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

The two distinct types of reaction behaviors, uniform-reaction vs moving-zone-reaction, can be illustrated by the discharge process of NMC111 and LFP cathodes, respectively. Figs. 1a–1b displays the P2D simulation of an NMC111 half cell (i.e. with Li metal anode) discharged at 5 mA cm⁻² in the electrolyte-transport-limited regime. Details on the implementation of the P2D model are described in Appendix. The NMC111 cathode is 200 μm thick and other simulation parameters are listed in Table A·I. Figure 1a shows that a large intercalation flux develops near separator at the beginning of discharge but is soon spatially uniform 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 in-plane-averaged SOC of electrode particles varies homogeneously within the porous electrode, see Fig 1b. Such uniform-reaction behavior as schematized in Fig. 1c is representative of electrode materials whose equilibrium potentials (${U}_{eq}$) have a strong SOC dependence such as NMC and NCA.

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The discharge process of an LFP half cell is shown in Figs. 1d–1e. 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 has a localized peak moving in the electrode during discharge. Fig. 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 Fig. 1e. Such moving-zone-reaction behavior is idealized by the schematic shown in Fig. 1e. It is representative of electrode materials with SOC-independent ${U}_{eq},$ 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 U_eq. The thermodynamic difference between electrodes such as LFP, which undergo first-order phase transitions upon (dis)charging, and electrodes with extended solid solution regime such as NMC is known to result in qualitatively different solid-state lithium transport kinetics, which has been extensively studied including numerical simulations.^13–15 However, there is little discussion in literature on the coupling between their thermodynamic phase behavior and the reaction distribution at the porous electrode level.

To illustrate the effect of the U_eq(SOC) curve on reaction uniformity, here we consider a model active material whose U_eq (unit: V) is given by

$$\begin{eqnarray}&&{U}_{eq}\left({f}_{Li}\right)=-\displaystyle \frac{{\rm{\Delta }}{U}_{eq}}{4}\,\mathrm{ln}\left(\displaystyle \frac{{f}_{Li}}{1-{f}_{Li}}\right)+3\end{eqnarray} \tag{ 4 }$$

where ${f}_{Li}$ is the occupancy fraction of available Li sites and f_Li = 1—SOC. As shown in Fig. 2a, parameter ${\rm{\Delta }}{U}_{eq}\equiv {\left.\tfrac{d{U}_{eq}}{d{f}_{Li}}\right|}_{{f}_{Li}=0.5}$ is equal the slope of U_eq at SOC = 0.5, which controls the steepness of the U_eq(SOC) curve. P2D simulations are performed with varied ${\rm{\Delta }}{U}_{eq}$ while all the other system properties are kept unchanged. Figs. 2b–2d present the evolution of the scaled Li concentration ${\tilde{c}}_{s}\equiv {c}_{s}/{c}_{s,{\rm{\max }}}$ in electrode particles in a half cell discharged at 0.5 C when ${\rm{\Delta }}{U}_{eq}$ = 0.001, 0.01 and 1 V, respectively. The electrode thickness ${L}_{cat}$ is 200 μm and other parameters are listed in Table A·I. The electronic conductivity ${\sigma }_{eff}$ and surface reaction rate constant k₀ used in the simulations are sufficiently large so that the discharging process is kinetically limited by electrolyte transport. The comparison shown in Figs. 2b–2d reveals that Li intercalation becomes more homogeneous as ${\rm{\Delta }}{U}_{eq}$ increases. Characteristics of moving-zone-reaction and uniform-reaction are evident at ${\rm{\Delta }}{U}_{eq}$ = 0.001 and 1 V, respectively. At the intermediate ${\rm{\Delta }}{U}_{eq}$ (0.01 V), electrode reaction exhibits a combination of moving-zone-reaction and uniform-reaction behaviors, where significant intercalation flux exists within a reaction zone of finite width. As illustrated in Fig. 2c, we may define a scaled reaction zone width W_RZ as the inverse of the slope of ${\tilde{c}}_{s}\left(\tilde{X}\right)$ at ${\tilde{c}}_{s}$ = 0.5 and SOC = 0.5: ${W}_{RZ}=\,d\tilde{X}/d{\tilde{c}}_{s}{| }_{{\tilde{c}}_{s}=0.5,{\rm{SOC}}=0.5}.$ Fig. 2e shows that W_RZ, which provides a measure of the reaction uniformity, increases monotonically with ${\rm{\Delta }}{U}_{eq}$ and should approach 0 and $\infty$ in the limiting moving-zone-reaction and uniform-reaction cases, respectively. The effect of ${\rm{\Delta }}{U}_{eq}$ on the reaction distribution has direct consequence on the rate performance of the electrodes. As shown in Fig. 2f, increasing ${\rm{\Delta }}{U}_{eq}$ from 0.001 to 1 V significantly improves the discharge performance at high rates, with a 74% increase in the normalized discharge capacity DoD_f at 2 C and a 103% increase at 5 C.

