Physics-based Modeling of Pulse and Relaxation of High-rate Li/CF_x-SVO batteries in Implantable Medical Devices

The basic electrode, material, and particle properties used in our model can be seen in Table 1. Our model is run in an isothermal setting at ambient temperature 37^∘C. The SVO material is assumed to be able to take in 6 stoichiometry of Li⁺, 2 from the Ag⁺ reaction and 4 from the V⁵⁺ reaction, resulting in volumetric energy densities $q_{{}_{\text{Ag}}}=\frac{1}{3}q_{{}_{\text{SVO}}}$, $q_{{}_{\text{V}}}=\frac{2}{3}q_{{}_{\text{SVO}}}$. Since inserted Li⁺ coexist in SVO, the particle volume and active reaction area is assumed to be shared across the Ag⁺ and V⁵⁺ reactions, and thus the capacity fraction weighted SVO filling fraction [29] is $\overline{c}_{{}_{\text{SVO}}}=\frac{1}{3}\overline{c}_{{}_{\text{Ag}}}+\frac{2}{3}\overline{c}_{{}_{\text{V}}}$. It is estimated that $x\approx 1$ in CF_x [18], and CF_x particles are assumed to consist of a single CF phase. The Li/CF_x-SVO batteries in this study have the same CF${}_{\text{x}}$-SVO cathode material, where the nominal capacity ratio between CF_x and SVO is 2.031:1, yielding nominal capacity fractions $\frac{1}{3}$ and $\frac{2}{3}$ for SVO and CF_x, and a nominal specific energy density of 434.12 mAh$\cdot\text{g}^{-1}$ for the electrode. For the representative battery, the measured electrode thickness is 140 $\upmu$m and measured electrode mass 3.055 g. Other batteries in the dataset have slight difference in cathode thickness, mass, and estimated active solid volume fraction and capacity due to manufacturing variation.

For electrolyte transport, a concentrated Stefan-Maxwell electrolyte model [48, 27] is used to represent the high-rate electrolyte 1.1M LiAsF₆ PC/DME 1:1. The properties of the binary electrolyte, including liquid-phase diffusivity $D_{l}(c_{{}_{l}},T)$, ionic conductivity $\sigma_{l}(c_{{}_{l}},T)$, transference number $t^{0}_{+}(c_{{}_{l}},T)$, and thermal factor $\gamma(c_{{}_{l}},T)$, are obtained from a corresponding Advanced Electrolyte Model (AEM) [49, 50, 51], and the exact values are shown in Supplementary Fig. 2.

The basic model formulation follows the schematic drawing in Fig.  1. The anode is a Li-foil (Li metal) anode foil and we assume the cell is cathode capacity limited. The porous separator used has thickness 20 $\upmu$m, porosity 0.4, tortuosity $\epsilon^{-0.85}$, and is discretized into 5 finite volumes. The CF_x-SVO cathode is spatially discretized into $N$ = 5 finite volumes, each containing $P_{{}_{\text{SVO}}}=5$ cylindrical SVO and $P_{{}_{\text{CF}}}=5$ spherical CF_x particles with sizes sampled from their respective distributions. To simulate a multi-active material electrode with limited number of particles and still ensure correct capacity fractions $\widetilde{Q}_{\text{CF}_{x}}$, we need to preserve $\widetilde{Q}_{\text{SVO}}$ at both electrode scale and finite volume scale, and thus introduce volume correction terms [29] to each finite volume $\mathcal{V}_{n}$ such that volume fractions $\widetilde{V}_{n,p_{{}_{\text{CF}}}}$, $\widetilde{V}_{n,p_{{}_{\text{SVO}}}}$ reflect correct active material contributions to the average state of charge and total reaction rate when the multi-active material electrode CF${}_{\text{x}}$-SVO has capacity fractions $\widetilde{Q}_{\text{CF}_{x}}$, $\widetilde{Q}_{\text{SVO}}$. As for the cell depth of discharge, i.e. cathode filling fraction, it can be both predicted from the model by averaging across finite volumes $\overline{c}=\frac{1}{N}\sum_{n=1}^{N}\overline{c}_{n}$ and estimated at each measured timestep from experimental data using $\bar{c}(t)=\frac{Q_{out}}{Q_{c}}=\frac{Q_{out}}{\rho_{\text{e},c}m}$. The initial depth of discharge of the active materials are set to $c_{{}_{\text{Ag}}}(t=0)=0.0383$, $c_{{}_{\text{V}}}(t=0)=0.0356$, and $c_{{}_{\text{CF}}}(t=0)=0.0156$ across the particles to account for cell capacity release from burn-in process ($\overline{c}(t=0)=0.024$) and ensure initial potential balance between active materials at 3.18 V, which matches the cell voltage first measured at the start of the pulse test for the representative battery.

