Research papersThree-dimensional experimental-scale phase-field modeling of dendrite formation in rechargeable lithium-metal batteries
Introduction
Global energy demand continues to rise due to industrial activity and the world’s population expansion, with an average growth rate of about 1% to 2% per year since 2010 (pre-Covid19 pandemic levels) [1]. The increasing consumption of non-renewable energy reserves, such as coal, gas, and oil [2], and awareness of climate change [3], [4], have triggered a steep growth in renewable energy sources (6% average annual growth worldwide over the past decade) [5], along with an urgent need for the development of improved energy storage systems [6]. Globally, around one-quarter of our electricity comes from renewables, which include hydropower, wind, solar, biomass, ocean energy, biofuel, and geothermal [1].
New chemistry and designs, such as metal anode batteries, are under active research to achieve a specific energy of 500 Wh/kg and manufacturing costs lower than $100/kWh [7]. Despite enormous efforts, today’s highest specific energy remains below 400 Wh/kg, with an average growth rate of about 5% per year since 1970 [8]. As the specific energy limitation (300 Wh/kg) of conventional lithium-ion batteries based on intercalated graphite anode cannot meet the current market demand, researchers are refocusing on lithium metal batteries (LMBs) [9]. LMBs can achieve ultra-high energy densities by avoiding the use of a graphite lattice to host (intercalation process), as illustrated by the comparative schematic of Fig. 1. The graphite material (host) drastically reduces the energy density of conventional Li-ion batteries by adding weight to the battery pack that does not participate in the electrochemical reaction [8]. For instance, a recent study by Mathieu et al. [10] analyzed the key materials that make up battery cells for medium-sized electric vehicles (weighted average of the battery chemistries commercialized in 2020 [10]). This study revealed that graphite material (anode) represented the largest share, accounting for 28% of the total weight of the battery cell. Furthermore, according to Lin et al. [11], the specific energy delivered by state-of-the-art Li-ion cells (250 Wh/kg) can be increased to approximately 440 Wh/kg once the graphite anode is replaced by lithium in a Li-LMO cell (lithium transition-metal oxide).
Although lithium metal has been an attractive anode alternative in rechargeable batteries since the early 1970s, its commercialization has been hindered due to several shortcomings. The greatest challenge to achieving the commercial realization of lithium-metal batteries is related to their stability and safety [11]. These issues are closely linked to the lithium anode: dendrite formation due to the uneven deposition of lithium, dead lithium formed after dendrites breakage, formation of unstable solid electrolyte interphase (SEI), and volume expansion of the metal anode. Additionally, these mechanisms interact, causing synergistic detrimental effects [12].
Dendrite formation in LMBs is the consequence of lithium’s uneven deposition, associated with thermodynamic and kinetic factors, such as the inhomogeneous distribution of Li-ion concentration and electric potential on the electrode surface. Furthermore, the morphology of the electrodeposited lithium is influenced by different factors such as the magnitude and frequency of the applied current density, electrolyte concentration, temperature, pressure, ion transport, and mechanical properties in the electrolyte [13], [14]. Understanding dendrite formation in LMBs combines theory, experiment, and computation [15]. Within computational research, various studies have demonstrated the use of phase-field (diffuse-interface) method to model the reaction-driven phase transformation within metal anode batteries, providing avenues for rationalization of morphological behaviors of dendrite formations observed experimentally [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45].
While progress has occurred in phase-field modeling of lithium dendrites in recent years, there are still several issues related to the evolution of dendritic patterns in lithium metal electrodes that remain unresolved [38]. The fundamental failure mechanism of lithium anode remains unclear and controversial [12]. A significant effort has gone into using 2D models to rationalize 3D dendritic patterns observed experimentally [43]. Furthermore, various strategies exist to suppress Li dendrites’ growth and weaken side reactions. Some of these strategies address the battery operating conditions, including pulse charging lithium dendrite suppression [41] and control of internal temperature [28]. Other alternatives focus on the electrode (anode), such as modeling of 3D conductive structured lithium metal anode [33], [44], and low porosity and stable SEI structure [30]. Besides, other approaches center on the electrolyte management and separator design, proposing a compositionally graded electrolyte [27], dendrite suppression using flow field (forced advection) [32], the study of separator pore size inhibition effect on lithium dendrite [45].
Given the inherent 3D nature of lithium dendrite morphologies [31], [46], [47], it is critical to develop phase-field models to understand the impact of 3D effects on triggering the formation of these patterns. Nevertheless, few papers attempt to simulate the full 3D lithium dendrite growth process using phase-field models. For instance, Mu at al. [37] performed parallel three-dimensional phase-field simulations of dendritic lithium evolution under different electrochemical states, including charging, suspending, and discharging states. Recently, Arguello et al. [43] presented 3D phase-field simulations using an open-source finite element library, to describe hazardous three-dimensional dendritic patterns in LMBs. The authors used time step adaptivity, mesh rationalization, parallel computation, and balanced phase-field interface thickness to mesh resolution ratio. The high computational cost of simulating the detailed lithium electrodeposition is a well-known challenge that has limited the domain size of phase-field simulations [26], [29]. Thus, higher-than-normal dendrite growth rates were reported in the literature for 3D phase-field modeling of dendrite growth, due to the short separation between electrodes used [37], [43]. Recently, experimental observations by Chae et al. [48] have revealed a change in the lithium deposition behavior and morphology from “hazardous” needle- and moss-like dendritic structures to “safer” morphologies (smooth and round shaped surface) as interelectrode spacing increases. Therefore, simulating the dendrite formation at the experimental scale has significant practical relevance.