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The effect of the slope of the U_eq(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 $\eta ={{\rm{\Phi }}}_{L}-{{\rm{\Phi }}}_{S}+{U}_{eq},$ where ${{\rm{\Phi }}}_{L}$ and ${{\rm{\Phi }}}_{S}$ are the electrical potentials of the electrolyte and active material, respectively. When discharging starts, U_eq is initially uniform across the porous electrode, but the presence of ionic/electronic resistances in electrolyte/solid phase generates spatial gradients in ${{\rm{\Phi }}}_{L}$ and ${{\rm{\Phi }}}_{S},$ which results in inhomogeneous η and therefore j_in according to the Butler-Volmer equation (Eq. 1). When U_eq has a strong SOC dependence, non-uniform j_in gives rise to an inhomogeneous spatial distribution of U_eq. Locations with higher j_in see a larger decrease in U_eq, which in turn causes η and j_in to drop more significantly than locations receiving lower j_in. Therefore, the SOC-dependent U_eq serves as a "rectifier" to the non-uniform reaction distribution: it reduces the spatial gradient of j_in until a constant reaction flux is reached within the porous electrode. The larger is the slope of U_eq(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 U_eq such as in LFP and LTO is not able to compensate the spatial gradients of ${{\rm{\Phi }}}_{L}$ and ${{\rm{\Phi }}}_{S}$ to help homogenize the reaction flux. When discharging is electrolyte-transport-limited, reaction will first occur at the separator, where η is the largest, similar to the development of shunt current in flow batteries.¹⁶ When electrode particles near the separator are fully intercalated, the reaction front will move into the porous electrode like a traveling wave.

In the next section, we analyze the dependence of reaction distribution on U_eq(SOC) in a more quantitative manner based on an equivalent circuit model.

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 U_eq(SOC) and other relevant parameters. Let L_R be the dimension of the reaction zone in which the intercalation flux is non-zero during discharging. As illustrated in Fig. 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, ${{\rm{\Phi }}}_{S}$ and ${{\rm{\Phi }}}_{L}.$ We treat the active material as a generalized capacitor, whose characteristic voltage–charge relation is given by U_eq(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, ${R}_{S}={L}_{R}/{\sigma }_{eff}$ and ${R}_{L}={L}_{R}/{\kappa }_{eff},$ respectively.

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Assuming small surface overpotential η, we use the linearized Butler-Volmer equation to express the reaction current in region I and II:

$$\begin{eqnarray}&&{I}_{k}=\displaystyle \frac{Fa{L}_{R}{i}_{0}}{2RT}{\eta }_{k}\,k=1,2\end{eqnarray} \tag{ 5 }$$

where $a$ is the volumetric surface area of the electrode particles and the exchange current density ${i}_{0}$ is taken as a constant. Accordingly, the polarization caused by surface reaction is represented by a resistor ${R}_{in}=2RT/Fa{i}_{0}{L}_{R}$ in each region. For galvanostatic discharging, I₁ and I₂ are subject to the constraint:

$$\begin{eqnarray}&&{I}_{1}+{I}_{2}=I\end{eqnarray} \tag{ 6 }$$

where I is the applied areal current density. Letting ${{\rm{\Phi }}}_{S}$ at the current collector be ${{\rm{\Phi }}}_{S}^{0}$ and setting ${{\rm{\Phi }}}_{L}$ at the cathode/separator interface to be 0, the surface overpotentials in region I and II are given by

$$\begin{eqnarray}&&{\eta }_{1}={U}_{eq,1}-{I}_{1}{R}_{S}-{{\rm{\Phi }}}_{S}^{0}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\eta }_{2}={U}_{eq,2}-{I}_{2}{R}_{L}-{{\rm{\Phi }}}_{S}^{0}\end{eqnarray} \tag{ 8 }$$

U_eq is assumed to vary linearly with SOC or ${c}_{s}:$ ${U}_{eq}\,={U}_{0}-{\rm{\Delta }}{U}_{eq}\left({c}_{s}-{c}_{s,0}\right)/\left({c}_{s,{\rm{\max }}}-{c}_{s,0}\right),$ where ${c}_{s,0}$ and ${c}_{s,{\rm{\max }}}$ are the Li concentrations in the fully delithiated and lithiated states, respectively, and U₀ is U_eq at SOC = 1. Accordingly, the evolution of U_eq in region I and II during discharge is governed by the following equations:

$$\begin{eqnarray}&&\displaystyle \frac{d{U}_{eq,1}}{dt}=-\displaystyle \frac{2{\rm{\Delta }}{U}_{eq}}{F\left(1-{{\epsilon }}_{cat}\right){L}_{R}\left({c}_{S,{\rm{\max }}}-{c}_{S,0}\right)}{I}_{1}\equiv -\xi {I}_{1}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{U}_{eq,2}}{dt}=-\xi {I}_{2}\end{eqnarray} \tag{ 10 }$$