With only minor adjustments to the exchange-current density rate constants, we adopt the Butler-Volmer (BV) [48, 52, 53, 54, 55, 56] reaction kinetics of Ag⁺, V⁵⁺ separately from our previous models of Li/SVO and Li/CF_x-SVO [29] to obtain their reaction current densities $i$ as functions of both electrolyte concentration and solid-phase filling fractions,

where $k_{{}_{\text{Ag}}}=1.5$$\times$$10^{-5}\ \text{A}$$\cdot$$\text{m}^{-2}$ and $k_{{}_{\text{V}}}=7$$\times$$10^{-1}\ \text{A}$$\cdot$$\text{m}^{-2}$. Note only the sides of the cylindrical SVO particle are active for reaction, and the SVO particle is further discretized in the radial direction by $\Delta r=0.02\ \upmu\text{m}$ to capture concentration gradients. From previous modeling studies of Li/CF_x-SVO batteries under low- and medium- current rates [18, 29], it was clear that the active material depletion order was Ag⁺, V⁵⁺, CF_x. While the pulse test data has quarterly pulses mixed in, we expect the thermodynamics to dominate again soon after the operation has been switched to background monitoring mode. As a result, the relatively fast decrease in cell voltage from $\sim$3.2 V to $\sim$3.0 V during the first few segments background monitoring mode in Fig.  2 can be directly attributed to increased Ag⁺ depletion, allowing us to refine our selection for its exchange current density rate constant $k_{{}_{\text{Ag}}}$. Compared to the conventional Li⁺ intercalation reaction of V⁵⁺, it is commonly believed that the Ag⁺ displacement reaction is kinetically slower [6]; such difference is in turn reflected in a much smaller $k_{{}_{\text{Ag}}}$. The transport of metallic silver atoms as well as nucleation and growth of silver nanoparticles [35, 12], which require statistical models such as Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory [57, 58, 59], is beyond the scope of our electrochemical porous electrode models. We thus assume they do not interfere with electrochemical reactions of Ag⁺ and V⁵⁺ since remarkably fast Ag atomic diffusion were measured during lithiation of silver doped vanadium oxide systems [60] with similar layered structures.

The electrochemical reduction reaction of CF_x has been experimentally observed to be irreversible [17] and obey Tafel kinetics [19, 16] even at very low current rates, indicating a very low exchange current density rate constant for CF_x [61]. The rest of background monitoring mode segments in Fig.  2 can be directly attributed to increased CF_x depletion, and allows us to obtain a matching $k_{{}_{\text{CF}}}$,

where $k_{{}_{\text{CF}}}=1.5$$\times$$10^{-5}\ \text{A}$$\cdot$$\text{m}^{-2}$. The small $k_{{}_{\text{CF}}}$ reflects the kinetics limitation of CF_x [19, 16, 61] as an electrode material, which has been traditionally only used for medium-rate batteries [1, 2, 5, 3, 4]. Its active material is assumed to be coated at the spherical CF_x particle surface, and thus CF_x has negligible mass transport limitations. The spherical $c_{{}_{\text{CF}}}$ can be treated as a homogeneous particle, and its active reaction area available decreases with increasing $c_{{}_{\text{CF}}}$ [18]. Since current is the product of current density and active reaction area, the effect of an decreasing active reaction area can be introduced to $k_{{}_{\text{CF}}}$ as an area term $f^{\text{CF}}_{A}(c_{{}_{\text{CF}}})=(1-c_{{}_{\text{CF}}})^{\frac{2}{3}}$ in Eq.  7. In our model of Li/CF${}_{\text{x}}$-SVO, the contributions to current from Ag⁺ and CF${}_{\text{x}}$ reactions during high-rate pulsing are small when compared to that of the V⁵⁺ reaction; thus, SVO, and more specifically its fast V⁵⁺ reduction reaction, is the source of high-power capability of Li/CF_x-SVO cells.

