Thermodynamic Origin of Reaction Non-Uniformity in Battery Porous Electrodes and Its Mitigation

The development of non-uniform reaction current distribution within porous electrodes is a ubiquitous phenomenon during battery charging / discharging and frequently controls the rate performance of battery cells. Reaction inhomogeneity in porous electrodes is usually attributed to the kinetic limitation of mass transport within the electrolyte and/or solid electrode phase. In this work, however, we reveal that it is also strongly influenced by the intrinsic thermodynamic behavior of electrode materials, specifically the dependence of the equilibrium potential on the state of charge: electrode reaction becomes increasingly non-uniform when the slope of the equilibrium potential curve is reduced. We employ numerical simulation and equivalent circuit model to elucidate such a correlation and show that the degree of reaction inhomogeneity and the resultant discharge capacity can be predicted by a dimensionless reaction uniformity number. For electrode materials that have equilibrium potentials insensitive to the state of charge and exhibit significant reaction non-uniformity, we demonstrate several approaches to spatially homogenizing the reaction current inside porous electrodes, including matching the electronic and ionic resistances, introducing graded electronic conductivity and reducing the surface reaction kinetics.

where the surface overpotential = Φ @ − Φ A + CD , Φ @ and Φ A are the electrical potentials of the electrolyte and solid matrix, respectively, and Ueq is the equilibrium or open-circuit potential of the active material.
Upon (dis)charging, jin is usually spatially inhomogeneous within porous electrodes especially in the depth direction of the electrode layer due to the ionic and electronic resistances of electrolytic and solid matrix phases. Such reaction non-uniformity limits the power output and is a main cause for the under-utilization of battery capacity at high rates. With the ever-growing need for higher energy density and the improvement in battery fabrication processes, the use of thick electrodes in Li-ion batteries has attracted increasing interest in recent years. However, reaction inhomogeneity becomes more severe with increasing electrode thickness, which leads to the inferior rate performance of thick electrodes and presents a major barrier to their commercial applications. A rational understanding of the origins of the non-uniform reaction distribution is thus critical for the design and optimization of battery systems at the cell level.
The importance of reaction non-uniformity to electrode performance has long been recognized since the early study of porous electrodes(1-4). Newman and Tobias theoretically examined the reaction current distribution in porous electrodes by deriving analytical solutions to the onedimensional porous electrode model (4). They show that the reaction distribution is controlled by two dimensionless numbers Reaction non-uniformity intensifies at large or , i.e. when the system has low effective ionic ( CQQ ) and electronic ( CQQ ) conductivities, large current (I) and/or exchange current density (i0).
On the other hand, the ratio between the electronic and ionic conductivities CQQ / CQQ controls the symmetry of the reaction distribution. Reaction occurs preferentially near the separator (or current collector) when CQQ / CQQ >>1 (or CQQ / CQQ ≈ 0), and develops on both sides of the electrodes when CQQ / CQQ ≈ 1.

In their analysis, Newman and Tobias treated the open-circuit potential Ueq of electrodes in
Eq. 1 as a constant. For electrode materials used in Li-ion batteries, however, it is common that Ueq depends on the extent of reaction or state of charge (SOC). A spatial gradient of Ueq will therefore result from the inhomogeneous reaction flux within porous electrodes, which will reversely influence the reaction distribution through the contribution of Ueq to jin in Eq. 1. In a recent study (5) In this work, we further the study on the thermodynamic origin of reaction non-uniformity in porous electrodes by considering a continuous spectrum of reaction behavior (UR, MZR and their intermediates) modulated by the Ueq -SOC relation. The effect of the average slope of the Ueq(SOC) curve, or Δ CD , on the reaction distribution and rate capability is examined by both pseudo-two-dimensional (P2D) porous electrode simulation (6)(7)(8)(9)(10) and an equivalent circuit model.
A dimensionless "reaction uniformity" number containing Δ CD is introduced to characterize the degree of reaction homogeneity. provides quantitative predictions of the reaction zone width and discharge capacity of battery cells. Based on insights obtained from the analysis, we propose several approaches to improving the reaction uniformity of MZR-type electrodes and demonstrate their effectiveness in P2D simulations.

