A Novel Method for Estimating the Charge Equilibrium within the Dendrites of Rechargeable Batteries
Graphical abstract
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 ( vs SHE) that likely provides the highest possible voltage, making it suitable for high-power applications such as electric vehicles. () [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 (). 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.
Section snippets
Computational method
Fig. 1 represents the dendritic evolution in the lab scale as well as in our computations. The ionic flux is generated in response to the variation of the electrochemical potential, which is per see the result of the variations (i.e. gradient) of concentration () or electric potential (). In the ionic scale, the regions of higher concentration tend to collide and repel more and, given enough time, diffuse to lower concentration zones, following Brownian motion. Such inter-collisions could
Results & discussions
Finding the minimum energy for randomly-formedmicrostructures is usually a non-convex problem, which makes it difficult to solve. This is merely due to stochastic allocation of the atoms. Since the reciprocal distance matrix R is symmetric (), for any given matrix B there exists matrix Z such that:
Due to symmetry, the trace of reciprocal matrix is the sum of it’s eigenvalues . Since, the distance of each atom to itself is zero, and hence:
Conclusions
In this paper, we developed a computational method for determining charge equilibrium distributionwithin the given stochastically-evolved dendritic microstructure. Our computationally affordable method, which has mainly been divided to simpler compartments, has been compared against the conventional method as well as the commercial package.
The significance of this method is the independence from the initial condition and very low computational cost. Our method could be used for determining the
CRediT authorship contribution statement
Asghar Aryanfar: Conceptualization, Validation, Formal analysis, Investigation, Data curation, Writing - original draft, Writing - review & editing, Visualization, Funding acquisition. Dimitri M. Saad: Methodology, Resources, Software, Writing - review & editing, Supervision, Project administration. William A. Goddard III: Supervision, Project administration.
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.
Acknowledgement
The authors would like to thank internal support from the Faculty of Engineering and Architecture at American University of Beirut.
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