where ${{\epsilon }}_{cat}$ is electrode porosity. Applying Eqs. 5–8 to eliminate I₁, I₂ and ${{\rm{\Phi }}}_{s}^{0}$ in Eqs. 9 and 10, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{d{U}_{eq,1}}{dt}=-\displaystyle \frac{\xi }{{R}_{L}+{R}_{S}+2{R}_{in}}\left[{U}_{eq,1}-{U}_{eq,2}+I\left({R}_{L}+{R}_{in}\right)\right]\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\displaystyle \frac{d{U}_{eq,2}}{dt}=-\displaystyle \frac{\xi }{{R}_{L}+{R}_{S}+2{R}_{in}}\left[{U}_{eq,2}-{U}_{eq,1}+I\left({R}_{S}+{R}_{in}\right)\right]\end{eqnarray} \tag{ 12 }$$

from which U_eq,1(t) and U_eq,2(t) can be solved:

$$\begin{eqnarray}&&\begin{array}{l}{U}_{eq,1}\left(t\right)={U}_{0}-\displaystyle \frac{I\xi t}{2}+\displaystyle \frac{I\left({R}_{S}-{R}_{L}\right)}{4}\left(1\right.\\ \left.\,\,\,\,-\,\exp \left(-\displaystyle \frac{2\xi t}{{R}_{L}+{R}_{S}+2{R}_{in}}\right)\right)\end{array}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\begin{array}{l}{U}_{eq,2}\left(t\right)={U}_{0}-\displaystyle \frac{I\xi t}{2}-\displaystyle \frac{I\left({R}_{S}-{R}_{L}\right)}{4}\left(1\right.\\ \left.\,\,\,\,-\,\exp \left(-\displaystyle \frac{2\xi t}{{R}_{L}+{R}_{S}+2{R}_{in}}\right)\right)\end{array}\end{eqnarray} \tag{ 14 }$$

The equilibrium potential difference between region I and II, which quantifies the difference in the reaction degree, is thus

$$\begin{eqnarray}&&{U}_{eq,1}-{U}_{eq,2}={\rm{\Delta }}{U}_{ss}\left[1-\exp \left(-\displaystyle \frac{t}{{t}_{c}}\right)\right]\end{eqnarray} \tag{ 15 }$$

in which we define

$$\begin{eqnarray}&&{\rm{\Delta }}{U}_{ss}\equiv \displaystyle \frac{I\left({R}_{S}-{R}_{L}\right)}{2}=\displaystyle \frac{I{L}_{R}}{2}\left(\displaystyle \frac{1}{{\sigma }_{eff}}-\displaystyle \frac{1}{{\kappa }_{eff}}\right)\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\begin{array}{l}{t}_{c}\equiv \displaystyle \frac{{R}_{L}+{R}_{s}+2{R}_{in}}{2\xi }=\displaystyle \frac{F\left(1-{{\epsilon }}_{cat}\right)\left({c}_{S,{\rm{\max }}}-{c}_{S,0}\right)}{4{\rm{\Delta }}{U}_{eq}}\\ \,\,\,\,\,\times \,\left(\displaystyle \frac{{L}_{R}^{2}}{{\kappa }_{eff}}+\displaystyle \frac{{L}_{R}^{2}}{{\sigma }_{eff}}+\displaystyle \frac{4RT}{Fa{i}_{0}}\right)\end{array}\end{eqnarray} \tag{ 17 }$$

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 t_c. In the steady state, region I and II have an equal reaction current (I₁ = I₂) so that their equilibrium potential difference ${U}_{eq,1}-{U}_{eq,2}$ remains at a constant value ${\rm{\Delta }}{U}_{ss}.$ ${\rm{\Delta }}{U}_{ss}$ is negative when discharge is electrolyte-transport-limited, or ${\kappa }_{eff}\lt {\sigma }_{eff},$ meaning that the active material near the separator will react first. For discharge limited by electronic transport (${\kappa }_{eff}\gt {\sigma }_{eff}$), ${\rm{\Delta }}{U}_{ss}$ is positive and the reaction will first occur near the current collector.

