To match the growing demand for grid scale energy storage in order to balance out renewable energies, the liquid metal battery (LMB) technology is highly promising. Key elements of these batteries are the liquid electrodes, which require LMBs operating at elevated temperatures. Hereby, the electrolyte can be liquid or solid state. Based on density differences, the battery is self-assembling since the components self-segregate when the system is heated. Additionally, many of the possible materials are earth-abundant and not expensive. This results in a very cost effective technology. Moreover, the molten electrodes feature superior kinetics and transport properties as well as high voltage efficiencies at high current densities. Otherwise, the liquidity has the disadvantage that high operating temperatures are necessary. Further, there is a risk of corrosion, self-discharge and short-circuiting. The latter makes the battery not applicable for portable applications [Kim, H. et al. Chem. Rev. 113 (3) (2013)]. Among all LMBs, the Li||Bi cell has been investigated to a large extend and therefore, most material properties are readily available. This is the reason why the present study focuses on this material pairing. A schematic of this cell during discharge is shown in figure 1a. Here, the anode metal Lithium is oxidized and the ion crosses the electrolyte layer. Then it alloys with the molten Bismuth. At charge, this process is reversed. A picture of a real cell can be found in figure 1b. Nevertheless, there are other very promising cell chemistries such as NaZn batteries [Xu, J. et al. J. Power Sources 332 (2016)].

For theoretical investigations and numerical simulations, electromagnetic fields, electrochemistry and fluid dynamics need to be considered. Coupling of the corresponding fundamental equations and the different regions in the battery is challenging. Further, there are multiple fluid flow phenomena – like surface instabilities, Tayler instability, electro-vortex flow, thermal convection,solutal convection or Marangoni convection [Kelley, D. H. and Weier, T. Appl. Mech. Rev. 70 (2) (2018)] – that might occur when the battery is operated. To determine the operating voltage of a cell, overpotentials need to be investigated. Generally, there are three types of overpotentials: the ohmic loss, the mass transfer and the charge transfer overpotential. However, the latter can be neglected [Newhouse, J. M. Ph.D. thesis, MIT (2014)]. Determining the ohmic overpotential is quite straightforward using Ohm’s law, while the concentration overpotential requires the calculation of the species distribution in the electrolyte and the positive electrode.

The present study will focus on the electrochemical behavior of an all liquid Li||Bi battery and neglects fluid flow. Hereby, the whole cell is modeled using the parent-child-mesh technique from Weber et al. [Weber, N. et al. Computers & Fluids 168 (2018)]. This implies that the species distribution in the electrolyte and the distribution of the dissolved species in the cathode are calculated locally and the potential and current density distribution are calculated globally. The equations to solve are coupled between the used meshes. To account for the potential distribution, it is necessary to consider the electrochemical double layer at the electrode-electrolyte interfaces. Macroscopically, the latter can be described as a potential jump at both interfaces in accordance to the Nernst equation. Weber et al. [Weber, N. et al. Electrochim. Acta 318 (2019)] introduced to model this jump using a source term. In the present study, the approach to use internal jump boundary conditions for modeling the potential jump is proposed. To the best knowledge of the authors, species distributions in the electrolyte have not been investigated previously. When there is no mixing of the electrolyte of LMBs, concentration gradients of each species might be significant [Vallet, C. E. et al. J. Electrochem. Soc. 130 (12) (1983)]. Based on the solution for the species distribution, the potential jump at each interface can be calculated. Further, the charge transfer overpotentials can be obtained.

The equations are solved quasi-one-dimensional using the finite volume method and are implemented in the open source library OpenFOAM. To validate this new approach, a comparative study with Comsol is performed. Lastly, an application case is simulated, which confirms that concentration layers in the electrolyte are indeed present. This is, for the case of an Li+ ion in a LiCl-KCl eutectic molten salt, exemplarily shown in figure 1d. Further, as it can be seen in figure 1c, the simulation is able to solve the potential distribution in the cell with respect to the potential jump at the interface.

Acknowledgment: This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 963599.