Pulse-Reverse Protocol for Efficient Suppression of Dendritic Micro-structures in Rechargeable Batteries

The ramified and stochastic evolution of dendritic microstructures has been a major issue on the safety and longevity of rechargeable batteries, particularly for the utilization metallic electrodes. We computationally develop pulse-reverse protocols for effective halting of the growing microstructures during extensive charging periods far beyond inter-ionic collisions. Our framework is based on the competitiveness of the dendritic growth during the pulse period versus the dissolution of the (sub)branches during the reverse charge. The detachment of the atoms from the structure preferentially occurs from the meta-stable sites which possess the lowest coordination number and require lowest activation energy for dissolution. We prove that even an infinitesimal pulse-reverse charge is more effective than an extended pulserelaxation method for the suppression of the dendritic structures.


Introduction
Metallic anodes are arguably popular materials for their applicability in the future rechargeable batteries [1,2,3]. Specifically, lithium metal possesses the lowest mass density which provides a very high energy density. It additionally has the highest electropositivity (E 0 = −3.04V vs SHE) making it suitable for high-power applications. (ρ = 0.53 g.cm −3 ) [4,5]. However, the quick formation of microstructures during the charge period leads to a tree-like formation of microstructures with high surface to volume ratio [6] which occupy a large volume, can reach the counter-electrode and short the cell. As well, they could dissolve from their thinner necks during the discharge period. Such a formation-dissolution cycle is particularly prominent for the metal electrodes due to lack of diffusion into the inner layers, as opposed to depositing in the surface [7].
Earlier model of the branched evolution have addressed the electric field and space charge as the main driving mechanism [27] for growth while the other models focused on ionic concentration causing the diffusion limited aggregation (DLA) [28,29,30]. Both mechanisms are part of the electrochemical potential [31,32], 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 rarely explored, especially in continuum scale and realistic time intervals, matching scales of the experimental time and space.
The main cause of dendritic formation is the non-uniformity of electrode surface morphology at the atomic scale and the Brownian nature of ionic motion during electrodeposition. A surface sharpness provides a large electric field that attracts the ions as a sink. The other contributing factor is the closeness of the asperity to the counter electrode, where more deposition accelerate the growth rate.
Pulse method has recently proven to be an effective mechanism for suppression of the dendrites [33,34]. The concentration of ions in a given region is the results of the cooperative role of the mass flux into/out of the region, as well consumption of ions due to reduction reaction. While the reaction process is probable during the interaction with any kind of surface, the influx of ions is in fact highly manipulated via the sharp and large electric fields( i.e. | E → 10 5 V.cm −1 | [33]). Therefore during each pulse period, the ions unfavorably accumulate at the dendrites tips and during the subsequent rest period they favorably diffuse away to lower concentration zones. Such relaxation, which occurs within the double layer (or stern layer [35]), provides a useful mechanism to achieve uniform morphology. In fact, after the equilibrium, there is still ionic exchange between the above regions, with equal rates, due to Brownian motion, which occurs in local and global scales. Particularly, during higher charge rates, the ionic concentration is completely exhausted on the electrode surface [36]; Nevertheless, the continuum-level approach could extend to a larger scale, beyond the double layer region. [37] While the pulse-relaxation method reduces the evolution of the dendrites in batteries [33,38,39,40] and during other applications [41,42], the pulse-reverse method has specific advantages including saving the total charge time and retaining charge capacity. A pulse-reverse protocol consists of a positive pulse charge followed by a negative charge, with a possible optional resting time [43,44] as shown in the Figure 2. This method reduces the concentration polarization [45] and is particularly used for controlling the thickness and uniformity of deposits [46]. With well chosen parameters, a larger cell's specific energy and specific power can be reached with this technique [47,48]. Pulse-reverse is a potentially good method to overcome mass transport limitations in  batteries [49]. This technique optimally decreases the possibly-formed concentration gradients.
Therefore, the pulse-reverse has adequate potentials in terms of health and efficiency [43].
In this paper, we elaborate further in the range of acceptable reverse-ratio for suppression of stochastically-grown dendrites and we develop new insight for the effective rest period on the curved boundary. Subsequently we carry out experimental investigation to verify our analytical developments on the pulse parameters. We perform dimensional analysis to set our formulation applicable to the large range of electrochemical devices.

