A Quantitative Analytical Model for Predicting and Optimizing the Rate Performance of Battery Cells

An important objective of designing lithium-ion rechargeable battery cells is to maximize their rate performance without compromising the energy density, which is mainly achieved through computationally expensive numerical simulations at present. Here we present a simple analytical model for predicting the rate performance of battery cells limited by electrolyte transport without any fitting parameters. It exhibits very good agreement with simulations over a wide range of discharge rate and electrode thickness and offers a speedup of>10$^5$ times. The optimal electrode properties predicted by the model are of less than 10% difference from simulation results, suggesting it as an attractive computational tool for the cell-level battery architecture design. The model also offers important insights on practical ways to improve the rate performance of thick electrodes, including avoiding electrode materials such as LiFePO$_4$ and Li$_4$Ti$_5$O$_{12}$ whose open-circuit potentials are insensitive to the state of charge and utilizing lithium metal anode to synergistically accelerate electrolyte transport within thick cathodes.


I. Introduction
Nearly thirty years after the debut of commercial rechargeable lithium-ion batteries (LIBs), LIBs was recognized by the 2019 Nobel Prize in Chemistry as a key energy storage technology for wide-ranging applications from portable devices, electrical vehicles to grid load balancing. In the drive towards developing better batteries with lower costs, there exists a perpetual trade-off between the energy and power densities. As illustrated by the well-known Ragone plot 1 , improvement in one metric usually leads to the degradation of the other. Recently, the use of thick electrodes has received significant interest as a way to enhance the gravimetric / volumetric energy of battery cells and cut materials and manufacturing costs by reducing the fraction of inactive components such as separator and current collectors [2][3][4][5][6][7][8] . However, a main challenge faced by thick electrodes is their inferior rate performance. Effective alleviation of this issue requires careful optimization of cell parameters for targeted applications and the development of novel electrode architectures (e.g. low tortuosity electrodes) 9-18 that can slow down the rate capability decay with increasing electrode thickness.
The current standard approach to predicting the rate performance of battery cells and optimizing their structures is numerical simulations 19-22 based on the porous electrode theory pioneered by Newman and co-workers 23,24 . Such simulations are often called pseudo-twodimensional (P2D) simulations because they involve solving the governing equations for ionic (or electronic) transport in electrolyte (or solid phase) along a macroscopic length scale, which are coupled with the lithium diffusion equation within electrode particles along a microscopic length scale. While the P2D model provides a comprehensive description of the kinetic processes during (dis)charge, it is also computationally intensive to solve. Various reformulation and reduced order modeling techniques [25][26][27][28] have been developed to accelerate P2D simulations, but the computational burden could still be considerable for battery cell optimization, which requires a large number of objective function evaluations and a time-dependent simulation for each evaluation.
In contrast to P2D simulations, (semi-)empirical battery models such as equivalent-circuit models 29,30 are simple and fast to execute and widely used in battery management systems. Parameters in these models need to be fitted against experimental data, e.g. from cycling or impedance measurements, and valuable insights can be obtained from the fitted parameter values [31][32][33] . However, it is often not easy to determine the quantitative relations between the fitting parameters and the physical properties of the systems, which makes the application of these models to the cell design and optimization not straightforward. Serving as an intermediate between the P2D and (semi-)empirical models, physics-based analytical models [34][35][36][37][38][39] have long been developed for predicting battery electrode performance. Even with the constantly improving capability of computers, this type of models are desirable because they not only are efficient to solve but also shed light on the structure-performance relation of battery cells in a more transparent way.
Analytical models are often derived by simplifying the P2D model for situation where one kinetic process is much more sluggish than the others and dominates the (dis)charge behavior. Examples include the single-particle model 38,39 for (dis)charging limited by solid-state diffusion, and the reaction zone model 35,40 that assumes ohmically dominated (dis)charge processes whereas the concentration gradients in electrolyte and electrode particles can be neglected. Nevertheless, advances in the LIB technology have rendered the limiting factors considered by these models less significant as slow solid-state diffusion can be addressed by particle size reduction and low electronic conductivity can be improved by conductive additives or coatings. Compelling evidence shows that the rate performance of LIBs based on today's commercial electrode materials and liquid electrolytes is mainly limited by electrolyte diffusion 3,5,[41][42][43][44] , which builds up a large salt concentration gradient in electrolyte during cycling. This limitation has become a major impediment to the on-going push for fast charging and the development of thick battery electrodes 10,11 .
Despite its technological relevance, there are few efforts in deriving simplified analytical models for (dis)charging in the electrolyte-diffusion-limited regime, partly because great complexity arises from the nonlinear coupling between the electrolyte diffusion, reaction flux, electrical potentials and state of charge (SOC) of electrode material in the governing equations of the porous electrode theory. In a notable work, Doyle and Newman 39 demonstrate a strategy to decouple electrolyte diffusion from other processes by assuming simple forms of reaction flux distributions. Gallagher et al. derived a simple expression for the penetration depth of electrolyte transport and used it as an indicator of the discharge capacity 5 . A similar expression was also given by Johns et al. 42 . While these early studies provide useful insights, the predictions are qualitative in nature and cannot replace the quantitative P2D simulations.
In this work, we present a quantitative analytical model for predicting the (dis)charge behavior of battery cells limited by electrolyte diffusion. Similar to the approach of Doyle and Newman 39 , we developed the model by decoupling the governing equations of the porous electrode theory but employed more reasonable assumptions of the electrolyte transport and electrode reaction behavior, which are generalized from P2D simulations. The model gives analytical expressions of the galvanostatic discharge capacity of both half and full cells consisting of common cathode (NMC, LiFePO4) and anode (graphite, Li4Ti5O12) materials. Without any fitting parameters, the predictions of the model exhibit very good agreement with P2D simulations over a wide range of electrode thickness and discharge rate. In particular, it accurately captures two important quantities for cell design, i.e. the critical discharge rate and critical electrode thickness, above which capacity utilization drops sharply due to electrolyte diffusion limitation. When applied to electrode optimization, the model is found to incur <10% error relative to P2D simulations while enabling >10 5 -fold speedup, which provides a very convenient and efficient alternative to P2D simulations for battery cell design. Furthermore, simple scaling relations derived from the model offer a number of significant insights on the dependence of discharge performance on the electrode properties. Notably, we discover that i) electrode materials that exhibit a strong SOC dependence of open-circuit potentials such as NMC and graphite have intrinsically better rate performance than those that do not (e.g. LiFePO4 and Li4Ti5O12) in the electrolyte-transport-limited regime, and ii) thick cathodes have significantly better rate capability when paired with Li metal than conventional graphite or Li4Ti5O12 anodes, which points to a clear synergy between the development of the Li metal anode and the wider adoption of thick-electrode batteries.

