Unified picture on temperature dependence of lithium dendrite growth via phase-field simulation

: Lithium dendrite growth due to uneven electrodeposition may penetrate the separator and solid electrolyte, causing inner short circuit and potential thermal runaway. Despite great electrochemical phase-field simulation efforts devoted to exploring the dendrite growth mechanism under the temperature field, no unified picture has emerged. For example, it remains open how to understand the promotion, inhibition, and dual effects of increased temperature on dendrite growth when using different electrolyte types. Here, by comprehensively considering the temperature-dependent Li + diffusion coefficient, electrochemical reaction coefficient, and initial temperature distribution in phase-field model, we propose that the activation-energy ratio, defined as the ratio of electrochemical reaction activation energy to electrolyte Li + diffusion activation energy, can be used to quantify the effect of temperature on dendrite morphology. Specifically, we establish a mechanism diagram correlating the activation-energy ratio, uniform initial temperature, and maximum dendrite height, which unifies the seemingly contradictory simulation results. Furthermore, results based on non-uniform initial temperature distribution indicate that a positive temperature gradient along the discharging current facilitates uniform Li + deposition and local hotspot should be avoided. These findings provide valuable insights into the temperature-dependent Li dendrite growth and contribute to the practical application of Li metal batteries.


Introduction
Lithium (Li) metal anodes are regarded as highly promising anode materials for lithium-based batteries due to their high theoretical specific energy (3860 mAh•g −1 ), low density (0.59 g•cm −3 ), and low redox electric potential (−3.04 V vs standard hydrogen electrode) [1]. However, the inevitable dendrite growth on the Li anode not only results in low Coulombic efficiency and reduced lifespan, but also may penetrate the separators and solid electrolyte, causing short circuit and even thermal runaway [2]. Diverse strategies to regulate Li dendrite growth have been developed, including modifications to a specific battery component [3][4][5][6] and applications of external fields such as pressure [7], magnetic field [8], acoustic wave [9], light field [10], electric field [11], and temperature field [12].
The temperature field within batteries, primarily influenced by the internal ohmic heating and surroundings, plays a significant role in various aspects of Li metal behavior. It affects the Li + diffusion coefficient and conductivity in electrolytes, Li atomic transfer within the Li metal bulk, and the physical properties of the solid electrolyte interphase (SEI) [13,14].
Furthermore, the uniformity of temperature distribution inside batteries has a substantial impact on the stability of Li electrodeposition and dissolution [15][16][17]. However, the mechanism underlying the temperature-dependent Li dendrite growth remains controversial, i.e., it is unclear how to understand the promotion, inhibition, and dual effects of increased temperature on dendrite growth [18][19][20][21]. These studies suggest the intricacy of temperature-dependent dendrite growth, signifying the necessity for building a unified picture. Various simulation methods are utilized to investigate Li dendrites across different scales, encompassing density functional theory [22], molecular dynamics [23], kinetic Monte Carlo [24], and phase-field [25][26][27][28] approach. Density functional theory and molecular dynamics serve as microscale techniques capable of accurately computing atom migration energy and pathways within Li metal under distinct physical conditions. However, their applicability is restricted in capturing dendrite morphological evolution due to inherent scale limitations.
Kinetic Monte Carlo facilitates the simulation of dendrite morphological evolution at a mesoscale. Nevertheless, they lack precision in depicting Li dendrite growth under diverse physical circumstances. In contrast, the phase-field simulation has emerged as a pivotal tool for comprehending temperature-dependent Li dendrite growth dynamics and multiple physical fields, and excels in addressing intricate morphological evolution and coupling of multiple physical fields. In 2018, Chen et al. [29] firstly established an electrochemical phase-field model combining heat transfer only by using the temperature-dependent Li + diffusion coefficient and found that both internal self-heating and elevated uniform initial temperature can inhibit the dendrite growth. Actually, the electrochemical reaction coefficient closely associated with the phase-field variable and temperature. Later, an improved thermal-coupled model by incorporating the reaction coefficient as a function of temperature was proposed, suggesting the either promoted or inhibited dendrite growth pattern due to internal self-heating, depending on the competition between the temperature-dependent diffusion and reaction [30]. Note that the improved model stays on the internal self-heating, without delving into the initial temperature distribution. Recently, Jeon et al. [31] fitted an accurate thermal-coupled model by formulating the experimental temperature-dependent conductivities of electrode and electrolyte, surface tension, reaction and diffusion coefficients, and found that the dendrite formation is promoted by elevating temperature. However, the reliance of this model on experimental data limits its applicability to other battery systems. In our previous work [32], we investigated the effect of initial temperature on Li dendrite morphology through temperature-dependent ionic diffusion coefficient, reaction coefficient, and conductivity, but did not couple the temperature field. In this paper, we will couple the temperature field to investigate the effect of initial temperature on Li dendrite morphology, and give a unified picture for the seemingly contradictory dendrite-promoting, dendrite-inhibiting, and dual effects of increased temperature in different electrolyte types.
The remaining parts of this paper are organized as follows: Section 2 provides the theoretical formulation of a thermal-coupled electrochemical phase-field model and the related simulation parameters. Three initial temperature distributions, i.e., uniform initial temperature, temperature gradient, and local hotspot, are adopted to investigate the effect of initial temperature on Li dendrite growth in Section 3. The conclusions are provided in Section 4.

