Open‐Source CFD Elucidating Mechanism of 3D Pillar Electrode in Improving All‐Solid‐State Battery Performance

Abstract All‐solid‐state batteries (ASSBs) have become an important technology because of their high performance and low‐risk operation. However, the high interface resistance and low ionic conductivity of ASSBs hinder their application. In this study, a self‐developed electrochemical model based on an open‐source computational fluid dynamics platform is presented. The effect of contact area reduction at the electrode/solid‐state electrolyte interface is investigated. Then, a new conceptual 3D structure is introduced to circumvent the existing barriers. The results demonstrate that the discharge time is shortened by over 20% when the area contact ratio reduces from 1.0 to 0.8 at 1 C‐rate, owing to the increased overpotential. By adopting the new 3D pillar design, the energy density of ASSBs can be improved. However, it is only when a 3D current collector is contained in the cathode that the battery energy/power density, capacity, and material utilization can be greatly enhanced without being limited by pillar height issues. Therefore, this work provides important insight into the enhanced performance of 3D structures.

increasing electrode thickness. L pe =8.08 mm L pe =12.12 mm L pe =16.16 mm L pe =20.2 mm Thin-fim structure I app = 10 A m -2 Figure S1. Effect of positive electrode thickness for the thin-film structure.

Note S2 Model development
In this model, the computational domain includes the negative current collector (Cu), negative electrode (Li metal), solid-state electrolyte (SSE) (LiPON), positive electrode (LiCoO 2 ), and positive current collector (Al) (Figure S2), among which the current collectors are necessary to be considered to simulate the non-uniform current distribution in the whole cell as well as the 3D electrode. During the discharge process of an ASSB, the lithium-ion extracted from the lithium metal anode at the negative electrode/SSE interface enters SSE, and is then inserted into the LiCoO 2 cathode through the positive electrode/SSE interface. The electrochemical reactions at the electrode/SSE interfaces can be written as follows: Herein, a three-dimensional layered electrochemical model is developed to simulate the discharge characteristics of ASSBs.

Assumptions
(1) The electrodes are assumed to be non-porous, and the electrochemical reactions only occur at the interfaces between SSE and electrodes.
(2) A uniform distribution of lithium-ion in the lithium metal anode is assumed, thus the related lithium-ion diffusion process at anode is not considered.
(3) Binary electrolyte and electroneutrality are assumed for SSE.
(4) All side reactions are neglected, and all domains are considered isothermal.

Governing equations
Intercalation lithium conservation (solved in positive electrode) where c Li,pe (mol m -3 ) is the molar concentration of intercalation lithium, D Li,pe (m 2 s -1 ) the diffusion coefficient of intercalation lithium, S Li,pe (mol m -3 s -1 ) the generation rate of intercalation lithium at the positive electrode/SSE interface.
Lithium-ion conservation (solved in SSE) [ is the ionic potential, σ ion (S m -1 ) the ionic conductivity, S ion (A m -3 ) the ion generation rate at the electrode/SSE interfaces.
The source terms of mentioned governing equations are listed in Table S1.

Electrochemical reaction kinetics
The electrochemical reaction rates (A m -2 ) at the negative and positive electrode/SSE interfaces are described by Butler-Volmer equation, i.e., where k ne (m s -1 ) and k pe (m 2.5 mol -0.5 s -1 ) are the reaction rate constants for negative and positive reactions, respectively, γ the contact area ratio, η ne/pe (V) the overpotential at the electrode/SSE interfaces, expressed by and pe e ion eq,pe where U eq,pe (V) is the equilibrium potential for the LiCoO 2 cathode, which can be fitted to a polynomial function, written as [3]   where soc, i.e., state of charge, is defined as The cut-off voltage, once reached which the discharge will be stopped, is set to be 3.5 or 3.0 V. Ground boundary and galvanostatic condition are adopted for the terminal boundary conditions of negative and positive current collectors, respectively, which are defined as, terminal,ne

Performance indicators
To assess the performance of the battery with 3D pillar electrode, three performance indicators, including area-specific energy density, power density and capacity, are defined as follows: Energy density (mWh cm -2 ) app 0

Numerical procedures
To simulate specific physical processes, the model governing equations are solved with the specific boundary conditions, geometric dimensions, operating conditions, and physicochemical properties listed in Table S2. An open-source software based on finite volume method, OpenFOAM, with SIMPLE algorithm is used to solve these governing equations. The convergence criterions of all the physical quantities are set to 1×10 -6 . The anode, SSE and cathode are divided into 5, 10 and 10 layers of grid along the thickness direction, respectively.
For the 3D electrode, the grid length is set to 1 μm along the height direction, and thus the total grid numbers change with the height of pillars. The grid independence test has been conducted successfully with the discharge time step of 1 s.

Note S3 Model validation
To validate this model, the discharge curves of partial cell and whole cell are compared with the experimental data [2] (Figure S3a Figure S3b shows the development of concentration profiles of the mobile lithium-ion in the SSE (on the left) and the intercalated lithium-ion in the cathode along the thickness direction of the partial cell at 1 C discharge, which exhibit the same trend as the published works. [1,6] It is seen that the distribution of lithium-ion concentration in the SSE almost remains steady in the process of discharge, except at the beginning. The mobile lithium-ion concentration gradient is relatively high near the SSE/electrode interfaces but almost zero in the inner region. The lithium concentration in the cathode increases with time due to the continuous intercalation at the SSE/positive electrode interface and inward diffusion of lithium-ion. Since the diffusion of lithium is driven by the concentration gradient, the lithium concentration in the cathode is small at the place away from the SSE/positive electrode interface, which could result in a low utilization rate of the active material when the electrode thickness or discharge rate is large. Figure S4 illustrates the distribution of electron potential within current collector. It is observed that the place with the highest potential in the anode is on the terminal of the negative tab, and the place with the lowest potential in the cathode is located at the terminal of the positive tab. Besides, the potential gradient is more significant at the place near tab terminal, which indicates a higher local electric current near and within the tab. The local distribution of overpotential at the electrode/SSE interfaces is shown in Figure S5.
It is found that the overpotential distribution is dominantly influenced by the potential distribution of positive electrode since the electrical conductivity is larger in the lithium metal anode. In other words, the potential gradient is lower in the anode. In addition, while the magnitude of the anode interface overpotential almost remains constant, the cathode interface overpotential increases with the depth of discharge. The interface overpotential of the cathode has a larger contribution than that of the anode to the total overpotential. The distribution of overpotential has a significant change with the progress of discharge. At the beginning and middle stage of discharge, the region located in the upper and right part of the cell has a higher overpotential value. However, at the end stage of discharge, the upper and right part of the cell turns out to have the lowest overpotential value of the whole cell.