II.1 Electrolyte transport and electrode reaction behavior
The development of our model starts from observations from P2D simulations, which inspire a simplified description of electrolyte transport and electrode reaction behavior during electrolytediffusion-limited discharge processes. Two key observations are that: i) electrolyte transport in this regime attains a (pseudo-)steady state during discharge, which allows the time-dependent simulations to be reduced to a time-independent problem; ii) the electrode reaction flux distribution can be categorized into two types of distinct behavior exhibited by different common electrode materials. The first type of electrode materials is characterized by a strong dependence of their equilibrium (or open-circuit) potentials Ueq on SOC. They include the mainstream cathode materials NMC and NCA. The second type of electrode compounds, exemplified by LiFePO4 (LFP) and Li4Ti5O12 (LTO), feature SOC-independent Ueq due to prominent first-order phase transition(s) upon (dis)charging. Below we use representative P2D simulations to elucidate these two types of discharging behavior and motivate the simplifying assumptions employed in the analytical model.
"Uniform-reaction"-type electrodes. Figure 1a-c presents the discharge process of an NMC111 half cell, which refers to battery cells consisting of a cathode and Li metal anode. The cathode is 250-µm thick and discharged at a current density 20mA/cm 2 or 1.5C, and electrolyte is 1M LiPF6 in EC:DMC (50:50 wt%). Other parameters employed in the simulation are listed in Supplementary Table S1 in Supplementary Information (SI). Figure 1a shows that salt depletion (c = 0) occurs in electrolyte near the current collector shortly after discharging starts and is responsible for capacity under-utilization. After an initial transient period of <10% depth of discharge (DoD), salt concentration establishes a steady state distribution in electrolyte, which persists up to ~50% DoD before discharging terminates at 62% DoD. During the steady-state period, the cathode region can be divided into a salt depletion zone (DZ), in which salt concentration c is nearly zero, and the complementary penetration zone (PZ). Figure 1b shows that the Li intercalation flux is mostly absent in DZ due to the unavailability of Li ions in the electrolyte, but a nearly constant flux is present inside PZ. As a result, electrode particles are lithiated relatively uniformly in PZ but remain largely delithiated in DZ, see Figure 1c. Near the end of discharge, salt concentration is transient again as salt depletion becomes more severe and causes the further expansion of DZ before discharging is terminated at the cut-off voltage (3 V) at DoD ≈ 62%. Figure 1c shows that the majority of electrode particles in PZ (DZ) are close to be fully discharged (charged) except for a transition region near the DZ/PZ boundary. As shown in Figure S1a-c, NMC cathode in a full cell paired with a graphite (Gr) anode displays similar discharge behavior, which may be approximated by a simplified "uniform reaction" (UR) model. As illustrated by the schematic in Figure 2a, it assumes that: i) Electrolyte transport maintains a steady state throughout the discharging process.
ii) Electrode consists of a salt PZ and DZ with uniform reaction flux in the former and zero flux in the latter.
iii) Discharging ends when electrode particles in PZ are fully discharged.  Supplementary Table S1. The salt concentration &, reaction flux at the electrolyte/electrode interface ' ( ) (* and the average Li concentration in cathode particles & + at different DoDs are shown in a-c for NMC half cell and d-f for LFP half cell, respectively. Cathode/current-collector interface is at , = 0 µm. The dashed lines in a and d represent the (pseudo-)steady-state salt distribution in electrolyte predicted by the analytical model. We emphasize that here "uniform reaction" implies a homogeneous reaction flux within PZ instead of the whole electrode. Doyle and Newman previously developed an analytical model that also assumes a uniform reaction distribution 39 . There are two major differences between the Doyle-Newman model and ours. First, the former assumes a uniform reaction flux across the entire cathode. Secondly, it assumes that the discharge process is terminated as soon as salt depletion occurs at the cathode/current-collector interface. The Doyle-Newman model thus only accounts for the discharge capacity obtained during the transient stage before the establishment of steady-state electrolyte transport. As simulations show, discharge