Electrochemical phase-field model combining heat transfer
During the charging process, Li + from the electrolyte and electrons from the electrode combine at the interface to form Li atoms, which is described by the following reaction equation . (1) There are two phases (electrode and electrolyte) and three components ( , anion and solvent) in the investigated system. The electrode phase is described by a non-conserved order parameter ( for the electrode, for the electrolyte, and for the electrode-electrolyte interface) and the components are described by two concentration variables (concentration of ) and (concentration of anion). The total free energy of the system is expressed as , where , , and are the chemical, interfacial, and electrostatic energy density, respectively; is a noise term representing where and are the reference chemical potentials of and anion, respectively; is the molar gas constant; is the absolute temperature; is the initial concentration of electrolyte. The interfacial energy density is expressed as , where is the barrier height; is the gradient energy coefficient; is the anisotropic strength; is the anisotropic modulus; is the argument of . The electrostatic energy density can be written as , where is the local charge density; is the electrostatic potential; is the Faraday constants; and are valence of and anion, respectively.
The phase-field equation can be written as [29] , where is the mobility of interface; is the electrochemical reaction coefficient; is the charge-transfer coefficient; is the overpotential; is the interpolating function. The Li + concentration equation is expressed as , where is the effective diffusion coefficient, with and being the diffusion coefficients of in the electrode and electrolyte, respectively; is the bulk concentration of electrode; The electric-potential equation is given by the charge conservation as follows: , where is the effective electric conductivity; and are where is the effective specific heat capacity, with and being the specific heat capacity of electrode and electrolyte, respectively; is the effective mass density, with and being the mass density of electrode and electrolyte, respectively; is the effective thermal conductivity, with and are the thermal conductivity of electrode and electrolyte, respectively. The heat source intensity in Eq. (9) is expressed as [29] , where is the ohmic heat; is the heat generated due to overpotential; is an empirical factor for converting the theoretical current density to the experimental current density [30].
The temperature-dependent reaction coefficient ( ) and diffusion coefficient ( ) can be expressed as follows [30,33,34]: , , where and are the electrochemical reaction and ionic diffusion activation energies, respectively; is the equilibrium temperature of the system; is the correction factor. The reaction activation energy can be modified by the Li + desolvation, nature of SEI, and surface structural features of electrodes [35][36][37], while the diffusion activation energy can be modified by additives in the electrolyte [38,39].

Numerical calculation method
The investigated electrochemical system is a half cell, with electrolyte being 1.0 M lithium bis(trifluoromethanesulfonyl)imide (LiTFSI) in volume ratio 1:1 1,3-dioxolane and 1,2-dimethoxyethane (DOL/DME). For the phase-field variable, Li + concentration, and where is the normal derivative on boundaries; is the heat convection coefficient; is the emissivity; is Stefan-Boltzmann constant. Eqs. (6)-(9) are numerically solved by finite difference method on a two-dimensional size of . The parameters used in electrochemical phase-field simulations are listed in Table 1. Interfacial mobility [40] Reaction coefficient [40] Gradient energy coefficient [26] Barrier height [26] The strength of anisotropy [41] A mode number of anisotropy [40] Charge-transfer coefficient [40] Initial concentration of electrode [40] Initial concentration of electrolyte [40] Diffusion coefficient of Li + in electrode [26] Diffusion coefficient of Li + in electrolyte [26] Electronic conductivity of electrode [26] Ionic conductivity of electrolyte [26] Specific heat capacity of electrode [42] Specific heat capacity of electrolyte [42] Mass density of electrode [42] Mass density of electrolyte [42] Thermal conductivity of electrode [42] Thermal conductivity of electrolyte [42] Ionic diffusion activation energy in electrolyte [29] 3. Results and Discussion