I. P2D Simulations of Electrode Reaction Distribution
The two distinct types of reaction behaviors, UR vs MZR, can be illustrated by the discharge process of NMC111 and LFP cathodes, respectively. Figure 1a-b displays the P2D simulation of an NMC111 half cell (i.e. with Li metal anode) discharged at 5mA/cm 2 in the electrolyte-tranportlimited regime. Details on the implementation of the P2D model are described in Appendix A. The NMC111 cathode is 200 µm thick and other simulation parameters are listed in Table A1. Figure   1a shows that a large intercalation flux develops near separator at the beginning of discharge but is soon homogenized across the electrode before the depth of discharge (DoD) reaches 0.1 and then remains uniform until the end of discharge. Consequently, the entire electrode is uniformly lithiated and the SOC of electrode particles varies homogeneously within the porous electrode, see Figure 1b. Such UR behavior as schematized in Figure 1c is representative of electrode materials whose equilibrium potentials ( CD ) have a strong SOC dependence such as NMC and NCA.
The discharge process of an LFP half cell is shown in Figure 1d-e. While the LFP electrode has the same thickness as the NMC111 cathode and is discharged at the same current density of 5 mA/cm 2 , its reaction flux is highly localized throughout the discharge process. Figure 1d shows that an intercalation flux peak first forms on the separator side and then migrates towards the current collector as discharging proceeds. The peak corresponds to a moving narrow reaction front, which separates a largely lithiated electrode region near the separator and an unreacted region near the current collector, see Figure 1e. Such MZR behavior is idealized by the schematic shown in Figure 1e. It is representative of electrode materials with SOC-independent CD , e.g. LFP and LTO that go through first-order phase transformation(s) upon (de)lithiation.
As the above simulations demonstrate, NMC111 and LFP half cells have very different reaction distributions during discharge, with the latter exhibiting much stronger inhomogeneity.
Because the two cells have similar electrode thickness, porosity and conductivities, such difference results from the intrinsic properties of the two active materials and specifically, their different SOC dependence of Ueq. To unambiguously illustrate the effect of the Ueq(SOC) curve on reaction uniformity, here we consider a model active material whose Ueq (unit: V) is given by where @) is the occupancy fraction of available Li sites (fLi = 1 -SOC) fraction. As shown in  Figure 2e shows that WRZ, which provides a measure of the reaction uniformity, increases monotonically with Δ CD and should approach 0 and ∞ in the limiting MZR and UR cases, respectively. The effect of Δ CD on the reaction distribution has direct consequence on the rate performance of the electrodes. As shown in Figure 2f, increasing Δ CD from 0.001 to 1 V significantly improves the discharge performance at high rates, with a 74% increase in the normalized discharge capacity DoDf at 2C and a 103% increase at 5C.
The effect of the slope of the Ueq(SOC) curve on the reaction uniformity in porous electrodes can be qualitatively understood as follows. Reaction flux at the electrode particle surface is controlled by the overpotential = Φ @ − Φ A + CD , where Φ @ and Φ A are the electrical potentials of the electrolyte and active material, respectively. When discharging starts, Ueq is initially uniform across the porous electrode, but the presence of ionic / electronic resistances in electrolyte / solid phase generates spatial gradients in Φ @ and Φ A , which results in inhomogeneous η and therefore jin according to the Butler-Volmer equation (Eq. 1). When Ueq has a strong SOC dependence, non-uniform jin gives rise to an inhomogeneous spatial distribution of Ueq. Locations with higher jin see a larger decrease in Ueq, which in turn causes η and jin to drop more significantly than locations receiving lower jin. Therefore, the SOC-dependent Ueq serves as a "rectifier" to the non-uniform reaction distribution: it reduces the spatial gradient of jin until a constant reaction flux is reached within the porous electrode. The larger is the slope of Ueq(SOC), the stronger is such rectifying effect and the faster the electrode can establish a uniform reaction during discharging.
On the other hand, SOC-insensitive Ueq such as in LFP and LTO is not able to compensate the spatial gradients of Φ @ and Φ A to help homogenize the reaction flux. When discharging is electrolyte-transport-limited, reaction will first occur at the separator, where η is the largest, and continue until electrode particles in the local region are fully intercalated, after which the reaction front will move away from the separator like a traveling wave.
In the next section, we analyze the dependence of reaction distribution on Ueq(SOC) in a more quantitative manner based on an equivalent circuit model.