continues after the onset of salt depletion and the steady-state stage could be responsible for the majority of the discharge capacity. "Moving-zone-reaction"-type electrodes. Figure 1d-f demonstrates the discharge behavior of an LFP half cell with a 250µm-thick cathode and discharged at 20mA/cm 2 or 1.8C. Other simulation parameters are listed in Supplementary Table S1. Unlike NMC111, Li intercalation in the LFP electrode is characterized by a moving reaction front. Figure 1e shows that a sharp reaction flux peak first develops near the separator and then travels towards the current collector upon discharging. As shown in Figure 1d and 1f, the reaction front divides the cathode into a PZ, in which the electrode is almost fully discharged, and a partial salt DZ, in which salt is partially depleted and electrode particles are barely reacted. The salt concentration continues to decrease inside the partial DZ when the PZ expands. Discharging is terminated when the DZ becomes completely depleted in salt because the electrolyte can no longer maintain the required salt concentration gradient to deliver Li flux to the reaction front if the PZ further expands. There is no steady-state electrolyte diffusion during discharging, but a pseudo-steady state is reached at the end of discharge, when the time derivative of salt concentration momentarily vanishes. Figure S1d-f shows that LFP in a full cell paired with Gr displays similar characteristics upon discharging. The observed discharging behavior can be idealized into a simple "moving-zone reaction" (MZR) model as illustrated in Figure 2b: i) Li intercalation occurs at an infinitely sharp, moving reaction front that defines the boundary between the salt PZ and partial DZ.
ii) Electrode particles in the PZ (partial DZ) are fully discharged (charged).
iii) Discharging ends when salt is completely depleted in the DZ and electrolyte diffusion reaches a pseudo-steady state.
Similar discharging behavior was considered in the reaction zone model by Tiedemann and Newman for ohmically limited electrodes 35,40 and also by Doyle and Newman 39 . However, a detailed explanation on the origin of the UR vs MZR behavior appears to be lacking in literature, which deserves some discussion here. Whether an electrode material exhibits UR or MZR behavior is related to its intrinsic properties, specifically the SOC dependence of Ueq. The reaction flux at the electrode/electrolyte interface is typically described by the Butler-Volmer equation, which shows that jin is controlled by the surface overpotential , where SOCs is the local SOC at the particle surface. Upon discharging, the ionic / electronic resistances of the electrolyte / electrode result in spatially varying , which contributes to an initial non-uniform jin (Figure 3a). When Ueq has a strong SOC dependence like NMC, however, the electrode is able to rectify the reaction non-uniformity. As illustrated in Figure 3b, electrode particles in regions with higher jin also see a larger decrease in local SOC, which in turn leads to a larger drop in Ueq and so are and jin. This self-regulating mechanism helps establish a uniform reaction flux across the PZ, where Li is available in electrolyte for intercalation. On the other hand, electrode materials like LFP have SOC-independent Ueq, which cannot compensate the spatial gradient of to homogenize the reaction flux as illustrated in Figure 3c.
Therefore, reaction will concentrate first at the separator, where η is the largest, and then propagate across the electrode like a traveling wave after electrode particles in the local region are fully intercalated.
Besides cathode, the reaction behavior of anode in a full cell may also be described as the UR and/or MZR type. Supplementary Figure S2 presents the discharge simulations of NMC/LTO and LFP/LTO full cells. Like LFP, LTO has a flat Ueq vs SOC curve, and Li deintercalation within LTO also proceeds through a moving reaction front. Compared to LTO, the discharging behavior of Gr anode is somewhat more complex and displays both UR and MZR features. Graphite's Ueq has a pronounced plateau at ~0.05 V due to the staging transition LiC6 → LiC12. Accordingly, Gr exhibits the MZR behavior at the early discharge stage, where a reaction flux peak can be seen traveling across the anode (Supplementary Figure S1b,e). After DoD reaches >30%, however, the anode's Ueq rises above the plateau and its reaction behavior switches to the UR type. Since there is no salt depletion within anode upon discharging, the reaction flux is uniformly distributed over the entire anode region.