Due to coexistence of 3 reduction reactions from 2 different particles, equations for both intra- and inter-particle scale parallel reactions are used in the Hybrid-MPET framework [29]. The sum of reactions from all particles in $n^{\text{th}}$ finite volume $\mathcal{V}_{n}$ yields its total finite volume average reaction rate,

where the corrected volume fractions $\widetilde{V}_{n,p_{{}_{\text{SVO}}}}$, $\widetilde{V}_{n,p_{{}_{\text{CF}}}}$ [29] ensure that the correct capacity ratio is preserved in all finite volumes and thus also the multi-active material electrode regardless of the number of particles used [29]. For SVO, the particle-scale average volumetric reaction rate $\frac{\partial\overline{c}_{n,p_{{}_{\text{SVO}}},_{\text{Ag}}}}{\partial t}$, $\frac{\partial\overline{c}_{n,p_{{}_{\text{SVO}}},_{\text{V}}}}{\partial t}$ are derived from the volume average of reaction rate of Ag⁺, V⁵⁺ reduction from the set of cylindrical SVO shells at different particle radii [62, 63, 64]. The macroscopic discharge current density $i_{\text{cell}}$ is thus supported by all 3 parallel reduction reactions at the cathode,

The OCV of Li/SVO [11, 12, 18] has a voltage plateau at 3.24 V covering the first $\frac{1}{3}$ cell capacity, strongly inferring coexistence of multiple stable phases throughout the Ag⁺ displacement reaction. As a result, to describe the thermodynamics of Ag⁺ reduction as a function of its depth of discharge $c_{{}_{\text{Ag}}}$, we utilize a double-well [65] like homogeneous free energy functional and a simple gradient penalty term to describe the non-homogeneous free energy [66, 67]. The resulting diffusional chemical potential $\mu_{{}_{\text{Ag}}}$ of Li⁺ inserted into SVO due to Ag⁺ reduction is

where

$k_{\text{B}}=1.38$$\times$$10^{-23}\ \text{J}$$\cdot$$\text{K}^{-1}$, $T=310.15\ \text{K}$, $\widetilde{\Omega}_{a}=3.0$, and $\kappa=5$$\times$$10^{-7}\ \text{J}$$\cdot$$\text{m}^{-1}$. As seen in Fig.  3(a), the corresponding OCV is thus

The OCV of Li/SVO [11, 12, 18] also has a second voltage plateau at 2.6 V that spans across the rest $\frac{2}{3}$ cell capacity; most studies attribute it to the V⁵⁺ reduction process that is accompanied by Li⁺ intercalation into an single-phase host structure with high degree of disorder [6, 11, 12, 10]. As seen in Fig.  3(b), the thermodynamics of V⁵⁺ reduction is described by a separate OCV $\Delta\phi^{\text{eq}}_{\text{V}}$ as a function of its filling fraction $c_{{}_{\text{V}}}$,

By assuming Tafel kinetics [19, 16] in the form of Eq.  7, we can extract [18] its OCV $\Delta\phi^{\text{eq}}_{\text{CF}}$ from experimental data as a function of its depth of discharge $c_{{}_{\text{CF}}}$,