In the last section, a steady-state equilibrium potential drop across the reaction zone, ${\rm{\Delta }}{U}_{ss},$ is determined from the two-block circuit model. The fact that $\left|{\rm{\Delta }}{U}_{ss}\right|$ cannot exceed ${\rm{\Delta }}{U}_{eq}$ places an upper limit on L_R that can be maintained during discharge, which is given by

$$\begin{eqnarray}&&{L}_{R,{\rm{\max }}}=\displaystyle \frac{2{\rm{\Delta }}{U}_{eq}}{I\left|\displaystyle \frac{1}{{\kappa }_{eff}}-\displaystyle \frac{1}{{\sigma }_{eff}}\right|}\end{eqnarray} \tag{ 18 }$$

We introduce a dimensionless reaction uniformity number and defined it as the ratio between ${L}_{R,{\rm{\max }}}$ and the electrode thickness ${L}_{cat}:$

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{L}_{R,{\rm{\max }}}}{{L}_{cat}}=\displaystyle \frac{2{\rm{\Delta }}{U}_{eq}}{I{L}_{cat}\left|\displaystyle \frac{1}{{\kappa }_{eff}}-\displaystyle \frac{1}{{\sigma }_{eff}}\right|}\end{eqnarray} \tag{ 19 }$$

$\lambda$ 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 $\lambda \gg 1,$ the entire electrode can establish a homogeneous reaction distribution and exhibit uniform-reaction behavior. When $\lambda \ll 1,$ reaction is confined to a narrow region much smaller than the electrode thickness, and so moving-zone-reaction-type behavior ensues. $\lambda$ reveals the roles of multiple factors (SOC dependence of U_eq, ${\kappa }_{eff},$ ${\sigma }_{eff},$ ${L}_{cat},$ I) in regulating the reaction distribution. Some electrode materials such as Li(Ni_0.5Mn_1.5)O₄ undergo more than one first-order phase transitions upon (dis)charging with two plateaus on their U_eq(SOC). The discharge process of these electrodes is characterized by two separate reaction zones, which form and propagate within the electrodes in sequence. The same analysis can be applied to each reaction zone individually, in which the reaction uniformity number should be calculated based on the potential drop across each plateau.

To examine the predicative power of $\lambda$ given by Eq. 19, we compare it against the P2D simulations of the model electrode system presented above in Fig. 2. In Fig. 4a, we plot the reaction zone width W_RZ, which is measured from the P2D simulations, against $\lambda$ estimated from the simulation parameters for electrodes with different ${\rm{\Delta }}{U}_{eq}.$ In evaluating λ, we use the electrolyte conductivity at c = 1 M for ${\kappa }_{eff}$ although concentration-dependent ${\kappa }_{eff}$ is used in simulations. It can be seen that the calculated $\lambda$ agrees very well with the measured ${W}_{RZ}$ over a wide range of values from 0.1 to 100, which shows that $\lambda$ can be used to accurately predict the extent of the reaction non-uniformity during discharge.

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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^7,17 and result in a large salt concentration gradient across the electrode, which will strongly influence the ionic conductivity. Although the simple circuit model does not take the concentration dependence of ${k}_{eff}$ into consideration, the reaction uniformity number $\lambda$ still provides a reliable indication of the electrode performance in the presence of salt depletion. In Fig. 4b, we plot the normalized discharge capacity DoD_f at 2 C, 3 C and 5 C from the P2D simulations of the model electrode system as a function of ${\rm{\Delta }}{U}_{eq}.$ It shows that DoD_f increases monotonically with ${\rm{\Delta }}{U}_{eq}$ and has two plateaus at small and large ${\rm{\Delta }}{U}_{eq}$ values, which correspond to the moving-zone-reaction and uniform-reaction behavior, respectively. Previously, we developed a quantitative analytical model to predict the discharge performance of uniform-reaction- and moving-zone-reaction-type electrodes in the electrolyte-diffusion-limited regime.⁷ It gives the expressions of the width of the salt penetration zone L_PZ (i.e. region with non-zero salt concentration and complementary to the salt depletion zone) as listed in Table I, and DoD_f is evaluated as L_PZ/L_cat. The dashed and dash-dotted lines in Fig. 4b represent DoD_f predicted by the analytical model for moving-zone-reaction (${{\rm{DoD}}}_{{\rm{f}}}^{{\rm{MZR}}}$) and uniform-reaction (${{\rm{DoD}}}_{{\rm{f}}}^{{\rm{UR}}}$) electrodes, respectively, which match the lower and upper limits of the simulated discharge capacity very well.

Table I.  Expressions of L_PZ and DoD_f for galvanostatic discharging of half cells predicted by an analytical model.⁷