II.2 Analytical model development
The simplified UR and MZR models described in the last section make it possible to analytically predict the discharge performance. We see that the width of salt PZ, or the penetration depth LPZ, is central to the rate performance of the electrolyte-diffusion-limited discharge process. The normalized discharge capacity DoDf can be estimated by the ratio of LPZ to the cathode thickness Lcat. As such, our goal is to derive an analytical expression of LPZ as a function of discharge current, electrode, electrolyte and separator properties.
We begin the derivation for half cells. In the porous electrode theory 22,23 , the mass balance and current continuity equations of a binary electrolyte are: where subscript i represents cathode (i = cat) or separator (i = sep). With the assumption of (pseudo-)steady-state electrolyte transport, Eq. 2 becomes a time-independent equation: 4) in which a constant t+ and homogeneous in-plane c are assumed. Assuming the UR or MZR behavior, jin within the PZ is given by

5)
Since c is 0 in DZ, Eq. 4 only needs to be solved within the PZ and separator regions with the boundary conditions at the PZ/DZ boundary, Setting x = 0 at the cathode/current-collector interface and treating Damb as constant, the solution to Eqs. 4-7 is given by 8)

9)
Eqs. 8 and 9 show that c varies parabolically within PZ in UR-type electrodes and has a linear profile in MZR-type electrodes. Finally, the unknown LPZ in the solution is determined by substituting Eq. 8 or 9 into the salt conservation equation

10)
The obtained LPZ expressions for the two types of electrodes are listed in Table 1.
To derive the expression of LPZ for full cells, Eq. 4 is extended to the anode region. Although the graphite anode exhibits the hybrid reaction behavior during discharging, it develops a uniform reaction flux distribution at the later stage as shown in simulations (Supplementary Figure S1). A uniform Li deintercalation flux can thus be assumed for the entire Gr anode region when DoDf is not too small. Details of the derivation are provided in Supplementary Note S1. The obtained expressions of LPZ for cathode/Gr full cells are given in Table 1. The analytical expression of LPZ for full cells containing MZR-type anodes like LTO is also derived in Supplementary Note S1 and the results are summarized in Supplementary Table S2 .
Given LPZ, DoDf is calculated as When LPZ > Lcat, Eq. 15 restricts DoDf to a maximal value of 1 since this means that electrolyte transport does not limits the discharge capacity at all. The model predicts negative LPZ for full cells with thick electrodes and discharged at high rates, in which case DoDf = 0 is assigned.
In the above derivation, Damb is assumed to be constant to render tractable analytical expressions. When considering the concentration dependence of Damb, Eqs. 8 and 9 will contain integrals of Damb(c) and Eq. 10 has to be solved numerically to determine LPZ, but the calculation is no more difficult and has comparable computational cost. With minor modification, the model is also applicable to constant-current charging limited by electrolyte diffusion. In practical applications, however, charging usually needs to be terminated before salt depletion occurs to prevent lithium plating on anode particle surface, which is a major concern for capacity fading. In this case, the model will give less conservative predictions. For this reason, here we focus on the application of the model to the prediction and optimization of discharge performance.

II.3 Comparison with P2D simulations
In this section, we examine how well the derived analytical model approximates P2D simulations.  Figure 4, the rate-dependent DoDf of NMC and LFP cathodes in half cells and full cells with Gr anode are calculated using the model and compared with the P2D simulation results. The model (dashed lines) and simulation again agree very well with each other over a wide range of C rates and electrode thickness values. Figure 4 shows that there exists a critical C-rate Ccrit for each cell configuration, which represents the C rate above which DoDf starts to deteriorate due to sluggish electrolyte diffusion. The discrepancy between the model predictions and simulations becomes more significant for full cells at large Lcat (250 and 300 µm) and high rates where DoDf < 0.3. The reasons are two-fold. First, Gr displays MZR behavior at small DoD, which deviates from the uniform reaction assumption employed in Eqs. 13 and 14. Second, discharging in these cases is prematurely terminated before electrolyte transport establishes the (pseudo-)steady state. Supplementary Figure S3 compares the analytical model with simulations for NMC/LTO and LFP/LTO full cells, which also show very good agreement.
To comprehensively test the model fidelity, 246 P2D simulations were performed for each type of cathode (NMC vs LFP) and cell configuration (half vs full) over a large range of discharge rates (0.1C -10C), electrode thickness (70 -300 µm) and other cell properties. The simulated DoDf is compared against LPZ/Lcat predicted by the model in Figure 5a-d. Overall, the model provides a satisfactory approximation to P2D simulations. In the case of half cells, >94% of the model predictions have a relative error of less than 10% (or 20%), and the average error is 5.1% (or 8.1%) for NMC (or LFP), see Supplementary Figure S4a Supplementary  Table S1.

II.4 Scaling relations
With additional approximations, the expressions of LPZ shown in Eqs. 11-14 can be further simplified to provide a less accurate but more revealing scaling description of the relation between LPZ and various cell properties and discharge rate. A simpler expression of LPZ for half cells can be obtained by neglecting the separator thickness (Lsep = 0) in Eqs. 11 and 12: (half cells) 15) where γh = 6 or 2 for UR-or MZR-type cathodes, respectively. To simplify the expression of LPZ for full cells with Gr anode, we start from the solution of c(x) given in Supplementary Note S1, from which one can write down:

16)
where is the salt concentration at the cathode/separator interface, and λ = 2 or 1 for UR-or MZR-type cathodes, respectively. In fact, Eq. 16 is valid for both full and half cells. Interestingly, P2D simulations (Supplementary Figure S1a,c) show that ccat/sep in NMC/Gr and LFP/Gr full cells does not deviate too much from the average salt concentration c0 during discharge and thus may be treated as a constant. In Supplementary Note S2, we derive an estimated value of ccat/sep ≈ c0 (or 12c0/11) for UR-(or MZR-)type cathodes paired with Gr anode. This leads to an approximate expression of LPZ for full cells containing Gr anode: where γf = 2 or 12/11 for UR-or MZR-type cathodes, respectively. We note that an expression of the penetration depth similar to Eq. 17 is given in Ref. 5, in which the proportionality factor γf is found to to ~1.8 numerically by fitting it to P2D simulations of NCA/Gr full cells. Since NCA is a UR-type electrode, our model naturally explains this result and shows that the obtained γf value is not limited to NCA alone. Supplementary Note S2 shows that Eq. 17 is also applicable to full cells with LTO anode but γf is different: γf = 12/7 or 1 for UR-or MZR-type cathodes, respectively.
Valuable insights can be obtained from the simple scaling relations given in Eqs. 15 and 17. First, they show that UR-type cathodes like NMC and NCA have inherently better rate performance than MZR-type cathodes like LFP in the electrolyte-transport-limited regime as the former group has a larger γh or γf. This is because the presence of a uniform reaction flux in URtype cathodes allows the salt concentration to decrease parabolicly instead of linearly with the distance to separator, which produces a wider PZ. At the same cathode thickness and C rate, NMC's DoDf is 1.7-2 times of that of LFP in both half and full cells according to Eqs. 15 and 17. The reaction behavior of the anode has a similar effect on the discharge performance since LTO has a smaller γf than Gr in Eq. 17. In addition to the poorer rate performance, MZR-type electrodes are subject to high local reaction flux during (dis)charging, which makes them prone to excessive stress concentration and localized heat generation that will accelerate battery degradation. Therefore, UR-type electrode materials are more advantageous than MZR-type electrodes in thick electrode applications.
Second, Eqs. 15 and 17 reveal that cathodes have intrinsically better rate capability in the half cell configuration because scales with or in half cells vs or in full cells. The different scalings predict that DoDf decays more rapidly with Lcat and I (or C) in full cells for identical cathodes, which is clearly seen in P2D simulations ( Figure 4) especially for large electrode thickness. We may use Eq. 16 to explain the superior performance of half cells over full cells, which relates LPZ to ccat/sep. While ccat/sep remains close to c0 in full cells during discharging, it rises significantly above c0 in half cells (Figure 1a,d). This is because salt depletion inside the cathode causes salt to be pushed towards the anode side. Unlike the porous anode in full cells, the Li metal anode in half cells cannot accommodate the influx of salt, which is instead accumulated in the separator. A higher salt concentration is thus built up at the cathode/separator interface in half cells, which allows salt to penetrate deeper into the cathode.
Because Eqs. 15 and 17 assume = 0, they tend to overestimate LPZ since a separator of finite thickness will further slow down electrolyte diffusion. The scaling exponents predicted by Eqs. 15 and 17, n = -1/2 for hall cells and -1 for full cells, hence represent an upper limit of the actual relation of LPZ ~ Lcat and I (or C). This is illustrated in Figure 6a,b, which plots DoDf from P2D simulations presented in Figure 4 against in logarithmic scale. It shows that n is very close to -1/2 (or -1) in half (or full) cells near DoDf = 1 but decreases with decreasing DoDf (or LPZ/Lcat). Eqs. 15 and 17 are thus most accurate for situation where salt depletion is not very severe and DoDf is high. In addition, Figure 6a shows that n is larger than -1 in half cells. This is because ccat/sep increases with C and Lcat in half cells due to the intensified salt accumulation in the separator, which according to Eq. 16 leads to a superlinear scaling between LPZ/Lcat and , or n > -1. Therefore, we can write down the following general scaling behavior:

18)
To test its validity, Eq. 18 is compared against a number of experimental works 3,5,8-10,41 that report the dependence of discharge capacity on the discharge rate and/or electrode thickness (see Supplementary Note S3 for the criteria used in selecting the experimental data.) Figure 6c and Supplementary Table S3, which lists the scaling exponents measured from the data, show that Eq. 18 is consistent with the experimental observations: while n < -1 holds for full cells, all of the half cell data exhibit -1 < n < -1/2.
The qualitative difference in the discharge characteristics of half vs full cells has significant practical implications. As a common practice, the rate capability of electrodes is often tested in half cells for convenience. However, our finding cautions that this approach may significantly overestimate the electrode performance in real applications that use full cells when discharge is kinetically limited by electrolyte transport. From another perspective, a half cell is essentially a cathode paired with a Li metal anode. Replacing conventional anodes with Li metal thus not only increases the specific/volumetric anode capacity but also improves the discharge performance of thick cathodes. Such a synergistic effect makes paring thick cathodes with Li metal anode a promising and perhaps necessary strategy to develop high-energy / high-power battery cells. This conclusion also applies to other high-capacity alloy anode materials such as silicon and its compounds. Because they require a much smaller anode thickness to match the cathode capacity, the asymmetric structure promotes salt enrichment in the separator upon discharging and improves salt penetration into the cathode.
Despite the difference, Eq. 18 suggests that both half and full cells obey the relation or at a fixed DoDf. In other words, cathode thickness should decrease parabolically with the C-rate, or conversely C should decrease quadratically with Lcat, to maintain the same level of capacity utilization. Experiments do show that both half and full cells follow this relation 3,5,41 , which is explained by our model.

