A Novel Method for Estimating the Charge Equilibrium within the Dendrites of Rechargeable Batteries

Abstract

The nonuniform formation and growth of microstructures during the electrochemical charging of a battery is the main reason for the short circuit and capacity fade. The charge distribution across the micro-structure is the result of both local and global equilibrium which is a non-convex problem merely due to random placement of the atoms. As such, obtaining the charge equilibrium (QEq) is critical, since the amount of withheldcharge determines the success rate of the bond formation for the ionic species approaching the microstructure which ultimately determines the morphology of the electrochemical deposits. Herein we develop a computationally-affordable method for estimating the charge allocation within such microstructures. The cost function and the span of the charge distribution correlates very closely with the trivial method as well as a conventional method, albeit having significantly less computational cost. The method can be used for optimization in non-convex problems, specially for those of randomly-formed morphology.

Introduction

Metallic anodes such as lithium, sodium and zinc are arguably highly attractive candidates for use in high-energy and high-power density rechargeable batteries [1], [2], [3]. In particular, lithium metal possess the lowest density and smallest ionic radius which provides a very high gravimetric energy density and possesses the highest electropositivity ($E^0=-3.04V$ vs SHE) that likely provides the highest possible voltage, making it suitable for high-power applications such as electric vehicles. ($\rho =0.53g.{\mathit{cm}}^{-3}$) [4], [5]. During the charging, the fast-pace formation of microstructures with relatively low surface energy from Brownian dynamics, leads to the branched evolution with high surface to volume ratio [6]. The quickening tree-like morphologies could occupy a large volume, possibly reach the counter-electrode and short the cell (Fig. 1a). Additionally, they can also dissolve from their thinner necks during subsequent discharge period. Such a formation-dissolution cycle is particularly prominent for the metal electrodes due to lack of intercalation¹ [1]. Previous studies have investigated various factors on dendritic formation such as current density [7], electrode surface roughness [8], [9], [10], impurities [11], solvent and electrolyte chemical composition [12], [13], electrolyte concentration [14], utilization of powder electrodes [15] and adhesive polymers [16], temperature [17], guiding scaffolds [18], [19], capillary pressure [20], cathode morphology [21] and mechanics [22], [23]. Some of conventional characterization techniques used include NMR [24] and MRI [25]. Recent studies also have shown the necessity of stability of solid electrolyte interphase (i.e. SEI) layer for controlling the nucleation and growth of the branched medium [26], [27].

Earlier model of dendrites had focused on the electric field and space charge as the main responsible mechanism [28] while the later models focused on ionic concentration causing the diffusion limited aggregation (DLA) [29], [30], [31]. Both mechanisms are part of the electrochemical potential [32], [33], indicating that each could be dominant depending on the localizations of the electric potential or ionic concentration within the medium. Nevertheless, their interplay has been explored rarely, especially in continuum scale and realistic time intervals, matching scales of the experimental time and space.

Recent works, have addressed the nucleation aspect of electrodeposition via tuning surface energy and the radius of the interface [34], [35].Dendrites instigation is rooted in the non-uniformity of electrode surface morphology at the atomic scale combined with Brownian ionic motion during electrodeposition. Any asperity in the surface provides a sharp electric field that attracts the upcoming ions as a deposition sink. Indeed the closeness of a dendritic spike to the counter electrode, as the source of ionic release, is another contributing factor. In fact, the same mechanism is responsible for the further semi-exponential growth of dendrites in any scale. During each pulse period the ions accumulate at the dendrites tips (unfavorable) due to high electric field in bulging geometry and during each subsequent rest period the ions tend to diffuse away to other less concentrated regions (favorable) [36]. The relaxation of ionic concentration during the idle period provides a useful mechanism to achieve uniform deposition and growth during the subsequent pulse interval. Such dynamics typically occurs within the double layer (or stern layer [37]) which is relatively small and comparable to the Debye length. In high charge rates, the ionic concentration is depleted and concentration on the depletion reaches zero [38]; nonetheless, our continuum-level study extends to larger scale, beyond the double layer region [39].

Various charging protocols have been utilized for the prevention of dendrites [40], which has previously been used for uniform electroplating [41]. We have proven that the optimum rest period for the suppression of dendrites correlates with the relaxation time of the double layer for the blocking electrodes which is interpreted as the RC time of the electrochemical system [42]. We have explained qualitatively how relatively longer pulse periods with identical duty cycles will lead to longer and more quickening growing dendrites [43]. We developed coarse grained computationally affordable algorithm that allowed us reach to the experimental time scale ($~\mathit{ms}$). Additionally, in the recent theoretical work we indicated that there is an analytical criterion for the optimal inhibition of growing dendrites [44].

Additionally, the ultimate morphology of the dendritic electrodeposits, depends on the possibility of the bond-formation when ion reaches the outer boundary of the microstructure. The success of electron transfer in such approach would highly be determined to the amount of the charge presents in the electron transfer site. Therefore, in this paper, we elaborate on the charge distribution in equilibrium across the dendritic microstructures, where the placement of the stochastically-grown dendrites. Subsequently, we verify our method via comparison with trivial method, which is far more computationally expensive as well as a conventional package. This affordable method of computation for charge distribution can be utilized for any given microstructure, specially those of large scales.