Literature reports show both benefits and negligible impact when including graded electrodes in battery design, depending upon the exact model and conditions used. In this paper, we use two different optimization approaches for a secondary current distribution porous electrode model with nonlinear kinetics to confirm that computed solutions are correct. We use these confirmed optimal solutions to probe several ways that graded porosity can improve electrode performance. Single objective optimization such as reducing the overall electrode resistance using a graded electrode design provides a modest 4-6% reduction in resistance for typical lithium-ion battery parameters. Multiple objective optimization-for example, simultaneously considering electrode resistance and the overpotential variance and eventually the overpotential average as well-shows that multilayer designs open up a much richer feasible design space for achieving multiple goals. The ultimate answer to the value of graded electrodes will be the techno-economic analysis that links the benefits of an expanded optimal design space to the detrimental costs associated with manufacturing multilayer electrodes. An open-access executable code that can give optimal porosity distribution of any specified chemistry and detailed explanation of the two approaches can be found on the Subramanian group's website. Modeling and mathematical optimization can significantly improve the efficiency of battery design, helping to meet the growing demands for various applications. The idea of using modeling for battery design was first introduced by Tiedemann and Newman in 1975. 1 They used an ohmically limited porous electrode model to maximize the cell's effective capacity by changing the electrode thickness and porosity. Newman later applied the reaction-zone model to maximize the specific energy of the system, taking mass into consideration as well.2 For these two models, the objective function can be directly related to the design variables, thus the optimum can be obtained by simply observing the plot or from the analytical solution. They further optimized the thickness and porosity of a lithium iron phosphate 3 electrode, where they maximized the specific energy using the Ragone plots. Ramadesigan et al.4 went one step further by including the linear electrode kinetics to minimize the internal resistance of the electrode. They used control vector parameterization (CVP) to minimize the ohmic resistance in the positive electrode by varying porosity.With the development of battery modeling, more physical processes have been included, and one of the most popular physics based models is the pseudo-2-Dimensional (P2D) model developed by the Newman group. 5 The P2D model involves a set of nonlinear partial differential equations (PDEs) that can only be solved numerically. Therefore, a numerical optimization approach is required to perform optimization on the system. Du et al. proposed a surrogate-modelbased approach, 6 and later developed a sophisticated framework based on th...

The main objective of this paper is to analyze transport models for lithium symmetric cells and arrive at an efficient model and code to simulate the same. Two one-dimensional models are considered. The first model uses the dilute solution Nernst-Planck equations in conjunction with the electroneutrality assumption (EN-NP model). For binary electrolytes, an analytical solution for electrolyte concentration is derived and compared with numerical solutions by an approximate finite volume method. The second approach relaxes the electroneutrality assumption by way of Poisson's equation for the electrostatic potential (PNP model). The computational difficulties of the PNP model are tackled using the approximate finite volume method, with demonstrated convergence characteristics even with bulk dimensions several orders of magnitude larger than the characteristic double layer size. A robust code is developed for the PNP model. The computationally efficient transport models can facilitate simulations, physical understanding, and analysis. This is illustrated by a case study in which these models are coupled with modified kinetic models, which are then parameterized with experimental voltage response data using a systems-level approach. The estimated parameters provide further insight into the electrochemical phenomena underpinning voltage transitions in symmetric cells.

Electrochemical models at different scales and varying levels of complexity have been used in the literature to study the evolution of the anode surface in lithium metal batteries. This includes continuum, mesoscale (phase-field approaches), and multiscale models. In this paper, using a motivating example of a moving boundary model in one dimension, we show how battery models need proper formulation for mass conservation, especially when simulated over multiple charge and discharge cycles. The article concludes with some thoughts on mass conservation and proper formulation for multiscale models.

A hybrid analytical-collocation approach for fast simulation of the impedance response for a Li-ion battery using the pseudo-two dimensional model is presented. The impedance response of the spherical diffusion equations is solved analytically and collocation is performed on the resulting boundary value problem across the electrode and separator thickness using an orthogonal collocation scheme based on Gauss-Legendre points. The profiles for a frequency range from 0.5 mHz to 10 kHz are compared with the numerical solution obtained by solving the original model in COMSOL Multiphysics. The internal variable profiles across a wide range of frequencies are compared between the two methods and the accuracy, robustness, and computational superiority of the proposed hybrid analytical-collocation approach is presented. The limitations of the proposed approach are also discussed. A freeware for academic use that reads the various battery parameters and frequencies of interest as input, and predicts the battery impedance for a half cell and full cell, is also developed and a means to access it is reported in this paper.

There has been significant recent interest in studying multiscale characteristics of current and next-generation batteries, including lithium-metal and lithium-sulfur batteries. Advances in computing power make researchers believe that the detailed multiscale models can be efficiently simulated to arrive at the insights for the degradation and performance loss; however, this is not true and special attention needs to be paid to local singularities, boundary layers, moving boundaries, etc. This article presents 2D examples that illustrate the importance of grid convergence studies, provides well-defined detailed models to test the efficiency of numerical schemes, and discusses the associated simulation challenges.

Lithium-sulfur (Li-S) batteries are one of the most promising next-generation energy storage technologies due to their high theoretical energy and low cost. However, Li-S cells with practically high energy still suffer from a very limited cycle life with reasons which remain unclear. Here, through cell study under practical conditions, it is proved that an internal short circuit (ISC) is a root cause of early cell failure and is ascribed to the crosstalk between the S cathode and Li anode. The cathode topography affects S reactions through influencing the local resistance and electrolyte distribution, particularly under lean electrolyte conditions. The inhomogeneous reactions of S cathodes are easily mirrored by the Li anodes, resulting in exaggerated localized Li plating/stripping, Li filament formation, and eventually cell ISC. Manipulating cathode topography is proven effective to extend the cell cycle life under practical conditions. The findings of this work shed new light on the electrode design for extending cycle life of high-energy Li-S cells, which are also applicable for other rechargeable Li or metal batteries.

In this study, a numerical modeling framework is developed to model and predict the morphological evolution in lithium metal batteries. A two-dimensional moving boundary model is used to simulate the dendritic growth from a nucleated lithium metal protrusion at the surface of the negative electrode. Depending on the geometric, kinetic, and transport parameters, the growth rate and shape of the lithium seed varies and in turn, affects the cyclability and capacity loss of the battery. Compared to conventional approaches, the proposed approach enables simulation of 100 cycles of charge-discharge in less than 1 minute. This robust model and algorithm for predicting metal deposition and stripping in lithium metal batteries brings together the mesoscale and electrochemical models and can pave the path towards specifically tailored dendrite-free morphological evolution to make lithium metal anodes viable in commercial systems. .

Electrochemical models at different scales and varying levels of complexity have been used to study the evolution of the anode surface in lithium metal batteries. This includes continuum, mesoscale (phase-field approaches), and multiscale models. Thermodynamics-based equations have been used to study phase changes in lithium batteries using phase-field approaches. However, grid convergence studies and the effect of additional parameters needed to simulate these models are not well-documented in the literature. In this paper, using a motivating example of a moving boundary model in one- and two-dimensions, we show how one can formulate phase-field models, implement algorithms for the same and analyze the results. An open-access code with no restrictions is provided as well. The article concludes with some thoughts on the computational efficiency of phase-field models for simulating dendritic growth.