II.5 Optimization of Battery Cells
The analytical model derived in this work provides a very fast and reliable computational tool for battery cell optimization. As a demonstration, we apply it to search for the optimal electrode porosity and thickness that maximize the areal or specific capacity of an NMC/Gr full cell at a given discharge rate. In the calculations, and are parameters subject to optimization, is fixed at 1.15Lcat and is correlated to to give an anode/cathode capacity ratio of 1.18. Other parameters are the same as in the P2D simulations reported above. The properties of various components used in calculating the cell-level specific capacity are given in Supplementary Table S4.  ) and specific (> @ ) capacity as a function of ! "#$ and 0 "#$ for 1C discharge. The anode/cathode capacity ratio is held constant at 1.18, and ! #* and 0 #* vary in proportion to ! "#$ and 0 "#$ , respectively. Red solid line represents ! "#$ that maximizes > ? or > @ at a given 0 "#$ . It is overlapped by the black dashed line, which corresponds to the critical cathode thickness ! "#$ "A($ at which ! -. = ! "#$ . The blue dash-dotted line is given by the approximate expression of ! "#$ "A($ in Eq. 19. The red and blue squares represent the global optimum (! "#$ CD$ , 0 "#$ CD$ ) predicted by the full analytical model and the approximate expression based on Eq. 19, respectively. c and d, > ? and > @ calculated from P2D simulations for NMC/Gr cells with different ! "#$ and 0 "#$ . Black dashed line is the maximal capacity at each ! "#$ predicted by the analytical model. Although it is possible to speed up P2D simulations via various techniques [25][26][27][28] , the computational saving achieved by the analytical model is still substantial. For example, benchmarking in ref. 28 reports that it takes 4-5 s or 46-156 ms to run a P2D simulation on a workstation (3.33 GHz Intel processor, 24 GB RAM) with or without employing state-of-the-art acceleration techniques. In comparison, it takes only 300 ms to complete 10 6 evaluations of DoDf based on the model, which were carried out in MATLAB on a laptop (2.2 GHz Intel Core i5 processor, 8 GB RAM). The analytical model thus delivers a speedup of at least 10 5 -10 6 times. Together with its ease of implementation, the model offers a very attractive alternative to P2D simulations for battery cell design in the electrolyte-diffusion-limited regime. It could be especially useful for exploring novel battery electrode architectures such as heterogeneous or graded electrodes that have a large number of design variables. Moreover, one may combine the model's efficiency and the accuracy of P2D simulations by first using the model to rapidly locate the approximate global optima and then employing P2D simulations to refine the predictions. This is already showcased in Supplementary Figure S6 and S7, where we use only 35 simulations in the neighborhood of ( , ) predicted by the model to confirm the optimal configurations, as against the 621 simulations performed over the entire parameter space ( Figure  7c,d). By leveraging the speed of the analytical model to broadly scan the parameter space, this hybrid approach can also prevent the search from being trapped by local optima, which is a common pitfall of gradient-based optimization algorithms.
Besides numerical efficiency, the analytical model illuminates an important quantity for cell optimization, i.e. the critical cathode thickness , which is defined as the cathode thickness above which salt depletion (i.e. LPZ < Lcat) occurs at given and C. In Figure 7a,b, the red solid lines represent that maximizes QA or Qw at a constant . They overlap the black dashed lines, which are at 1C. Therefore, is the optimal cathode thickness at a fixed . This conclusion is valid for not only full cells but also half cells as shown in Supplementary Figure S8. It suggests that a key consideration in the optimization of battery cell structure in the electrolyte-limited regime is to prevent salt depletion during discharge. Because

21)
In Figure 7a, b and Supplementary Figure S8, given by Eq. 19 is plotted as blue dash-dotted lines and the optimal ( , ) predicted by Eqs. 19-21 is marked by blue squares. It can be seen that they are close to the predictions of the full model. The reason that these approximate expressions work well is that Eqs. 15 and 17 are most reliable near DoDf ≈ 1 or LPZ ≈ Lcat as discussed in Sec. II.4.
Taking advantage of its simplicity, we employ the analytical model to gain insights on the effects of various design parameters on the cell performance. Figure 8 Areal vs specific capacity: A notable observation of Figure 8 is that QA and Qw are optimized at very different electrode porosity and thickness. When the objective is to maximize QA, the optimized cells are much thicker and more porous than cells optimized for Qw. Such a difference can be explained by the fact that QA increases with more pronouncedly than Qw does (Eq. 20 vs 21), and thicker electrodes also require larger porosity to facilitate ion transport. Interestingly, Figure 8c shows that for QA is insensitive to the C rate, electrode material and cell type and remains around 0.6 (excluding the low-tortuosity electrodes). To explain this behavior, we use Eqs. 19 and 20 to maximize QA. Assuming a general Bruggeman-like relation between tortuosity and porosity, , one obtains