From our modeling perspective, the cell voltage is a macroscopic and collective representation of chemical potentials [55, 68] and concentrations of inserted Li⁺ in the electrode solid phase. The pulse waveforms and cell voltage recovery during the initial 10 s relaxation, or any cell voltage evolution in general, can be attributed to Li⁺ chemical potential and concentration changes in the solid phase and electrolyte, which must involve both reaction kinetics and diffusion. As a result, rather than relying on unrealistically large pseudo-capacitances [30], it is more sensible to use mass transfer limitations at various stages of Li/CF_x-SVO cell operation to capture cell voltage transient behavior during high-rate pulses and post-pulse relaxations. While there is yet no consensus on the exact dependency of the diffusion coefficient of Li⁺ in SVO on depth of discharge [6, 7, 35, 69, 36] from existing experimental studies of Li/SVO cells, these studies collectively showed that solid-phase diffusion in SVO is dependent on both operation conditions and SVO cathode filling fraction: (1) The internal resistance of Li/SVO was dominated by mass transfer limitations at higher current density pulses and longer pulse duration. (2) SVO as a cathode generally faces increasing diffusion limitations as more Ag⁺ is displaced and V⁵⁺ becomes only active material in the remaining oxide with high degree of disorder [6, 11, 12, 10]. However, note that in previous studies on solid-phase diffusion in SVO did not account for the fact that Li⁺ can insert both into octahedral and tetrahedral sites of SVO lattice [13] due to the parallel Ag⁺ and V⁵⁺ reduction reactions, respectively. As a result, we believe that using a single diffusion coefficient to represent mass transport of SVO overlooks the fact that Li⁺ insertion takes place in parallel along two different pathways. Based on observations from existing experimental studies of SVO and similar modeling studies [70], we hypothesize that solid-phase diffusion in SVO can be more accurately captured by two parallel diffusion processes across distinct pathways, separately for Li⁺ insertion due to Ag⁺ and V⁵⁺ reactions.

SVO has a monoclinic unit cell [71, 11] under space group C$2\text{/}\text{m}$: vanadium oxide layers are formed by sharing edges and corners of distorted VO₆ octahedra, and the layered structure is stabilized by inserted Ag⁺ between them [9]. In similar monoclinic SVO structures, the silver interlayer between the vanadium oxide sheets was found to allow for more facile lithium diffusion than the octahedral coordination of the vanadium [14, 72, 73]. The expectation that the Li⁺ diffusion coefficient from the Ag⁺ reaction $D_{{}_{\text{Li,Ag}}}$ is larger than Li⁺ diffusion coefficient from V⁵⁺ reaction $D_{{}_{\text{Li,V}}}$ matches the experimental measurements [35, 28] of solid-phase diffusion in SVO at different cell voltages, where the diffusion coefficient of Li⁺ in SVO at $V_{\text{cell}}=3.2$ V (before Ag⁺ depletion) was larger than that at $V_{\text{cell}}=2.6$ V (full Ag⁺ depletion). In addition, for a Li/SVO cell under high-rate pulsing, considering the reaction of Ag⁺ is much slower than V⁵⁺, it is expected that it is mostly V⁵⁺ reaction supporting the large current. As a result, it is also expected that the greater mass transfer limitations in Li/SVO cell measured at higher current density pulses and longer pulse duration can be mostly attributed to the lower $D_{{}_{\text{Li,V}}}$, which is supported by the measured or calculated poor Li⁺ diffusivity in various vanadium oxides electrode materials [74, 75, 76, 77, 78, 79].

Based on the analysis above, Li⁺ diffusion in SVO is described by two independent and parallel diffusion processes,

The flux of Li⁺ due to Ag⁺ reaction $\textbf{F}_{{}_{\text{Ag}}}$ is proportional to the gradient in chemical potential $\mu_{{}_{\text{Ag}}}$ because of its reaction’s association with phase transition processes; Li⁺ insertion due to V⁵⁺ reduction is modeled as intercalation into solid solution material and thus the flux of Li⁺ due to V⁵⁺ reaction $\textbf{F}_{{}_{\text{V}}}$ is governed by Fick’s law. For species conservation at different SVO particle depths,

Under high-rate pulsing, due to the kinetic limitations of Ag⁺ reaction, the impact of $D_{{}_{\text{Li,Ag}}}$ on cell voltage is overwhelmed by that of the smaller $D_{{}_{\text{Li,V}}}$; the impact of $D_{{}_{\text{Li,Ag}}}$ is also minimal after the pulse mode because the background monitoring mode operates under such small currents ($\sim$11 $\upmu$A) that the system is not diffusion limited. Our model is thus insensitive to $D_{{}_{\text{Li,Ag}}}$, and we use a generic solid-phase Li⁺ diffusion coefficient $D_{{}_{\text{Li,Ag}}}=5$$\times$$10^{-13}\ \text{m}^{2}$$\cdot$$\text{s}^{-1}$ [35]. For Li⁺ intercalation from V⁵⁺ reaction, as seen in Fig.  4(a), we use the pulse waveforms and initial 10 s of post-pulse relaxations of the representative cell to formulate an empirical $D_{{}_{\text{Li,V}}}(c_{{}_{\text{SVO}}})$ input to our model, with