Methodology
The ionic motion is generated either from the variation of the concentration (∇C) or the electric potential (∇φ). Typically, the ions in the high-concentrated zones collide and repel more and gradually move to less populated regions. In the larger scales such displacement can be represented by the diffusion length. On the other hand, ions tend to acquire drift velocity in the electrolyte medium when exposed to electric field. Therefore, as represented in the Figure 1, the total displacement of the ions in the given coarse time interval of δt is given by 1 where D + is the ionic diffusion coefficient in the electrolyte, andĝ is a normalized vector in a random direction, representing the Brownian dynamics, µ + is the mobility of cations in electrolyte, E is the local electric field, which is the gradient of electric potential ( E = −∇φ ).
The diffusion length represents the average progress of a diffusive wave in a given time, obtained directly from the diffusion equation. [50] 1 δt = n i=1 δt i where δt k is the inter-collision time, typically in the range of f s. The pulse charging in its simplest form consists of trains of square active period t ON , followed by a square rest interval t OF F in terms of current I or voltage V as shown in Figure 2. The period P is the time lapse of a full cycle defined as: We define the reverse ratio f REV as: which is solely true for the intended galvanostatic (i.e. constant current density) charging. In fact f REV shows the fraction of the discharge-to-charge amount (i.e. f REV = 0.1 means for every 10 attached atoms, there is 1 removal). As well we define the relaxation-ratio f OF F as: Based on the Equations 2, 3 and 4, the pulse reverse curve is uniquely characterized.
The reverse ratio f REV is characterized based on charge amount whereas the relaxation-ratio f OF F based on the amount of time. The algorithm defined in the Figure 3 sets to enter the either. Hence, to enter the discharge phase, i.e. the reverse, a certain amount of charge needs to be met.
This means that after a certain number of depositions, the program will set the phase to reverse and it will discharge the specified charge amount based on the reverse-ratio f REV . In the ideal  case, one can reduce the time to an infinitesimal amount and magnify the current i indefinitely.
Therefore, the reverse phase could be explained as a Dirac delta function function below: where the higher the current i is, the less time is needed to obtain the same discharge amount Q.
The electrodeposition-dissolution process for the formation of the dendritic morphologies is described in detail below: 1. Define the values for the time intervals composing the period P : they are the pulse period t ON , rest period t OF F and the reverse period t REV which is defined through the reverse-ratio f REV .
The initial values are determined as the relaxation time scale for blocking electrodes, such that: [42] t on = t of f = κl D where κ is the scale of the double-layer, l is the scale of the domain and D is the diffusivity coefficient of the bulk medium.
2. Determine the displacement of the each ion ∆ r in the given time interval δt, where the pulse period carries the diffusion and migration effect, while the rest period only carries the sole diffusion effect. τ is the remainder of the total time divided by the period P and is the measure for determining the time zone, such that:

Experimental
We performed experiments within a manually-fabricated sandwich cell [51], which provides the possibility of in-situ observation of growing dendrites from the periphery (Figure 6a 2. The grayscale image I i,j is binarized to J i,j via the grayness threshold I c as below: the threshold value I c has been chosen to minimize the weighted intra-class variance σ 2 defined as:  where ω 0 and ω 1 are the total fraction of element divided by the value of I c and σ 2 0 and σ 2 1 are their respective variances. [52] Such minimization ensures that the resulted 3. The circular sandwich cell with the radius R has been divided of 3 arcs with the angle of 2π 3 and width incremental length of δx, which is supposed to be projected to a 2D plane with the incremental width of δx . From Figure 6a due to geometry we have: where D is diameter of the sandwich cell [53].
4. Starting from the electrode surface, the occupied space by the dendrites has been calculated by the square site percolation paradigm [54].