22)
Eq. 22 shows that is a function of only and independent of the C rate. Consistently, it predicts = 3/5 for the Bruggeman relation (β = 1/2, 0 = 1) used in Figure 8c. In contrast, and C are not separable in Eq. 21, which is why for Qw is rate-dependent (Figure 8f). in NMC half cells, more than twice of that in NMC/Gr full cells. This shows that the use of thick cathodes will be essential for cell optimization when Li metal anodes are deployed in rechargeable batteries.
Effect of cathode materials: Figure 8a shows that an optimized NMC/Gr full cell can achieve ~40% higher QA than an optimized LFP/Gr cell, and the difference is even larger (~90%) for half cells. We emphasize that such difference is not only because of the higher volumetric capacity of NMC (734 vs LFP's 611 mAh/cm 3 ), but also related to the inherently better rate performance of UR-type electrodes like NMC in the electrolyte-diffusion-limited regime, which is reflected by the larger f / h of NMC than LFP in Eqs. 15 and 17. Eqs. 19 and 20 estimate QA,max for NMC to be 48% (or 90%) higher than that for LFP in full (or half) cell configuration, in good agreement with Figure 8a. In comparison, NMC and LFP have a smaller difference in Qw,max (Figure 8d) because LFP has a higher intrinsic specific capacity than NMC111 (170 vs 150 mAh/g). Nevertheless, NMC still delivers 7% and 12% more Qw,max at 5C in full and half cells, respectively.
Effect of electrode tortuosity: In addition to electrode thickness and porosity, tortuosity is another important electrode property, which characterizes how tortuous porous channels inside electrodes are. The effective electrolyte diffusivity in a porous electrode is inversely proportional to as . While its theoretical minimum is 1, in conventional porous electrodes usually has much higher values [45][46][47] . In a number of recent studies, low-tortuosity (LT) electrodes containing straight pore channels were fabricated by various methods 10,13,14,16 . They demonstrate notably enhanced rate performance over regular electrode structures. To assess the ultimate potential of LT electrodes, it is desirable to determine the maximal improvement achievable through tortuosity reduction. For this purpose, we use the analytical model to estimate QA,max and Qw,max of NMC/Li and NMC/Gr cells with an assumed = 1, and in the case of full cells, = 1 too. Figure 8a shows that LT electrodes can deliver a modest (14 -15%) increase in QA,max over regular electrodes, consistent with the prediction of Eqs. 19 and 20. Decreasing has a lesser effect on Qw,max, giving a maximal increase of 4 -8% for both full and half cells. However, we caution that the above comparison may underestimate the benefits of LT electrodes because Figure 8 assumes the Bruggeman relation 48 ( ) for regular electrodes, which is often found to underpredict the tortuosity of calendered porous electrodes 49,50 . Thorat et al. 45 report a relation from the AC impedance measurements of LFP and LCO cathodes, which is later corroborated by Li et al. 11 . If using this relation for regular electrodes in the calculation, Figure S9 shows that LT electrodes offers a much more significant increase over the baseline, providing 50 -60% higher QA,max and 7 -16% higher Qw,max in the best case scenario ( = 1).
Electrode tortuosity reported in literature spans a large range of values, with some supporting the Bruggeman relation 48 and others not, which indicates its sensitivity to electrode preparation. Our analysis shows that the introduction of LT pores is highly effective when the tortuosity of untreated electrode structures is inferior to the Bruggeman relation.

III. Conclusions
In summary, an analytically solvable model is developed in this work to predict the rate performance of battery cells limited by electrolyte diffusion. The model simplifies the governing equations of the porous electrode theory by assuming (pseudo-)steady-state electrolyte transport and two types of electrode reaction flux distributions (UR vs MZR), with the former applicable to common electrode materials such as NMC and the latter to LFP given their different SOC dependence of open-circuit potentials. We derive analytical expressions of the galvanostatic discharge capacity for different electrodes and cell configurations (Eqs. 11-14, Supplementary  Table S2), which exhibit very good agreement with P2D simulations over a wide range of electrode thickness and discharge rates. When applied to maximize the areal or specific capacity of battery cells, the model is able to identify the optimal electrode thickness and porosity with <10% difference from the P2D simulation results but at negligible computational cost. With additional approximations, the model sheds light on revealing scaling dependence (Eqs. 15, 17 and 19) of the cell performance on electrode and electrolyte properties. In addition to being a very efficient tool for battery cell design as a replacement or compliment to P2D simulations, the analytical model provides useful insights on how to improve battery rate performance when it is mainly limited by electrolyte diffusion, including: i) Uniform-reaction-type electrodes (e.g. NMC) have inherently better rate capability than moving-zone-reaction-type electrodes (e.g. LFP) by alleviating salt depletion in electrolyte during discharge.
ii) Replacing conventional porous anodes with Li metal not only increases the specific/volumetric capacity of the anode but also significantly improves the cathode's rate capacity to facilitate the use of thicker cathodes.
iii) The optimal electrode thickness that maximizes the cell-level areal or specific capacity corresponds to the critical electrode thickness above which salt depletion occurs during discharge.
iv) To optimize a battery cell's areal capacity, the electrode porosity should be much higher (~60%) than in typical commercial cells.
v) Reducing electrode tortuosity is especially beneficial for enhancing rate performance when the electrode tortuosity-porosity relation is inferior to the commonly assumed Bruggeman relation. where subscript i = cat or an. The reaction flux on electrode particle surface jin obeys the Butler-Volmer equation (Eq. 1), in which the exchange current density i0 is given by

26)
The reaction flux on the Li metal anode surface is similarly described. Solid-state diffusion of Li in electrodes at the particle scale is approximated as a radial diffusion process in spherical particles:

27)
Eq. 27 is coupled to mass transport in electrolyte through the boundary condition at the particle surface

Data Availability
The data that support the findings of this study are available from the corresponding author upon request.
Supplementary Note S1 -Derivation of the analytical model for full cells For full cells, the half-cell solution (Eqs. 8 or 9) to the steady-state electrolyte transport equation (Eq. 4) remains valid in the cathode and separator regions. The salt concentration distribution inside the anode depends on the reaction behavior of the anode and is derived for graphite and LTO anodes separately below.