The $D_{{}_{\text{Li,V}}}$ values are also shown as functions of Ag⁺ depth of discharge $c_{{}_{\text{Ag}}}$ and cell voltage in background monitoring mode before each pulse in Fig. 4(b)(c), respectively. We see that more drastic change of $D_{{}_{\text{Li,V}}}$ occurs when considerable fraction of Ag⁺ has been depleted and cell voltage during background monitoring mode falls below $\sim$3.0 V. We hypothesize that the evolution of $D_{{}_{\text{Li,V}}}$ can be explained by the increased disorder, amorphization, and structural defects between vanadium oxide layers [6, 7, 8, 9] caused by the size difference between Ag⁺ and Li⁺; the more Ag⁺ displacement, the worse deterioration of the structural integrity of the crystalline SVO. While we assumed that the two Li⁺ parallel but distinct insertion pathways coexist, since they share the same particle volume, it is possible that structural changes in SVO particles due to substantial Ag⁺ displacement could affect the Li⁺ intercalation pathways from V⁵⁺ reaction, and subsequently impact $D_{{}_{\text{Li,V}}}$ evolution.

It has been widely observed in the literature that formation of highly conductive metallic silver can considerably improve conductivity of SVO and silver vanadium phosphorus oxide (Ag₂VO₂PO₄) electrodes [11, 12, 9, 35, 80] and lower the cell resistance during operation. The increase in cathode conductivity, again, usually takes place during the first $\frac{1}{3}$ of cell depth of discharge, and is directly associated with the extent of Ag⁺ depletion. While CF_x has extreme low conductivity [81, 82, 83], its reduction reaction can contribute to cathode conductivity change: the leftover carbon is a good conductive aid yet LiF is a insulator that forms inside both the carbon matrix and the free volume of the electrode [84]. As a result, the evolution of the CF_x-SVO cathode conductivity is affected by the depth of discharge of active materials. An evolving cathode conductivity is thus another property that needs to captured, where smaller or larger cathode conductivity lowers and raises cell voltage during high-rate pulses, respectively, but both have limited impact on the pulse waveform shape.

For the specific CF_x-SVO hybrid cathode material in this study, its electronic conductivity is measured using the experimental configurations described in Gomadam et al.  [85] at different cell depth of discharge as seen in Fig.  4(d). The evolution of the cathode conductivity $\sigma_{s}$ includes contributions from the aforementioned factors, and is passed into our model as a function of cell depth of discharge $\overline{c}$,

Existing studies have observed [12, 11, 35, 42] that Li metal anode’s contribution to cell resistance could not be neglected in Li/CF_x-SVO and Li/SVO cells when tested over long periods of time. We noticed how previous assumptions [29, 18, 86] on Li metal anode having fast BV kinetics and thus negligible kinetic limitations might not apply to our pulse test. The quarterly high-rate defibrillation pulses have current densities reaching $\sim$$0.04\ \text{A}$$\cdot$$\text{cm}^{-2}$), which is higher than the current densities ($\sim$$0.01\ \text{A}$$\cdot$$\text{cm}^{-2}$) capable of causing significant Li stripping and pitting [87] for the Li metal anode surface despite the presence of a solid electrolyte interphase (SEI) passivation layer. The surface damage from Li stripping and pitting was also found to be larger for higher current densities. As a result, the Li metal anode surface should not be always treated as an ideal rectangular cuboid or cylinder with even surfaces. The Li metal anode is damaged by repeated high current density pulses, ending with large amounts of $\sim\upmu\text{m}$ voids and stripped regions at its uneven surface, which yields a significant larger electrochemically active surface area [87, 45] compared to that of a pristine Li metal anode.

