Thermoelectricity at a gallium-mercury liquid metal interface

Abstract

We present experimental evidence of a thermoelectric effect at the interface between two liquid metals. Using superimposed layers of mercury and gallium in a cylindrical vessel operating at room temperature, we provide a direct measurement of the electric current generated by the presence of a thermal gradient along a liquid-liquid interface. At the interface between two liquids, temperature gradients induced by thermal convection lead to a complex geometry of electric currents, ultimately generating current densities near boundaries that are significantly higher than those observed in conventional solid-state thermoelectricity. When a magnetic field is applied to the experiment, an azimuthal shear flow, exhibiting opposite circulation in each layer, is generated. Depending on the value of the magnetic field, two different flow regimes are identified, in good agreement with a model based on the spatial distribution of thermoelectric currents, which has no equivalent in solid systems. Finally, we discuss various applications of this effect, such as the efficiency of liquid metal batteries.

Keywords: energy; heat transfer; magnetohydrodynamics; thermoelectric; turbulence.

Fig. 1.

Sketch of the experiment. A cylindrical vessel made of two concentric, electrically insulating cylinders with radii $R_i=37$ mm and $R_o=100$ mm and height 50 mm is filled with half mercury, half gallium, forming a liquid metal interface. All boundaries are electrically insulating, ensuring complete electrical insulation of the two liquid metals from the outside world. The fluids are subjected to a thermal gradient due to a temperature difference between the two cylinders $\mathrm{\Delta }{T}_0=T_i-T_o$. Thermoelectric potential and flow velocities are measured in the middle of the gap (see text). A vertical magnetic field up to 80 mT can be applied to the experiment. $J_{\mathit{TE}}$ represents a simplified distribution of thermoelectric currents, but only in the limit of very low thermal gradients or solidified metals (see text).

Fig. 2.

Radial temperature profile (measured at the top endcap) for different applied temperature differences $\mathrm{\Delta }{T}_0=T_i-T_o$, for $B_0=0$. Temperatures at the first and last radial positions are measured inside the cylinders. Inset: Time-averaged temperature difference $\mathrm{\Delta }{T}_B$ as a function of $\mathrm{\Delta }{T}_0$, where $\mathrm{\Delta }{T}_B=T_{15}-T_2$ is obtained using temperatures measured in gallium, at $5$ mm from the cylinders. Legend: $\mathrm{\Delta }{T}_0=\text{0 K}$ (◦), $\mathrm{\Delta }{T}_0=\text{4 K}$$(\mathrm{\nabla })$, $\mathrm{\Delta }{T}_0=\text{7 K}$ ($▵$), $\mathrm{\Delta }{T}_0=\text{11 K}$$(◃)$, $\mathrm{\Delta }{T}_0=\text{15 K}$$(▹)$, $\mathrm{\Delta }{T}_0=\text{18 K}$$(·)$, $\mathrm{\Delta }{T}_0=\text{22 K}$$(-)$, $\mathrm{\Delta }{T}_0=\text{26 K}$$(\star )$, $\mathrm{\Delta }{T}_0=\text{30 K}$$(\times )$, $\mathrm{\Delta }{T}_0=\text{33 K}$$(◊)$, $\mathrm{\Delta }{T}_0=\text{37 K}$$(\square )$. The red curve is a linear fit of the piece-wise linear temperature profile in the case $\mathrm{\Delta }{T}_0=\text{37 K}$.

Fig. 3.

Thermoelectric potential as a function of the applied temperature difference $\mathrm{\Delta }{T}_0$, for $B_0=0$. The error bars correspond to the SD of the time signal of the electric potential and therefore reflect a certain degree of unsteadiness induced by turbulent convection.

Fig. 4.

Numerical integration of Eq. 1 using the parameters of the experimental setup (Materials and Methods) and a piecewise linear thermal gradient, for $B_0=0$. (A) colorplot of the induced magnetic field and associated current streamlines, in the case of a purely conductive temperature solution. (B) Radial profiles of the corresponding temperature (black) and the radial current induced at $z=1$ mm from the interface (red). (C) and (D) are the same, but for a piecewise temperature gradient typical of convection. Near the cylinders, the thermal boundary layers generate a very large current density, 10 times larger than the value expected with solid-state conventional thermoelectricity. The dashed (resp. dashed-dotted) line shows the simple prediction [4] for bulk (resp. boundary) density currents.

Fig. 5.

Time-averaged azimuthal velocity as a function of the product $B_0\mathrm{\Delta }{T}_B[T.K]$. $\mathrm{\Delta }{T}_0=23$ K $(▵)$, $\mathrm{\Delta }{T}_0=29$ K $(◊)$, $\mathrm{\Delta }{T}_0=33$ K $(\square )$, $\mathrm{\Delta }{T}_0=37$ K (◦). Two distinct regimes are observed. In the first regime, $u_{\phi }$ increases with the magnetic field $B_0$. In the second regime, $u_{\phi }$ remains constant despite the increase in the magnetic field $B_0$. Both regimes are relatively well fitted by our theoretical predictions [6], represented by the red dotted line, and [7], represented by the blue dotted line.

Fig. 6.

Time-averaged velocity as a function of the bulk temperature difference $\mathrm{\Delta }{T}_B[K]$ for $B_0=56$ mT (black circles), compared to prediction [7] (dashed line). Points above $\mathrm{\Delta }{T}_B=6$ K were performed at constant imposed temperature difference in the domain where evolving $B_0$ let the velocity invariant. The error bar corresponds to the SD of the velocity.

Fig. 7.

Radial profile of the azimuthal velocity $u_{\phi }$ measured at the surface of the gallium for $B_0=36$ mT and $\mathrm{\Delta }{T}_0=37$ K, when the top endcap is removed, using particle tracking of surface oxides. Near the outer cylinder, azimuthal velocity increases significantly with radius, due to the high current density generated at the boundaries.