I. Full cells with graphite (Gr) anode
Assuming that the reaction flux is uniform in Gr anode upon discharge, we apply to Eq. 4 within the entire anode region. Using given by Eq. The obtained expressions of LPZ are given in Table 1.

II. Full cells with LTO anode
Because LTO exhibits MZR behavior, we assume that a penetration zone also exists near the separator inside the anode during discharge. All the anode particles within this zone are fully deintercalated, those outside the zone remain fully intercalated, and an infinitely sharp reaction flux is present at the boundary between these two regions but zero elsewhere. The width of the penetration zone in anode LPZ,an is correlated with the penetration depth LPZ in the cathode because of the conservation of Li in the solid phase: Additionally, we assume that c(x) in the anode region is uniform outside the penetration zone ( ). Substituting the above solution of c(x) into Eq. S3 and solving for LPZ, we obtain the analytical expressions of LPZ as given in Supplementary Table S2. where is equal to DoDf when 0 < LPZ < Lcat. The ccat/sep(f) functions above are plotted in Supplementary Figure S5, which shows that ccat/sep is close to c0 when f is not too small. For example, ccat/sep/c0 varies in the range of 1 -0.7 (UR) and 1.09 -0.98 (MZR) when r decreases from 1 to 0.4. Supplementary Figure S5 also shows that the approximate ccat/sep given by Eq. S7 agrees well with the exact value of ccat/sep. If replacing ccat/sep in Eq. 16 with its approximate value at f = 1, we obtain a simple expression of LPZ for full cells with Gr anode as given in Eq. 17 in the main text.

II. Full cells with LTO anode
An approximate expression of ccat/sep in full cells with LTO anode can be found in a similar way as for full cells with Gr anode by using Eq. 16 and the analytical expression of LPZ given in Supplementary Supplementary Note S3 -Selection criteria of experimental data presented in Figure 6c The analytical model developed in this work assumes that the cathode is in fully deintercalated state at the beginning of the discharge process so that the reaction non-uniformity stems solely from electrolyte diffusion limitation. Among the experimental works we found in literature that systematically report the dependence of the discharge capacity on discharge rate and electrode thickness, many employed a symmetric cycling protocol, in which the discharge capacity was measured after battery cells were first charged at the same rate. At relatively high rates, this will cause the cathode particles to have a non-uniform initial SOC distribution at the beginning of discharge, which deviates from the model assumption. Furthermore, the discharge performance measured in symmetric cycling may be limited by the charging capacity preceding the discharge step, which causes the measurements to reflect the kinetic limitations during charging instead of discharging. As such, we include in Figure 6c only data from works that conducted asymmetric cycling tests, in which battery cells were first charged at a low C rate (<C/3) and then discharged at different rates. This ensures that the initial cell state is close to be fully charged prior discharging. We also limit the DoD range of the selected data to between 0.2 and 0.9 to ensure that they are in the electrolyte-diffusion-limited regime.     (1 −^7) /" #$% 0 #$% + " *+, 0 *+, + " $-0 $-1 + _ 1 2 0 *+, + " $-" *+, 0 $-`V*+, 0 *+,  found by simulations is located at the square with red edges, and the red circle is the model prediction.

ΔL cat
Δε cat (L cat opt ,ε cat opt ) Supplementary Figure S7. Optimization of NMC/Gr full cells at 5C discharge. a and c, contour plots of cell-level areal ( ) and specific ( ) capacity at 5C discharge as a function of Lcat and . The anode/cathode capacity ratio is held constant at 1.18, and 0 $-and " $-vary in proportion to 0 #$% and " #$% , respectively. Red solid line represents 0 #$% that maximizes q r or q s at a given " #$% . It is overlapped by the black dashed line, which corresponds to the critical cathode thickness 0 #$% #tb% at which 0 HI = 0 #$% . The blue dash-dotted line is given by the approximate expression of 0 #$% #tb% in Eq. 19. The red and blue squares represent the global optimum predicted by the full analytical model and the approximate expression based on Eq. 19, respectively. b and d, comparison of 0 #$% u,% and " #$% u,% for 5C discharge identified by P2D simulations vs. the analytical model. P2D simulations are performed at increments of = 10 µm and = 0.025. The filled color of square symbols represents the areal (b) and specific capacity (d) from simulations. ( , ) from simulations is located at the square with red edges, and the red circle is the model prediction. respectively. The anode/cathode capacity ratio is held constant at 1.18, and 0 $-and " $-vary in proportion to 0 #$% and " #$% , respectively. Red solid line represents 0 #$% that maximizes QA or Qw at a given " #$% . It is overlapped by the black dashed line, which corresponds to the critical cathode thickness 0 #$% #tb% at which 0 HI = 0 #$% . The blue dash-dotted line is given by the approximate expression of 0 #$% #tb% in Eq. 19. The red and blue squares represent the global optimum predicted by the full analytical model and the approximate expression based on Eq. 19, respectively.