The larger active area greatly boosts the intrinsic kinetics at Li metal anode and lowers cell resistance during high-rate pulsing, whose impact can again be captured by an area factor $f^{\text{foil}}_{A}$ and parameterized into the effective exchange current density rate constant $k^{\text{eff}}_{\text{Li}}=k_{\text{Li}}f^{\text{foil}}_{A}$ in the BV equation,

where $k^{\text{eff}}_{\text{Li}}$ has been measured to be as high as $\sim$$400\ \text{A}$$\cdot$$\text{m}^{-2}$ in carbonate-based electrolytes during ultrafast cyclic voltammetry [88], which is substantially higher than the $k_{\text{Li}}\sim 5$$-$$30\ \text{A}$$\cdot$$\text{m}^{-2}$ measured for pristine Li metal anodes in similar electrolytes. Note that $k^{\text{eff}}_{\text{Li}}$ does not increase infinitely and tends to converge after enough cycles [44]. In our case, we also expect $k^{\text{eff}}_{\text{Li}}$ to rapidly increase with the first few pulses and then reach a plateau because Li metal anode surface morphology change cannot be infinite. Under the same pulse current density, due to the increase in electrochemically active surface area, the average current density at the surface of Li metal anode decreases, and thus further repeating pulses at similar current density and pulse duration create lesser and lesser surface damage. As a result, there is a limit on how much anode surface morphology change takes place to accommodate for specific high-rate pulse current density and pulse duration. Since the quarterly defibrillation pulses always demand the very similar current density and all last for $\sim$4 s, as seen in Fig.  4(e), we formulate a sigmoid-type function to capture the impact of change in Li metal anode surface morphology on Li metal anode $k^{\text{eff}}_{\text{Li}}$,

where the $k^{\text{eff}}_{\text{Li}}$ plateau at $400\ \text{A}$$\cdot$$\text{m}^{-2}$ is in reference to the measured effective exchange current density rate constants from Boyle et al.  [88].

Film resistance buildup [11, 12, 9, 35, 42] at the Li metal anode in Li/CF_x-SVO and Li/SVO cells has been widely observed in experimental studies [37, 38, 89, 39, 40, 41, 39, 43]. The formation of multi-layered resistive films on the Li metal anode surface includes contribution from both traditional Li metal anode solid electrolyte interface (SEI) and deposition of dissolved vanadium and silver species from the cathode, whose presence was found via anode surface mapping using X-ray absorption spectroscopy and X-ray microfluorescence of the Li metal anode [40, 41]. Cathode dissolution leads to loss of electroactive material to non-aqueous electrolyte [40, 41, 37, 38, 39], where dissolved cathodic species are then able to travel to the anode and deposit. In most secondary batteries, cathode dissolution is considered as loss of active material [90, 91, 92] and more emphasis is put on its impact on capacity fade. In our Li/CF_x-SVO batteries, since we have sufficient Li from the anode, we focus more on the impact of cathode dissolution on increasing cell resistance and diminishing pulse power capability of the cell. As a result, we introduce an additional film resistance term $R^{\text{foil}}_{\text{film}}$ to the effective overpotential $\eta^{\text{eff}}_{{}_{\text{Li}}}=\eta_{{}_{\text{Li}}}+i_{{}_{\text{Li}}}R^{\text{foil}}_{\text{film}}$ to replace $\eta_{{}_{\text{Li}}}$ in Eq. 22. Assuming the buildup of film resistance at the anode has a limit, as seen in Fig.  4(f), we again formulate a sigmoid-type function to capture the growth of $R^{\text{foil}}_{\text{film}}$,

As seen in Fig.  5, we overlay the pulse test experimental voltage measurements from the representative cell and its Hybrid-MPET model predictions; we also highlight the accuracy of our model in predicting battery pulse performance using the representative cell’s pulse 1, 7, and 18. The corresponding cell voltage versus cell depth of discharge can be seen in Supplementary Fig. 3, and model predictions for all 25 pulses can be seen in Supplementary Fig. 4. The model is robust under vastly different time and current scales, and shows strong agreement with experimental cell voltage measurements across 3-month low-rate background monitoring mode segments, $\sim$4 s high-rate pulsing, and initial 10 s of post-pulse relaxation. Note at the start of a pulse, the experimental instant switching of $R_{\text{ext}}(t)$ from 270 k$\Omega$ to 0.65 $\Omega$ is normally difficult to replicate in a model; it involves many orders of magnitude transition of current and creates challenges for the Hybrid-MPET’s differential algebraic equation (DAE) solver [93]. At the end of the pulse, the same challenge exists for the switching of $R_{\text{ext}}(t)$ from 0.65 $\Omega$ back to 270 k$\Omega$. For numerical stability purposes, we introduce a stairway transition of $R_{\text{ext}}(t)$ that involves multiple 0.004 s timesteps, which are only $\sim$$0.1\%$ of most pulse durations. For visual clarity, some of these transition timesteps are removed from the cell voltage versus month plot in Fig. 5, but can be more clearly observed in the more zoomed in version of pulse waveform and initial 10 s relaxation plots in Fig.  5 and Supplementary Fig. 4. Variable time stepping is also used to save computation time by keeping lower time resolution during 3-month background-monitoring and higher time resolution during pulsing and post-pulse relaxation.