II. Equivalent Circuit Model for Reaction Uniformity Analysis
While P2D simulations provide detailed predictions of the reaction distribution within porous electrodes, its numerical nature makes it less straightforward to illuminate the general relation between the degree of reaction uniformity and various battery cell properties. As an alternative, we consider the discharge process in a considerably simplified circuit model, with the goal to derive a tractable expression to quantify the reaction uniformity in terms of the slope of Ueq(SOC) and other relevant parameters. Let LR be the dimension of the reaction zone in which the intercalation flux is non-zero during discharging. As illustrated in Figure 3b, the model represents this portion of the electrode with an equivalent circuit, which divides the zone into two regions (region I and II). For simplicity, electrode particles in each region is assumed to undergo reaction uniformly and have the same SOC, Φ A and Φ @ . We treat the active material as a generalized capacitor, whose characteristic voltage-charge relation is given by Ueq(SOC). Its internal resistance is neglected as solid diffusion is assumed to be facile. The electronic resistance of the solid phase and ionic resistance of the electrolyte are represented by two resistors connecting region I and II, A = 5 / CQQ and @ = 5 / CQQ , respectively.
Assuming small surface overpotential η, we use the linearized Butler-Volmer equation to express the reaction current in region I and II: where is the volumetric surface area of the electrode particles and the exchange current density is taken as a constant. Accordingly, the polarization caused by surface reaction is represented by a resistor )* = 2 / -5 in each region. For galvanostatic discharging, I1 and I2 are subject to the constraint: where I is the applied areal current density. Letting Φ A at the current collector be Φ A and setting Φ @ at the cathode/separator interface to be 0, the surface overpotentials in region I and II are given by

8)
Ueq is assumed to vary linearly with SOC or p : where p,-and p,sXt are the Li concentrations in the fully delithiated and lithiated states, respectively, and U0 is Ueq at SOC = 1. Accordingly, the evolution of Ueq in region I and II during discharge is governed by the following equations: where vXw is electrode porosity. Applying Eqs. 5 -8 to eliminate I1, I2 and Φ p in Eqs. 9 and 10, from which Ueq,1(t) and Ueq,2(t) can be solved: The equilibrium potential difference between region I and II, which quantifies the difference in the reaction degree, is thus in which we define The above result characterizes the time evolution of the reaction distribution within the reaction zone. As a main result of the circuit model, Eq. 15 shows that intercalation flux inside the reaction zone will reach a steady state distribution after a transient period with a characteristic time tc. In the steady state, region I and II have an equal reaction current (I1 = I2) so that their equilibrium potential difference CD,: − CD,‰ remains at a constant value Δ pp . Δ pp is negative when discharge is electrolyte-transport-limited, or CQQ < CQQ , meaning that the active material near the separator will react first. For discharge limited by electronic transport ( CQQ > CQQ ), Δ pp is positive and the reaction will first occur near the current collector.