${L}_{PZ}$	Uniform-reaction (UR) electrodes	$-\displaystyle \frac{3{{\epsilon }}_{sep}}{2{{\epsilon }}_{cat}}{L}_{sep}+\sqrt{\displaystyle \frac{6F{D}_{amb}{c}_{0}}{{\tau }_{cat}I\left(1-{t}_{+}\right)}\left({{\epsilon }}_{cat}{L}_{cat}+{{\epsilon }}_{sep}{L}_{sep}\right)+\left(\displaystyle \frac{9{{\epsilon }}_{sep}^{2}}{4{{\epsilon }}_{cat}^{2}}-\displaystyle \frac{3{\tau }_{sep}}{{\tau }_{cat}}\right){L}_{sep}^{2}}$
	Moving-zone-reaction (MZR) electrodes	$-\displaystyle \frac{{{\epsilon }}_{sep}}{{{\epsilon }}_{cat}}{L}_{sep}+\sqrt{\displaystyle \frac{2F{D}_{amb}{c}_{0}}{{\tau }_{cat}I\left(1-{t}_{+}\right)}\left({{\epsilon }}_{cat}{L}_{cat}+{{\epsilon }}_{sep}{L}_{sep}\right)+\left(\displaystyle \frac{{{\epsilon }}_{sep}^{2}}{{{\epsilon }}_{cat}^{2}}-\displaystyle \frac{{\tau }_{sep}}{{\tau }_{cat}}\right){L}_{sep}^{2}}$
${{\rm{DoD}}}_{{\rm{f}}}=\left\{\begin{array}{cc}{L}_{PZ}/{L}_{cat}, & 0\leqslant {L}_{PZ}\leqslant {L}_{cat}\\ 1 & {L}_{PZ}\gt {L}_{cat}\end{array}\right.$

While it is challenging to directly extend this analytical model to electrodes with the reaction behavior intermediate between uniform-reaction and moving-zone-reaction, we find that $\lambda$ serves as a very good descriptor of DoD_f. When using Eq. 19 to calculate $\lambda$ in the presence of salt depletion, we replace ${L}_{cat}$ with L_PZ for uniform-reaction 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 Fig. 4c, we replot Fig. 4b by rescaling DoD_f as ${\tilde{{\rm{DOD}}}}_{{\rm{f}}}\,=\left({{\rm{DoD}}}_{{\rm{f}}}-{{\rm{DoD}}}_{{\rm{f}}}^{{\rm{MZR}}}\right)/\left({{\rm{DoD}}}_{{\rm{f}}}^{{\rm{UR}}}-{{\rm{DoD}}}_{{\rm{f}}}^{{\rm{MZR}}}\right),$ which always varies between 0 and 1. It clearly shows that the ${\tilde{{\rm{DOD}}}}_{{\rm{f}}}$ ~ $\lambda$ relations at different C rates collapse onto a single S-shaped curve, which can be fitted by an analytical function:

$$\begin{eqnarray}&&T\left(\lambda \right)=\displaystyle \frac{1}{2}\left[1+\,\tanh \left(1.963{\mathrm{log}}_{10}\left(\lambda \right)-0.104\right)\right]\end{eqnarray} \tag{ 20 }$$

To test the generality of Eq. 20, we performed 100 additional simulations by varying different cell parameters (${D}_{amb},$ ${L}_{cat},$ ${{\epsilon }}_{cat},$ ${\tau }_{cat}$ and ${L}_{sep}$) and compared the simulated ${\tilde{{\rm{DOD}}}}_{{\rm{f}}}$ against $T\left(\lambda \right).$ As shown in Fig. 4d, overall the simulation results are in very good agreement with predictions by $T\left(\lambda \right),$ which likely represents a universal ${\tilde{{\rm{DOD}}}}_{{\rm{f}}}$ ∼$\lambda$ relation. The discharge capacity of electrodes with intermediate reaction behavior in the electrolyte-transport-limited regime may therefore be predicted as

$$\begin{eqnarray}&&{{\rm{DoD}}}_{{\rm{f}}}={{\rm{DoD}}}_{{\rm{f}}}^{{\rm{MZR}}}+T\left(\lambda \right)\left({{\rm{DoD}}}_{{\rm{f}}}^{{\rm{UR}}}-{{\rm{DoD}}}_{{\rm{f}}}^{{\rm{MZR}}}\right)\end{eqnarray} \tag{ 21 }$$

Our work reveals that moving-zone-reaction-type electrodes, i.e. electrodes whose U_eq 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. Based on Eqs. 17 and 19 from the circuit model and somewhat counter-intuitively, we show that reducing the electronic conductivity and/or surface reaction rate is beneficial to improving the reaction uniformity in moving-zone-reaction-type electrodes.

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 ${\kappa }_{eff}\ll {\sigma }_{eff},$ electrode reaction first occurs near the separator upon discharging,^18,19 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 ${\sigma }_{eff}\approx {\kappa }_{eff}$ to render a large $\lambda .$ To test this prediction, a P2D simulation is performed for a model system with ${\rm{\Delta }}{U}_{eq}=0.001{\rm{V}},$ in which ${\sigma }_{eff}$ is set to $\kappa \left(c=1M\right)={\kappa }_{0}{{\epsilon }}_{cat}^{1.5}=$ 0.291 s m⁻¹. As shown in Fig. 5a, two reaction fronts form on both sides of the electrode and propagate towards the electrode center during 0.5 C discharge. Accordingly, the intercalation flux is split into two peaks of lower intensities and does become more uniformly distributed compared to the higher ${\sigma }_{eff}$ case, see Fig. 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. Fig. 5c shows that DoD_f increases monotonically with decreasing ${\sigma }_{eff}$ upon discharging at higher rate (1–3 C) and can even reach the discharge capacity of uniform-reaction-type electrodes (dash-dotted lines) when ${\sigma }_{eff}$ approaches ${\kappa }_{eff}.$ Upon further decreasing ${\sigma }_{eff},$ however, the discharge kinetics will be increasingly limited by the sluggish electronic transport, resulting in eventual deterioration in rate performance as indicated by the discharge capacity at 3 C. There exists an optimal ratio of ${\sigma }_{eff}$ and ${\kappa }_{eff}$ around 1.

