim 1600\mathrm{K}$ radiance temperature, indicative of the nonphysical cooling response for times greater than 20 s. The minimum temperature that the pyrometer would register was $\sim 900\mathrm{K}$ radiance temperature, thus leading to this reading at all times when the sample was below this temperature. (c) Close-up of data from (b) around the melting temperature. (d) Close-up of data from (b) around the freezing temperature. The temperature at which this thermal arrest occurs is used to determine the melting temperature of the sample.

Figure 2

Thermograms of the radiance temperature response of W subjected to various power perturbations above the baseline power $Q_0$, which was used to bring the sample up to a quasi-steady baseline temperature. The $\mathrm{\Delta }\mathrm{Q}$ perturbations result in a $\mathrm{\Delta }T$ above the baseline temperature that is used to calculate the thermal conductivity of the sample. We note that over the course of collecting a series of data (various thermograms of $\mathrm{\Delta }T$), slow temperature rises due to delayed equilibration of the chamber lead to increases in the baseline temperature of the sample ($T_0$). This is apparent from the difference in radiance temperature of the W from $Q_{0,\text{initial}}$ to $Q_{0,\text{final}}$, which were collected using identical powers after 10 successive scans spanning about 30 minutes. Thus, $\mathrm{\Delta }{T}_{\mathrm{rad}}$ is determined by interpolating between temperatures at times before and after the change in temperature driven by $\mathrm{\Delta }Q$.

Figure 3

Measured thermal conductivity of solid and molten W from this work (fill and open circles, respectively) compared to accepted values from the literature for both solid [1] and molten states [10]. The previously reported accepted value for the thermal conductivity of molten W from Tolias et al. [10] is derived from the Wiedemann-Franz law applied to electrical resistivity data.

Figure 4

ML-MD simulated vibrational thermal conductivities of solid (phonons) and molten (vibrations) W compared to data presented in Fig. 3 (data symbols the same as in Fig. 3). The vibrational component to the thermal conductivity of W only contributes $<5\%$ to the total thermal conductivity above 2000 K with the phononic contribution decreasing with increasing temperature up to the melting temperature, indicating the phononic thermal conductivity at these elevated temperatures is decreasing due to anharmonic phonon-phonon interactions. In the molten state, the vibrational thermal conductivity is $\sim 0.5\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$, contributing $<1\%$ to the thermal conductivity. At these elevated temperatures above $T_{\mathrm{melt}}/2$, the vibrational contribution is effectively suppressed leading to the thermal conductivity of W being dominated by electrons, thus resulting in the Wiedemann-Franz law proving an accurate route to calculate the thermal conductivity.