III. Reaction Uniformity Number
In the last section, a steady-state equilibrium potential drop across the reaction zone, Δ pp , is determined from the two-block circuit model. The fact that |Δ pp | cannot exceed Δ CD places an upper limit on LR that can be maintained during discharge, which is given by We introduce a dimensionless reaction uniformity number and defined it as the ratio between 5,sXt and the electrode thickness vXw : bears the physical meaning of the maximum normalized reaction zone width. Its magnitude provides a measure of the degree of the reaction uniformity within the porous electrode. When ≫ 1, the entire electrode can establish a homogeneous reaction distribution and exhibit UR behavior. When ≪ 1, reaction is confined to a narrow region much smaller than the electrode thickness, and so MZR-type behavior ensues. reveals the roles of multiple factors (SOC dependence of Ueq, CQQ , CQQ , vXw , I) in regulating the reaction distribution.
To examine the predicative power of given by Eq. 19, we compare it against the P2D simulations of the model electrode system presented above in Figure 2. In Figure 4a, we plot the reaction zone width WRZ, which is measured from the P2D simulations, against estimated from the simulation parameters for electrodes with different Δ CD . In evaluating λ, we use the electrolyte conductivity at c = 1 M for CQQ although concentration-dependent CQQ is used in simulations. It can be seen that the calculated agrees very well with the measured 5} over a wide range of values from 0.1 to 100, which shows that can be used to accurately predict the extent of the reaction non-uniformity during discharge.
When the anion in the electrolyte has a non-zero transference number, local salt depletion (i.e. zero salt concentration) will occur in electrolyte near the current collector at high discharging rates or large electrode thickness (5,11) and result in a large salt concentration gradient across the electrode, which will strongly influence the ionic conductivity. Although the simple circuit model presented in Section II does not take the concentration dependence of CQQ into consideration, the reaction uniformity number still provides a reliable indication of the electrode performance in the presence of salt depletion. In Figure 4b, we plot the normalized discharge capacity DoDf at 2C, 3C and 5C from the P2D simulations of the model electrode system as a function of Δ CD . It shows that DoDf increases monotonically with Δ CD and has two plateaus at small and large Δ CD values, which correspond to the MZR and UR behavior, respectively. Previously, we developed a quantitative analytical model to predict the discharge performance of UR-and MZR-type electrodes in the electrolyte-diffusion-limited regime (5). It gives the expressions of the width of the salt penetration zone LPZ (i.e. region with non-zero salt concentration and complementary to the salt depletion zone) as listed in Table 1, and DoDf is evaluated as LPZ/Lcat. The dashed and dash-dotted lines in Figure 4b represent DoDf predicted by the analytical model for MZR (DoD ¤ ¥¦ § ) and UR (DoD ¤ § ) electrodes, respectively, which match the lower and upper limits of the simulated discharge capacity very well. for UR behavior, which is a more appropriate reference length scale as it represents the maximum thickness of the electrode region that can be fully discharged at the given discharge condition. In To test the generality of Eq. 20, we performed 100 additional simulations by varying different cell parameters ( Xs® , vXw , vXw , vXw and pCª ) and compared the simulated DOD · ¤ against ( ). As shown in Figure 4d, overall the simulation results are in very good agreement with predictions by ( ), which likely represents a universal DOD · ¤ ~ relation. The discharge capacity of electrodes with intermediate reaction behavior in the electrolyte-transport-limited regime may therefore be predicted as

IV. Approaches to homogenizing reaction distribution in MZR-type electrodes
Our work reveals that MZR-type electrodes, i.e. electrodes whose Ueq is insensitive to SOC, have inferior performance at high rates and/or large electrode thickness due to the strong reaction inhomogeneity during discharge. In addition, the highly localized intercalation flux within the narrow reaction front may accelerate battery degradation by causing excessive stress concentration and local heat generation. Based on the insights from the P2D simulation and circuit model, we discuss in this section how reaction in this type of electrodes can be homogenized to make them more suitable for high rate and thick electrode applications. Somewhat counter-intuitively, we show that reducing the electronic conductivity and/or surface reaction rate is beneficial to improving the reaction uniformity in MZR-type electrodes.