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Grade electronic conductivity

Further improvement in the reaction uniformity can be realized by allowing ${\sigma }_{eff}$ to vary spatially within the electrode. This is because having a uniform reaction flux requires a constant surface overpotential everywhere, or

$$\begin{eqnarray}&&{\rm{\nabla }}\eta ={\rm{\nabla }}{{\rm{\Phi }}}_{L}-{\rm{\nabla }}{{\rm{\Phi }}}_{S}+{\rm{\nabla }}{U}_{eq}=0\end{eqnarray} \tag{ 22 }$$

In Eq. 22, a significant ${\rm{\nabla }}{{\rm{\Phi }}}_{L}$ is usually present upon (dis)charging at relatively high rates due to the low ionic conductivity of the electrolyte, but ${\rm{\nabla }}{U}_{eq}\,\approx$ 0 for moving-zone-reaction-type materials. Replacing ${\rm{\nabla }}{{\rm{\Phi }}}_{S}$ and ${\rm{\nabla }}{{\rm{\Phi }}}_{L}$ with the current densities in the solid phase (I₁) and electrolyte (I₂), respectively, Eq. 22 becomes:

$$\begin{eqnarray}&&\displaystyle \frac{{I}_{2}}{{\kappa }_{eff}}-\displaystyle \frac{{I}_{1}}{{\sigma }_{eff}}=0\end{eqnarray} \tag{ 23 }$$

In the presence of a uniform flux, both ${I}_{1}$ and ${I}_{2}$ vary linearly with the distance to the current collector X: ${I}_{1}=I\left({L}_{cat}-X\right)/{L}_{cat}$ and ${I}_{2}=IX/{L}_{cat}.$ Therefore, Eq. 23 is satisfied if the following relation between ${\sigma }_{eff}$ and ${\kappa }_{eff}$ holds:

$$\begin{eqnarray}&&{\sigma }_{eff}=\displaystyle \frac{{L}_{cat}-X}{X}{\kappa }_{eff}\end{eqnarray} \tag{ 24 }$$

According to Eq. 24, the optimal ${\sigma }_{eff}$ is a hyperbolic function and varies monotonically from infinity at the current collector (X = 0) to 0 at the separator (X = ${L}_{cat}$). We confirm the effectiveness of such conductivity distribution via P2D simulation, in which ${\sigma }_{eff}$ is set as ${\kappa }_{0}{{\epsilon }}^{1.5}\left({L}_{cat}-X\right)/X$ and other parameters are the same as those for Fig. 5a. As shown in Figs. 5b and 5d, lithium intercalation indeed has a much more uniform distribution across the electrode throughout the discharge process than in systems with constant ${\sigma }_{eff}.$

We note that Palko et al.²⁰ 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 ${\sigma }_{eff}$ expressions in Ref. 20 and here demonstrates the analogy in the behavior of moving-zone-reaction-type battery electrodes and capacitors. On the other hand, uniform-reaction-type electrodes behave in a very different way and Eq. 22 highlights such difference. The ability of uniform-reaction-type compounds to sustain a non-zero spatial gradient in ${U}_{eq}$ in porous electrodes makes it possible to offset ${\rm{\nabla }}{{\rm{\Phi }}}_{L}$ with ${\rm{\nabla }}{U}_{eq}$ to maintain a uniform overpotential without the need for spatially varied ${\sigma }_{eff}.$ In the absence of a non-zero ${\rm{\nabla }}{U}_{eq}$ in moving-zone-reaction-type electrodes, however, ${\rm{\nabla }}{{\rm{\Phi }}}_{L}$ can only be balanced by the ${{\rm{\Phi }}}_{s}$ gradient to satisfy Eq. 22.