The infinitesimal calculations have been integrated and normalized to inter-electrode dis-
tance (λ i := λ i /l ) to get the dendrite measureλ as: where h max is the height of the tallest peak in the observed dendrite. The integral Equation 6 has been obtained by incremental sum from experimental data.

Results & Discussions
During the excessive pulse period, which imposes the ionic flux on the dendrite body, the concentration order in the interface regions turns to the following due to manipulation from the electric field:

Concentration: T ip > Bulk > V oids
whereas, during a typical diffusion process, the flow of the ions occurs from high-to-low concentrated regions (i.e. J = −D∇C). Therefore, given enough time for the relaxation period, the concentration gradient should vanish, ensuring the equilibrium in ionic kinetics. Figure 4 shows that the reverse ratio f REV correlates directly with the dendritic tendency.
Starting from the uniform charging in the Figure 4a the average height reduces consecutively through Figures 4b, 4c and 4d where atoms leave the highly condensed sites and reattach on the inner unoccupied regions. This phenomenon is explained by the fact that the reverse current density at the tips is larger than in other locations which makes the high spots' atoms more prone to detachment [55] due to Arrhenius relationship as [56]: Therefore, due to lowest coordination number (i.e. N = 1), such sites require the lowest activation energy ∆G for dissolution and the preferential detachment of the high-spots with sharp interfaces leads to the melting of the peaks and the residue of smoother surface. The density results in the Figure 5b with identical reverse ratio f REV are slightly higher in the presence of relaxation ratio f OF F , which is additionally effective in vanishing the concentration gradients in the over-occupied zones until a certain limit. Nevertheless, the density decreases after a peak merging to smaller values by the reverse-ratio. The dissolution of the atoms is supposed to cut and shorten the branches in the structure and, in the best cases, completely remove them and flatten-out the interface. However, excessive dissolution will lead to the disturbing of the flat interface where the extra cavities form. Figure 7 schematically explains such over-dissolution process, where the disruption occurs again. The green arrows indicate an increase in the f REV f OF F ratio and their intersection with the flat interface is at the critical value f REV f OF F ≈ 0.15. The extended advantage of this the reverse-charing is the facilitation of the accessibility in ions to inner layer during the subsequent charge-discharge cycles. Comparing the Figures 5a and 5b the cooperation of pulse-reverse (f REV ) and pulse-relaxation (f OF F ) is constructive, albeit pulse reverse shows much more efficacy for increasing the density. This has collectively been shown in the Figure 8 where the relatively much smaller reverse ratio f REV leads to much larger density values, even compared to extended relaxation ratio f OF F . Thus, in order to reach comparable densities to the pulse-reverse technique, pulse-relaxation obliges a very long and unnecessary waiting time [49].
In practice, the partial discharge process can be imposed by programming the charger electronics and the entire pulse-reverse protocol could be automated to the infinitesimal (i.e. finite) time intervals, scaling down up to ∼ µs. Needless to mention that, the results obtained are suitable and effective during a single charge period and short-term application, whereas the long term performance parameters such as coulombic efficiency and cycle stability will be studied in future works. Ultimately, tuning the morphology adjusts the distribution of the electric field and the concentration profile in the favor of uniform growth and since they are controlled not to reach extreme values, there will possibly be less side reactions.

Conclusion
In this paper, we have developed a reverse feedback control loop for more effective prevention of formation of dendritic growth during charge period of battery. The controlling parameters is mainly the charge-based reverse ratio f REV accompanied with time-based relaxation ratio f OF F . The preferential detachment of ions during the reverse charge correlates with the respective coordination number as well as their location in the microstructure. Both pulse-reverse and pulse-relaxation were investigated and compared with variant charge ratios. Pulse-reverse has shown to be the most efficient method for morphology control during dendritic evolution.