i) Reduce electronic conductivity
The rate performance of today's Li-ion battery cells is typically limited by sluggish ionic transport in the electrolyte, whereas the electronic conductivity can be made sufficiently high with conductive additives or coatings on active materials. When CQQ ≪ CQQ , electrode reaction first occurs near the separator upon discharging (12,13), to which electrons travel a longer distance from the current collector to meet slow-moving Li ions from the anode. However, Eq. 19 predicts that the reaction uniformity can be improved by reducing the electronic conductivity to CQQ ≈ CQQ to render a large . To test this prediction, a P2D simulation is performed for a model system with Δ CD = 0.001V, in which CQQ is set to ( = 1 ) = -vXw :.m = 0.291 S/m. As shown in Figure 5a, two reaction fronts form on both sides of the electrode and propagate towards the electrode center during 0.5C discharge. Accordingly, the intercalation flux is split into two peaks of lower intensities and does become more uniformly distributed compared to the higher CQQ case, see Figure 5b, although the reaction distribution is not entirely homogenized as predicted by the circuit model. This is because the model oversimplifies the situation by dividing the reaction zone into only two blocks and neglecting the non-uniformity within each block. Figure 5c shows that DoDf increases monotonically with decreasing CQQ upon discharging at higher rate (1 -3C) and can even reach the discharge capacity of UR-type electrodes (dash-dotted lines) when CQQ approaches CQQ .

ii) Grade electronic conductivity
Further improvement in the reaction uniformity can be realized by allowing CQQ to vary spatially within the electrode. This is because having a uniform reaction flux requires a constant surface overpotential everywhere, or ∇ = ∇Φ @ − ∇Φ A + ∇ CD = 0 22) In Eq. 22, a significant ∇Φ @ is usually present upon (dis)charging at relatively high rates due to the low ionic conductivity of the electrolyte, but ∇ CD ≈ 0 for MZR-type materials. Replacing ∇Φ A and ∇Φ @ with the current densities in the solid phase (I1) and electrolyte (I2), respectively, Eq. 22 becomes: In the presence of a uniform flux, both : and ‰ vary linearly with the distance to the current collector X: : = ( vXw − )/ vXw and ‰ = / vXw . Therefore, Eq. 23 is satisfied if the following relation between CQQ and CQQ holds:

24)
According to Eq. 24, the optimal CQQ is a hyperbolic function and varies monotonically from infinity at the current collector (X = 0) to 0 at the separator (X = vXw ). We confirm the effectiveness of such conductivity distribution via P2D simulation, in which CQQ is set as -:.m ( vXw − )/ and other parameters are the same as those for Figure 5a. As shown in Figure 5b and d, lithium intercalation indeed has a much more uniform distribution across the electrode throughout the discharge process than in systems with constant CQQ .
We note that Palko et al. (14) recently describe a similar approach of tailoring spatially varied electrode matrix resistance to homogenize electrolyte depletion in electrical double layer capacitors (EDLC). The similarity in the derived CQQ expressions in ref. (14) and here demonstrates the analogy in the behavior of MZR-type battery electrodes and capacitors. On the other hand, UR-type electrodes behave in a very different way and Eq. 22 highlights such difference. The ability of UR-type compounds to sustain a non-zero spatial gradient in CD in porous electrodes makes it possible to offset ∇Φ @ with ∇ CD to maintain a uniform overpotential without the need for spatially varied CQQ . In the absence of a non-zero ∇ CD in MZR-type electrodes, however, ∇Φ @ can only be balanced by the Φ p gradient to satisfy Eq. 22.
Experimentally, the electronic conductivity of porous electrodes can be tuned by adjusting the amount of conductive additives or applying coatings to active materials to either increase or decrease the conductivity, and graded electrodes may be prepared via layer-by-layer deposition processes. In Ref. (15) They report that graded TiO2(B)/RGO electrodes with the high CQQ layer placed adjacent to the current collector deliver more than 70% capacity at 20C than uniform electrodes with the same average RGO weight fraction. The theoretical analysis presented here explains why such an approach is effective. Using graded and heterogeneous architecture to enhance the rate performance of thick electrodes has been explored theoretically (16,17) and experimentally (18)(19)(20)(21)(22)(23) in recent years. Most existing studies focus on tailoring the porosity distribution to enhance electrolyte transport. Here we demonstrate a different strategy based on reducing the electronic conductivity to match the low ionic conductivity to improve the rate performance.