Experimentally, the electronic conductivity of porous electrodes can vary several orders of magnitude by adjusting the types and amount of conductive additives^21–24 or applying coatings to active materials to either increase or decrease the conductivity. Electrodes with graded electronic conductivity may be prepared via layer-by-layer deposition processes. In Ref. 24, Zhang et al. fabricated layer-graded electrodes consisting of TiO₂(B) and reduced graphene oxide (RGO) and varied the RGO:TiO₂(B) ratio to control ${\sigma }_{eff}$ in each layer. They report that graded TiO₂(B)/RGO electrodes with the high ${\sigma }_{eff}$ layer placed adjacent to the current collector deliver more than 70% capacity at 20 C than uniform electrodes with the same average RGO weight fraction. The theoretical analysis presented here explains why such an approach is effective. In recent years, using heterogeneous architecture to enhance the rate performance of thick electrodes has been explored theoretically^25,26 and experimentally.^27–32 These studies focus on improving ionic transport in electrolyte by decreasing the electrode tortuosity or introducing graded porosity in electrodes. In contrast, the approach we consider here enhances the reaction uniformity by tailoring the electronic transport in the solid phase. The two different strategies could potentially be combined in the same electrode structures to achieve synergistic improvement to the battery rate performance.

Reduce surface reaction rate

The circuit model presented above shows that during discharge a battery cell first goes through a transient period with a characteristic time t_c 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 moving-zone-reaction-type electrodes is to increase t_c 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 t_c can be increased by reducing i₀. We demonstrate this approach via P2D simulations of the model system with ${\rm{\Delta }}{U}_{eq}$ = 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 ${\tilde{c}}_{S}\left(\tilde{X}\right)$ is plotted in Figs. 6a and 6b, respectively. Compared to Fig. 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 (1–3 C), Fig. 6c, which is similar to the effect of reducing ${\sigma }_{eff}$ 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. Like reducing ${\sigma }_{eff},$ having a too small k₀ will lead to inferior discharge capacity because the induced large surface overpotential diminishes the capacity utilization before the cut-off potential is reached. An optimal k₀ thus exists, which may be estimated by the criterion that the corresponding t_c be comparable to the discharge time. Experimentally, surface reaction kinetics may be tailored by "artificial" SEI such as ALD coatings of various inorganic compounds (e.g. Al₂O₃, TiO₂, ZrO₂^33–35), whose insulating nature could retard the intercalation process and cause higher surface polarization.

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While reducing ${\sigma }_{eff}$ 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. Figure 7a shows the cell potential curves from the simulations of discharging four types of 200 μm-thick model electrodes (${\rm{\Delta }}{U}_{eq}$ = 0.001 V) at 1 C: high ${\sigma }_{eff}$ (100 s m⁻¹, baseline), low and uniform ${\sigma }_{eff}$ (0.291 s m⁻¹), graded ${\sigma }_{eff}$ (Eq. 24), and low surface reaction rate k₀ [10^–13 mol·m⁻²·s⁻¹·(mol·m⁻³)^−1.5]. The electrodes with low and graded ${\sigma }_{eff}$ only see a small drop (∼0.02 V) in the discharge 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 Fig. 7b, the total energy loss in a half cell upon 1 C 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 ${\sigma }_{eff}$ 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.

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detailed description of the P2D model can be found in literature.^8–12 The governing equations implemented in the half cell simulations are summarized as follows.

The concentration $c$ and ionic current ${\bf{i}}$ in a binary electrolyte are given by

$$\begin{eqnarray}&&{{\epsilon }}_{p}\displaystyle \frac{\partial c}{\partial t}={\rm{\nabla }}\cdot \left(\displaystyle \frac{{{\epsilon }}_{p}}{{\tau }_{p}}{D}_{amb}{\rm{\nabla }}c\right)+{\rm{\nabla }}\cdot \left(\displaystyle \frac{\left(1-{t}_{+}\right){\bf{i}}}{F}\right)\end{eqnarray} \tag{ A·1 }$$

$$\begin{eqnarray}&&{\bf{i}}=-\displaystyle \frac{{{\epsilon }}_{p}}{{\tau }_{p}}\kappa \left(c\right){\rm{\nabla }}{{\rm{\Phi }}}_{L}-\displaystyle \frac{{{\epsilon }}_{p}}{{\tau }_{p}}\displaystyle \frac{RT\kappa \left(c\right)\left(2{t}_{+}-1\right)}{Fc}\left(1+\displaystyle \frac{\partial \,\mathrm{ln}\,{f}_{\pm }}{\partial \,\mathrm{ln}\,c}\right){\rm{\nabla }}c\end{eqnarray} \tag{ A·2 }$$

where the subscript $p=cat$ or $sep$ to represent cathode or separator. Let X = 0 be at the interface between the current collector and cathode. The boundary conditions are