Effect of uniform initial temperature on Li dendrite growth
The promotion or inhibition of dendrite growth depends on the competition of ionic diffusion and reaction [43]. The activation energy associated with diffusion and reaction signifies the magnitude of ionic diffusion and reaction rates, respectively [30]. As the activation energy for diffusion or reaction increases or decreases, the magnitude of the corresponding chemical process also experiences an increase or decrease. Therefore, the change of activation-energy ratio denotes the magnitude change of diffusion and reaction processes. Here, the activation-energy ratio ( ) is proposed to describe temperature-dependent reaction-diffusion competition. Fig. 2 [44] They observed that at the same temperature, an increase in current density results in a decrease in the Li-core size and an increase in the nucleation density. Additionally, higher temperatures correspond to a higher density of Li deposition. Therefore, these findings suggest that selecting a lower applied voltage and maintaining a higher uniform initial temperature are effective strategies for achieving dendrite-inhibiting deposition.  Fig. 3 is the mechanism diagram correlating the uniform initial temperature, maximum dendrite growth height ( , calculated by the total height of electrodeposition minus the average thickness of uniform deposition) [32], and . The red, green, and blue curves show that is respectively monotonically increasing, monotonically decreasing, and non-monotonic with respect to temperature, denoting the dendrite-promoting, dendrite-inhibiting, and dual effects of the increased temperature, respectively. Higher temperature can either mitigate or exacerbate dendrite growth, depending on the relationship between and temperature. The is influenced by various factors such as the electrolyte concentration and solid electrolyte interphase (SEI). Typically, the unstable formation of SEI tends to increase [30,45]. From the temperature characteristics, an electrolyte corresponds to a curve in the mechanism diagram. The red curve qualitatively Their observations show that from 273 K to 333 K, the increased temperature results in a stabler organic/inorganic composite lamellar SEI, i.e., slightly varying SEI, thereby causing a decrease in and a dendrite-inhibiting effect. The temperature gradient is defined as , where and represents the temperatures on the top and bottom boundaries of the simulation region, respectively, with in this subsection. Here ， the . Fig. 4(a) displays the dendrite morphology under a negative temperature gradient ( ). In contrast to the uniform temperature of 298 K (Fig. 2,  ), dendrite growth is intensified under a negative temperature gradient. As the negative temperature gradient increases, the dendrites growth becomes more pronounced. Fig. 4(b) displays the dendrite morphology under a positive temperature gradient ( ). Obviously, increasing the positive temperature gradient is beneficial to inhibit dendrite growth. Here, the maximum height and space utilization [32] ( ) are utilized to quantify dendrite morphology. Fig. 4(c) presents the maximum height and space utilization, indicating that a positive temperature gradient inside batteries help to reduce dendrite height. These findings align with the work conducted by Atkinson et al. [17], who observed that in symmetrical lithium metal batteries, a negative temperature gradient ( ) exhibits less uniform electrodeposition and more electrochemically inactive Li builds up on the Li metal surface . In contrast, a positive temperature gradient ( ) exhibits more uniform electrodeposition, enhances working stability, and prolongs the cycle life. From Fig. 5, under a negative initial temperature gradient, the temperature at the interface is lower, impeding Li + diffusion at the interface. Conversely, under a positive initial temperature gradient, the temperature at the interface is higher, facilitating Li + diffusion and bot top

Local hotspot
To investigate the impact of local hotspot on dendrite growth, we set an applied voltage and a uniform initial temperature of 298 K as simulation conditions, which facilitates a smooth electrodeposition (Fig. 2). Fig. 6  120 μm towards the electrolyte and electrode, causing an uneven distribution of temperature ( Fig. 6(d)) at the electrolyte-electrode interface. The heat accumulating at the interface, results in an uneven distribution of Li + concentration ( Fig. 6 (b)) and potential (Fig. 6(c)) at the interface and induces Li dendrite growth ( Fig. 6(a)). Figs. 7(a-c) depict the dendrite morphology under different hotspots. Dendrites in the hotspot area grow as the temperature of the hotspot increases (ranging from 324 K to 372 K). The simulation results align with the experimental observations of Zhu et al. [16], who discovered that hotspots can induce significant growth of Li metal if the surrounding has lower temperature, due to the locally enhanced exchange current density. Figs. 7(d-f) are SEM photos of Li deposition on Cu at different hotspot temperatures (324 K, 356 K, and 372 K), clearly demonstrating that Li deposition is considerably faster in the hot zone (at the center of the SEM image). As the hot spot temperature increases, more Li deposition can be observed in that region.

Conclusions
This paper presents a comprehensive analysis of initial temperature on Li dendrite growth using an electrochemical phase field model combining heat transfer. For uniform initial temperature, we provide a unified picture for temperature-dependent Li dendrite growth, and utilize the activation-energy ratio to unify the seemingly contradictory simulation results (different electrolyte types exhibit the promotion, inhibition and dual effect of increased temperature on dendrite growth). A promotion (inhibition) effect emerges when the activation-energy ratio is monotonically increasing (decreasing) with respect to temperature, and a dual effect emerges when the activation-energy ratio is not monotonic with respect to temperature. For non-uniform initial temperature, simulation results show that a positive temperature gradient along the discharging current facilitates uniform Li + deposition and local hotspot should be avoided. The findings of this study provide valuable insights for future advancements in temperature regulation to control dendrite growth.

Author Contributions
Yajie Li and Wei Zhao wrote the initial draft of the manuscript. Geng Zhang and Wei Zhao jointly developed the program. Geng Zhang, Siqi Shi and Yajie Li helped to revise the manuscript. Siqi Shi was responsible for the conceptualization, writing, and management. All