iii) Reduce surface reaction rate
The circuit model presented in Section II shows that during discharge a battery cell first goes through a transient period with a characteristic time tc before establishing a steady-state reaction zone within the porous electrode. Since the active material has a uniform SOC at the beginning of discharge, another way to improve the reaction uniformity in MZR-type electrodes is to increase tc to delay the establishment of the narrow reaction front and let the system remain in the transient period for the majority of the discharge process. Eq. 17 shows that tc can be increased by reducing i0. We demonstrate this approach via P2D simulations of the model system with Δ CD = 0.001 V, in which the surface reaction rate constant k0 is decreased from the default value [10 -8 mol·m -2 ·s -1 ·(mol·m -3 ) -1.5 ] to 10 -11 and 10 -13 mol·m -2 ·s -1 ·(mol·m -3 ) -1.5 . The corresponding time evolution of ̃A( z ) is plotted in Figure 6a and 6b, respectively. Compared to Figure 3b, a smaller k0 indeed slows down the development of the sharp reaction front and results in a more uniform reaction across the electrode during discharge. As expected, decreasing k0 can increase the discharge capacity by 20 -35% at high rates (1 -3C), Figure 6c, which is similar to the effect of reducing CQQ although the improvement is not as pronounced. The reason that reducing the surface reaction kinetics is beneficial is that it prevents the localization of the intercalation flux and forces the reaction current to spread out over a larger electrode region. Experimentally, surface reaction kinetics may be tailored by "artificial" SEI such as ALD coatings of various inorganic compounds (e.g. Al2O3, TiO2, ZrO2 (24-26)), whose insulating nature could retard the intercalation process and cause higher surface polarization.
While reducing CQQ and k0 promotes the reaction uniformity, such approaches may lead to increased energy loss and degrade energy efficiency, the severity of which needs to be examined. potential while delivering 30% more capacity than the baseline electrode at the same time. On the other hand, the electrode with reduced k0 has a larger depression in the discharge potential (~0.1 V). In Figure 7b, the total energy loss in a half cell upon 1C discharging, which is the sum of losses due to the ionic, electronic and surface reaction resistances, is plotted as a function of DoD for the four cases. It can be seen that reducing or grading CQQ only slightly increases the energy loss by less than 15% compared to the baseline case while decreasing k0 doubles the energy loss. Therefore, tailoring the electronic conductivity of the solid electrode phase is a more attractive strategy to enhance the discharge performance of thick electrodes.

Conclusion
In this work, we employ P2D simulations and equivalent circuit model to elucidate the important role of the SOC dependence of the open-circuit potential Ueq, an intrinsic thermodynamic property of battery compounds, in controlling the reaction uniformity within porous electrodes. Electrode reaction becomes increasingly homogeneous with the slope of the Ueq(SOC) curve, which has a direct impact on the battery discharge performance at high rates. The limiting cases can be described by the "uniform reaction" or UR behavior for electrodes whose Ueq has strong SOC dependence (e.g. NMC and NCA), and the "moving-zone reaction" or MZR behavior for electrodes with SOC-independent Ueq (e.g. LiFePO4, Li4Ti5O12 / vXw , is introduced to capture the effects of electrode and cycling parameters on the degree of reaction inhomogeneity. In the electrolytetransport-limited regime, accurately predicts the reaction zone width and exhibits a universal correlation with the rescaled discharge capacity, making a useful indicator of the electrode performance. We show that the reaction distribution in MZR-type electrodes can be homogenized by several approaches including 1) matching the ionic and electronic conductivities, 2) grading the electronic conductivity, and 3) slowing down the surface reaction kinetics, of which the first two do not significantly reduce the energy efficiency of the discharging process.

Appendix A
P2D simulations: detailed description of the P2D model can be found in literature (6)(7)(8)(9)(10). The governing equations implemented in the half cell simulations are summarized as follows.
The concentration and ionic current in a binary electrolyte are given by For the model electrode material, Li diffusion is assumed to be very facile so that p is constant within each electrode particle. All of the simulations are implemented in COMSOL Multiphysics® 5.3a.

Symbol list
Volumetric  Li diffusivity in active materials (m 2 ·s -1 )  Table A1. a and d. Reaction flux vXw )* on NMC111 and LFP particle surface, respectively. b and d. Average Li concentration A in NMC111 and LFP particles, respectively. c and f. Schematics of idealized UR vs MZR behavior.        (Table 1).