Electrical conductivity of Sn at high pressure and temperature

To date, temperature and conductivity have many outstanding implications in extreme environments but are yet to be fully understood under high pressure and temperature dynamic conditions. Here, we introduce an approach to provide high quality electrical conductivity results under dynamic loading conditions. Emphasis is given to address the skin depth effect's influence in a dynamic loading experiment by using thin films. The thin film samples in this study were at least 100 times thinner than previous samples in dynamic electrical conductivity experiments, increasing the current density to its full potential across the sample's entire cross section. Consideration of the skin depth accounts for at minimum a $4\times$ scaling factor to the final electrical conductivity result that has been neglected in previous dynamic electrical conductivity studies. We also obtained improved signal-to-noise ratio with custom diagnostics optimized for better electrical impedance matching. These considerations were applied to Sn to assess electrical conductivity at elevated pressure and temperature. The high signal-to-noise ratio with reduced skin depth influence results in Sn allowing observation of the conductivity changes related to solid-to-solid and solid-to-liquid phase transitions. Additionally, we calculate the Sn thermal conductivity using the Wiedemann-Franz law for our experiments and compare against thermal transport dependent temperature measurements from previous work.

Figure 1

Skin depth effect shown in (a) a thick sample, larger than the skin depth ($\delta$), and (b) a thin sample with a thickness less than the skin depth ($\delta$). As shown by the color gradient, dark orange demonstrates the full current density while white is a region of no current flow

Figure 2

(a) Schematic of the thin film tin sample coated upon a $38.2\text{-mm-diam}\times 2\text{-mm}$-thick ${\mathrm{Al}}_2{\mathrm{O}}_3$ anvil. (b) Schematic of the experimental assembly for the six dynamic experiments. The flyer is launched at the sample from the left and impacts the Al baseplate. (c) Schematic of the target assembly as seen with the projectile coming out of the page. This schematic displays the placement of the Cu leads, silver epoxy, and the downrange ${\mathrm{Al}}_2{\mathrm{O}}_3$ anvil relative to the coated uprange ${\mathrm{Al}}_2{\mathrm{O}}_3$.

Figure 3

Voltage, current (Rogowski coil), and shorting pin signals for an ambient sample dry run. Panel (a) shows the signals from when the circuit triggers until the experiment takes place. The red box in (a) is enlarged in (b) to show the ambient signals during the duration of the shock loading experiment. The vertical gray lines are presented to show the experimental timings of the shock wave's location within the experimental assembly for experiment 019. As the shock wave reaches the back surface of the downrange ${\mathrm{Al}}_2{\mathrm{O}}_3$ anvil, the cap pins that are in contact with the back surface are triggered. As shown in (b), the current from the Rogowski coil and subsequent sensing voltage remains effectively constant on this time scale.

Figure 4

(a) Computationally derived pressure and temperature profiles within the tin sample for shot 019. (b) Raw experimental voltage from the differential amplifier and the Rogowski coil voltage, both from shot 019. The computationally expected times for the shock transit location are labeled in both (a) and (b).

Figure 5

(a) Corrected resistance vs time from shot 019. (b) Corrected electrical resistivity vs time from shot 019. (c) Corrected electrical conductivity vs time shot 019. The red markers are the regions where the dynamically loaded experimental data was analyzed.

Figure 6

(a),(d) Electrical resistivity, (b),(e) electrical conductivity, and (c),(f) thermal conductivity vs pressure and temperature, respectively. The thermal conductivity is calculated from the Wiedemann-Franz law and the computationally derived temperature using LEOS. Error bounds are shown with transparent region accompanying each data set.

Figure 7

Computationally derived pressure and temperature conditions of the sample at equilibrium for each experiment in this study overlayed upon the Sn phase diagram (gray lines). Beside each experimental point from this study is its electrical conductivity value as derived in the experiments herein. Additional data points of the melt line from shock melt on release experiments [77], phase boundaries from shock sound speed experiments [64, 65], and various DAC experiments [76, 78, 79].

Figure 8

Data from Brantley et al. [41], where Sn was shocked to $\sim 120\mathrm{GPa}$. This experiment was modeled with three different thermal conductivity values: $66\text{W/mK}, 300\text{W/mK}$, and the Wiedemann-Franz law derived $1200\text{W/mK}$ from this study.