In this paper, we seek to present a 3D phase-field model of lithium dendrite formation in an experimental scale battery. The domain sizes simulated here (with up-to inter-electrode separation) represent a significant extension towards practical application compared to previous 3D phase-field electrodeposition works, where separations of only about could be achieved due to computational cost limitation [37], [43].
Here we expand on our previous work on the phase-field model [42], [43]; we use small-scale 3D simulations to analyze the sensitivity of the model on spatial resolution and phase-field interface thickness to determine the numerical requirements of the experimental scale simulations.
We also look into the incorporation of a modified 3D representation of the surface anisotropy based on the cubic crystal structure of lithium [29], as well as the model of William et al. [49]. Besides the 2D work by Wang et al. [50] on dendrite formation in zinc-air batteries, our work presents the application of the corresponding surface anisotropy approach in a 3D phase-field model of dendrite electrodeposition for the first time in the literature.
We organize the paper as follows: Section 2 presents the basic equations describing the lithium-battery dendrite growth process and details its implementation where we introduce a modified representation of the surface anisotropy of lithium metal. Section 3 describes the system layout and properties, together with the implementation of symmetric boundary conditions for a detailed study of symmetric dendritic patterns [31], [47]. We discuss numerical simulations of spike-like (small scale) lithium-battery dendrites growth in Section 4, where we analyze the sensitivity of the simulation results for a series of spatial resolutions and phase-field interface thickness. Section 5 evaluates the behavior of the surface anisotropy representation model for metal anode battery simulations through different numerical tests. We compare the dendritic patterns with the results obtained in preceding simulation work [43]. Following this, we present our implementation of the modified surface anisotropy model under a larger interelectrode distance (experimental scale) 6. We show that a significant modification in lithium electrodeposition behavior is obtained with increasing interelectrode distance. We study and describe the lithium dendrite propagation rates and morphologies obtained under different charging voltages. Finally, we draw conclusions in Section 7.
Section snippets
Surface anisotropy representation for phase-field electrodeposition models
In this section, we present a modified representation of the 3D surface anisotropy of crystalline lithium. We start by considering the surface energy expression, following [42], [43]: , where its variational derivative (surface anisotropy of lithium crystal) is: , consistent with most recent phase-field models of dendritic electrodeposition [24], [25], [26], [27], [30], [38]. However, a more accurate representation of may include an additional term, as
System layout & properties
Consider a battery cell with a traditional sandwich architecture, with an anode on one side of an electrolyte-filled hexagonal domain and a cathode on the other side (represented by a current collector boundary condition — see Fig. 2). This numerical problem has previously been considered in [43] and has been reproduced here for readability. The battery cell undergoes a recharging process under fixed applied electro potential. An artificial nucleation site in the form of an
Phase-field interface thickness to mesh resolution ratio: A sensitivity analysis
The phase-field interface thickness significantly affects the simulated reaction rates [42]. Wider interfaces (larger ) increase the reactive area in the simulation, which induces faster electrodeposition rates. Numerical evidence shows that 1D interface-thickness-independent growth (convergent results) are possible well before reaching the physical nanometer interface width [42], [66].
In this section, we perform a sensitivity analysis to study possible mesh-induced effects on the simulated
3D simulations using modified surface anisotropy
We evaluate the performance of the surface anisotropy representation model for metal anode battery simulations (see Section 2.1). We perform numerical tests to gain insight into the benefits of this modification compared with the results previously obtained in Section 4 and the preceding 3D simulation work [43]. These studies consist of 3D phase-field simulations of lithium electrodeposition during battery charge state to explore three-dimensional highly branched “spike-like” dendritic patterns
Experimental-scale 3D simulations of lithium dendrite formation
This section evaluates the performance of the modified surface anisotropy model (see Section 2.1) in experimental-scale interelectrode distances. We map the nodal distribution concentrating the nodes in the region of interest, inspired by experimental and simulation results. The increased domain size affects the lithium electrodeposition behavior by increasing the interelectrode distance. We discuss the lithium dendrite propagation rates and morphologies for different charging voltages.
Conclusions
We use phase-field modeling to investigate the electrodeposition process that forms dendrites within lithium-metal batteries (LMB). We analyze the dendrite formation in domains with various sizes using both, short () and experimental-scale () interelectrode separation. Through a resolution sensitivity analysis, we asses the mesh-induced effect on the simulated 3D dendrite morphology, propagation rates (dendrite’s height vs. time), electrodeposition rates (dendrite’s volume vs. time),
CRediT authorship contribution statement
Marcos E. Arguello: Methodology, Investigation, Writing – original draft. Nicolás A. Labanda: Methodology, Investigation, Writing – original draft. Victor M. Calo: Methodology, Resources, Writing – review & editing. Monica Gumulya: Conceptualization, Writing – review. Ranjeet Utikar: Conceptualization, Supervision. Jos Derksen: Conceptualization, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the sponsorship of a Curtin International Postgraduate Research Scholarship (CIPRS), Australia and the Aberdeen-Curtin Alliance PhD Scholarship, Australia. This publication was also made possible in part by the Professorial Chair in Computational Geoscience at Curtin University. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie, Australia grant agreement No 777778 (MATHROCKS).
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