Table 2 quantitatively compares our model predictions with experimental measurements using RMSE as a goodness-of-fit measure for describing Li/CF_x-SVO battery pulse performance. Our model, accurate in predicting pulse test performance of the representative cell, also generalizes well to the rest of the 80 cells in the dataset, obtaining $\text{RMSE}\approx 0.03$ V for both pulse minimum voltage $V^{\text{min}}_{\text{pulse}}$ and pulse average voltage $V^{\text{avg}}_{\text{pulse}}$. We also show residuals, i.e., the difference between experimental measurement and model prediction, of the four metrics in Supplementary Fig. 5; each of their distributions resembles a normal distribution with almost zero mean, which strongly infers that our model is representative of the high-rate Li/CF_x-SVO cells from the pulse test dataset and capable of predicting their pulse performance.

For the multi-active material electrode CF_x-SVO, we are most interested in examining active materials contribution to pulsing current as well as their interactions. In Fig.  6, to more concisely visualize these results, we show the model-predicted active materials contributions to released capacity during high-rate pulses and the evolution of active material depth of discharges throughout the test duration of the representative cell. $\overline{c}_{{}_{\text{Ag}}}$,$\overline{c}_{{}_{\text{V}}}$,$\overline{c}_{{}_{\text{CF}}}$ are obtained by averaging across corresponding particle types within each finite volume $\mathcal{V}$. Finite volume $\mathcal{V}_{1}$ is the one closest to separator and finite volume $\mathcal{V}_{5}$ closest to the current collector. We observe how V⁵⁺ supports the majority of the pulse current across all pulses and the reduced contribution from Ag⁵⁺ as it gradually becomes more depleted. Due to the constant resistive load operation mode, as soon as the pulsing mode is turned on, a decreasing cell voltage means that less current is required; the reaction rate and thus the reaction rate of each active materials changes at every timestep. To more clearly observe the change in reaction rates at the second timescale, we also show an expanded view of the model-predicted evolution of active material depth of discharge and their reactions rates during pulses 1, 7, 18 and corresponding initial 10 s of post-pulse relaxations in Fig. 7. The reaction rates throughout the test duration of representative cell can be seen in Supplementary Fig. 6. Under high-rate pulsing, the outstanding spikes in V⁵⁺ depth of discharge in Fig.  6(c) and significant higher reaction rates in Fig.  7 again show the contribution from fast V⁵⁺ reaction to support the majority of the pulse current. Despite their inherent slow reaction kinetics and limited contribution to supporting pulse current, Ag⁺, CF_x reactions are still active as seen in Fig. 6(a) and Fig. 7, and are driven by greater overpotentials after cell voltage drops significantly due to pulsing. The schematic picture describing the Ag⁺, CF_x reactions in parallel with a more dominant V⁵⁺ reaction under high-rate pulsing can be seen in Fig.  8(a). If the high-rate pulsing mode were kept for longer, V⁵⁺ is expected to deplete first. Thus, our model predictions do not contradict the common experimental observations [9, 11, 18, 6, 35, 94] that, when Li/CF_x-SVO is under lower discharge rates, the active materials depletion order follows first Ag⁺, then CF${}_{\text{x}}$, and then V⁵⁺. Rather, our modeling study offers more quantitative insights on how the utilization rate of different active materials is strongly dependent on operation conditions and the individual active material reaction kinetics.

