# Fileset

[Supporting_Information.pdf](https://mdr.nims.go.jp/filesets/67b26e05-cd7e-4ac4-835c-01005be5ddd2/download)

## Creator

[Atsushi Tamura](https://orcid.org/0000-0002-9858-0853), [Toshihiko Mandai](https://orcid.org/0000-0002-2403-7794), [Shunsuke Yagi](https://orcid.org/0000-0003-1675-650X)

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Modeling for homogeneous Mg electrodeposition on Mg metal negative electrode in rechargeable Mg batteries](https://mdr.nims.go.jp/datasets/2c5a0143-fbb9-42ea-b0d2-5cdc15a5c018)

## Fulltext

Instructions for Preparing Camera-Ready Summaries for SSDMS-1  Supporting Information  Modeling for Homogeneous Mg Electrodeposition on Mg Metal Negative Electrode in Rechargeable Mg batteries Atsushi Tamura1 Toshihiko Mandai2 and Shunsuke Yagi1,*  1Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan 2Research Center for Energy and Environmental Materials, National Institute for Materials Sci-ence (NIMS), 1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan *Corresponding author, E-mail: syagi@iis.u-tokyo.ac.jp   S-2   Governing equations   The equations used in the simulation that were not shown in the paper are pre-sented here. The diffusion of Mg2+ ions in the positive electrode can be described by the following equation:  𝜕𝑐𝑠𝜕𝑡= −𝛻(−𝐷𝑠𝛻𝑐𝑠) (S-1) where 𝑐𝑠 is the concentration of Mg2+ ions in the positive electrode, and 𝐷𝑠 is the dif-fusion coefficient of Mg2+ ions in the positive electrode. Thermal balance is expressed as follows:  𝜌𝐶𝑝𝜕𝑇𝜕𝑡+ 𝛻𝑞 = 𝑄𝑗ℎ + 𝑄𝑐ℎ𝑒𝑚 + 𝑄𝑚 (S-2) where 𝜌 is the density, 𝑞 is the heat flux, 𝐶𝑝 is the constant pressure heat capacity, 𝑄𝑗ℎ is the joule heat, 𝑄𝑐ℎ𝑒𝑚 is the heat due to electrochemical reactions, and 𝑄𝑚 is the heat due to mixing. The bulk values of the diffusion coefficient were modified using the Bruggeman relationship for the positive electrode and the separator.  𝐷 𝑠𝑒𝑝 = 𝜀𝑠𝑒𝑝1.5 𝐷 (S-3)  𝐷 𝑝𝑜𝑠 = 𝜀𝑝𝑜𝑠1.5 𝐷 (S-4) where 𝐷𝑠𝑒𝑝 and 𝐷𝑝𝑜𝑠 are the diffusion coefficients in the separator and positive elec-trode, respectively, and 𝜀𝑠𝑒𝑝 and 𝜀𝑝𝑜𝑠 are tortuosity and porosity, respectively.     S-3   Generalization using a dimensionless number Figure S1 shows the Mg2+ ion concentration after charging at position 1 shown in Figure 4 with different diffusion coefficients varying from 7.5×10−11 to 7.5×10−9 m2 s−1. As shown in the figure, the differences in the concentrations at positions 1, 2, and 3 are due to their different distances from the Mg2+ ion source, which is the active material of the positive electrode. In Figure S1, the values of a dimensionless number 𝐴, defined by equation S1, to quantify the suppression of the ion concentration depletion, are also plotted together with the ion concentration after charging. 𝐴 is defined according to  𝐴 =𝑐0𝐷𝐹𝑖𝑙 (S1)  Figure S1. Mg2+ ion concentration and corresponding dimensionless number A after charging with different diffusion coefficients of Mg2+ ions in the electrolyte solution at positions 1, 2, and 3. The red broken line represents the dimensionless number value of 150 below which the difference in the concentration of Mg2+ ions inside and outside the well-type structure is less than 25 mol m−3. S-4  where 𝑐0 is the initial concentration inside the well-type structure, 𝐷 is the ion diffu-sion coefficient in the electrolyte solution, 𝑖 is the current density on the electrode sur-face and 𝑙 is the depth of the well-type structure. A decreasing value of 𝐴 corresponds to an increased concentration of the Mg2+ ions inside the well-type structure, which is favorable for suppressing inhomogeneous Mg electrodeposition. For example, in Figure S1, the red broken line represents the 𝐴 value of 150 below which the difference in the concentration of Mg2+ ions inside and outside the well-type structure is less than 25 mol m−3. It should be noted that this threshold value of 𝐴 = 150 was empirically determined.  Above 𝐴 = 150, the concentration at the bottom of the well-type structure, where the concentration decreases the most, retains 95% or more of the initial concentration (500 mol m−3 in this simulation) after charging. Below 𝐴 = 150, an apparent concentration gradient appears. Although this definition is derived from simulation results, it provides a useful criterion for assessing diffusion-limited regimes.   S-5  Here, the depth of the well-type structure 𝑙 is used as the representative length for the dimensionless number rather than the cubic root of volume 𝑉13 based on the fol-lowing considerations. In this study, the uniform Mg deposition was observed at a higher diffusion coefficient and a lower current density, which means the dimensionless param-eter A should be higher to obtain a uniform Mg electrodeposition. As shown in Figure S2, the Mg2+ ion concentration in the well increases with decreasing the well depth l, which is favorable for a uniform Mg electrodeposition and corresponds to the increase in the dimensionless parameter A. In contrast, the Mg2+ ion concentration in the well decreases but the dimensionless parameter A increases with decreasing V1/3, which is the opposite direction change in A. The above is the reason why the well depth 𝑙 was chosen as the characteristic length in the definition of the dimensionless parameter 𝐴 partly because diffusion into the well-type structure is primarily governed by longitudinal transport along the depth direction, which dominates the ion supply limitation. (a)                                 (b)  Figure S2. Mg2+ ion concentration at position 1 in Figure 4 after charging up to 2.2 V plotted together with the dimensionless number 𝐴 as a function of (a) the volume of the well-type structure, (b) the depth of the well-type structure. The diffusion coeffi-cient in the electrolyte is set to 7.5×10−11 m2 s−1.