During the initial 10 s of relaxation, as soon as the cell switches from pulsing mode to background monitoring mode, the cell voltage recovers and the demand of current decreases drastically. The overpotentials for the Ag⁺, CF_x reactions become very small, resulting in rapid decrease of $\overline{i}_{{}_{\text{Ag}}}$,$\overline{i}_{{}_{\text{CF}}}$. While three parallel reduction reactions across the active materials are still be taking place to support the low-rate background monitoring current, the spikes in V⁵⁺ depth of discharge disappear after pulsing, and corresponds to the existence of $\overline{i}_{{}_{\text{V}}}<0$ for finite volumes close to the separator in Fig.  7. This indicates that some V⁴⁺ is oxidized back to V⁵⁺ at the cathode for some duration despite the overall cell being under discharge mode and net outward cell current. This phenomena corresponds to the Li⁺ redistribution uniquely seen in multi-active material electrodes. We define Li⁺ redistribution as: uniformly mixed active materials in a conductive solid phase are naturally in favor of thermodynamic balance, and will exchange Li⁺ through the liquid electrolyte pathway to arrive at compositions at which the active materials have more balance potentials. Such phenomena has been observed experimentally from a more macroscopic perspective as CF_x recharging the SVO [3] after high-rate pulses, and our model offers more comprehensive explanation. During a defibrillation pulse, rapid depletion of V⁵⁺ close to the separator noticeably lowers the its potential and creates potential imbalance across the active materials; during initial 10 s of post-pulse relaxation and subsequent 3-month background discharge, the cell is held at such a low-rate current that it is nearly open circuit, and therefore thermodynamics control the reaction directions.

As seen in Fig. 8(b), such process could be equivalently viewed as a previously inserted Li⁺ due to V⁵⁺ reaction leaves its site in SVO, diffuses to the SVO particle surface, enters the electrolyte, and then participates in either Ag⁺, V⁵⁺, or CF_x reduction reaction that shift the compositions of active materials such that their potentials become more balanced. Note that the Li⁺ redistribution can occur: (1) intra- or inter-particle, i.e., between active materials in the same particle or different particles; (2) intra- or inter-material, i.e., between the same or different active materials in the porous electrode, as long as there are potential imbalances. After a discharge pulse, if we consider relaxation under full open-circuit or near open-circuit low currents, the Li⁺ redistribution process at the cathode involves multiple processes, including an electrochemical oxidation reaction for active materials at low potentials, Li⁺ transport in solid, Li⁺ transport in electrolyte, and a reduction reaction for active materials at higher potentials. As a result, during the initial 10 s of post-pulse relaxation as seen in Fig.  5, the rate of voltage recovery is collectively determined by pre-existing concentration gradients in solid and electrolyte, solid- and liquid-phase diffusion coefficients, and active material reaction rates at their respective compositions and overpotentials. Normally, the contribution from each physical process would be very difficult to decouple; yet in our model, because V⁵⁺ reaction contributed most to the pulsing current and created Li⁺ concentration gradient in SVO at the end of pulse, the the rate of post-pulse voltage recovery is found to be most sensitive to $D_{{}_{\text{Li,V}}}$, leading us to use $D_{{}_{\text{Li,V}}}$ to capture the pulse waveforms and initial 10 s of post-pulse relaxation.

Despite the introduction of quarterly pulses in the pulse test, their pulse duration is too short compared to the 3-month duration of near open-circuit low current background discharge. The previous reaction kinetics- and mass transport-limited system is soon overturned by thermodynamics control. The estimated diffusion time scales $\tau_{{}_{\text{Ag}}}=\frac{L^{2}}{D_{{}_{\text{Li,Ag}}}}$, $\tau_{{}_{\text{V}}}=\frac{L^{2}}{D_{{}_{\text{Li,V}}}}$ are both negligible compared to the timescale of months, leading to quickly diminishing Li⁺ concentration gradients inside the SVO particles, which are shown in animated videos in Supplementary Information. The active materials reach more balanced potentials after considerable amount of Li⁺ redistribution; the three reactions will take place in parallel as seen in Fig.  8(c). The depth of discharge increase of V⁵⁺ due to high-rate pulses are mostly reverted before the next pulse, keeping V⁵⁺ at low compositions that not only allow more balanced potentials across the three active materials but also preserves the pulse capability of SVO material throughout battery operation. As a result, because low-rate background discharge between pulsing intervals is long enough, the general active material depletion order seen across Fig.  6(b)(c)(d) matches those in medium-rate Li/CF_x-SVO cells [9, 11, 18, 6, 35, 94] and those directly predictable from active material OCV: first Ag⁺, next CF${}_{\text{x}}$, and then V⁵⁺.