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial c}{\partial X}\right|}_{X=0}=0\end{eqnarray} \tag{ A3·1 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\epsilon }}_{cat}}{{\tau }_{cat}}{D}_{amb}{\left.\displaystyle \frac{\partial c}{\partial X}\right|}_{X={L}_{cat}^{-}}=\displaystyle \frac{{{\epsilon }}_{sep}}{{\tau }_{sep}}{D}_{amb}{\left.\displaystyle \frac{\partial c}{\partial X}\right|}_{X={L}_{cat}^{+}}\end{eqnarray} \tag{ A3·2 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{{\epsilon }}_{sep}}{{\tau }_{sep}}{D}_{amb}{\left.\displaystyle \frac{\partial c}{\partial X}\right|}_{X={L}_{cat}+{L}_{sep}}=\displaystyle \frac{I\left(1-{t}_{+}\right)}{F}\end{eqnarray} \tag{ A3·3 }$$

$$\begin{eqnarray}&&{\left.{\bf{i}}\right|}_{X=0}=0\end{eqnarray} \tag{ A4·1 }$$

$$\begin{eqnarray}&&{\left.{\bf{i}}\right|}_{X={L}_{cat}^{-}}={\left.{\bf{i}}\right|}_{X={L}_{cat}^{+}}\end{eqnarray} \tag{ A4·2 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\left.{\bf{i}}\right|}_{X={L}_{cat}+{L}_{sep}}={i}_{0}^{Li}\left[\exp \left(\displaystyle \frac{\alpha F\left({{\rm{\Phi }}}_{L}-{{\rm{\Phi }}}_{{\rm{Li}}}\right)}{RT}\right)\right.\\ \left.\,\,\,\,\,-\exp \left(-\displaystyle \frac{\left(1-\alpha \right)F\left({{\rm{\Phi }}}_{L}-{{\rm{\Phi }}}_{{\rm{Li}}}\right)}{RT}\right)\right]=-I\end{array}\end{eqnarray} \tag{ A4·3 }$$

where superscripts (+ and −) denote the right and left side of the cathode/separator interface, respectively, and the potential of Li anode ${{\rm{\Phi }}}_{Li}$ is fixed at 0. The electronic current in solid electrode matrix ${{\bf{i}}}_{{\bf{S}}}$ is

$$\begin{eqnarray}&&{{\bf{i}}}_{{\bf{S}}}=-{\sigma }_{eff}{\rm{\nabla }}{{\rm{\Phi }}}_{S}\end{eqnarray} \tag{ A·5 }$$

with the boundary conditions

$$\begin{eqnarray}&&{\left.{{\bf{i}}}_{{\boldsymbol{S}}}\right|}_{X=0}=-I\end{eqnarray} \tag{ A6·1 }$$

$$\begin{eqnarray}&&{\left.{{\bf{i}}}_{{\boldsymbol{S}}}\right|}_{X={L}_{cat}}=0\end{eqnarray} \tag{ A6·2 }$$

The ionic and electronic currents are coupled by surface reaction as

$$\begin{eqnarray}&&{\rm{\nabla }}\cdot {\bf{i}}=-{\rm{\nabla }}\cdot {{\bf{i}}}_{{\bf{S}}}=-Fa{j}_{in}\end{eqnarray} \tag{ A·7 }$$

where $a=3\left(1-{{\epsilon }}_{cat}\right)/{r}_{cat}$ in cathode and reaction flux density ${j}_{in}$ follows the Butler-Volmer equation (Eq. 1), in which ${i}_{0}$ is given by

$$\begin{eqnarray}&&{i}_{0}=F{k}_{0}{c}^{1-\alpha }{c}_{S}^{\alpha }{\left({c}_{S,{\rm{\max }}}-{c}_{S}\right)}^{\alpha }\end{eqnarray} \tag{ A·8 }$$

In the P2D simulations of NMC and LFP half cells, lithium diffusion in active materials is simplified as a radial diffusion in spherical particles as

$$\begin{eqnarray}&&\displaystyle \frac{\partial {c}_{S}}{\partial t}=\displaystyle \frac{1}{{r}^{2}}\displaystyle \frac{\partial }{\partial r}\left({r}^{2}{D}_{S}\displaystyle \frac{\partial {c}_{S}}{\partial r}\right)\end{eqnarray} \tag{ A·9 }$$

where the boundary conditions are

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial {c}_{S}}{\partial r}\right|}_{r=0}=0\end{eqnarray} \tag{ A10·1 }$$

$$\begin{eqnarray}&&{\left.Ds\left(\displaystyle \frac{\partial {c}_{S}}{\partial r}\right)\right|}_{r={r}_{cat}}={j}_{in}\end{eqnarray} \tag{ A10·2 }$$

For the model electrode material, Li diffusion is assumed to be very facile so that ${c}_{s}$ is constant within each electrode particle. All of the simulations are implemented in COMSOL Multiphysics® 5.3a.

Table A·I.  Parameters used in P2D simulations (unless otherwise stated).

Parameter	Symbol	Value
Electrode properties
		NMC	LFP	Model cathode material
Cathode particle radius (μm)	${r}_{cat}$	1	0.1	0.1
Cathode porosity	${{\epsilon }}_{cat}$	0.25	0.25	0.25
Separator thickness (μm)	${L}_{sep}$	25
Separator porosity	${{\epsilon }}_{sep}$	0.55
Tortuosity	$\tau$	$\tau ={{\epsilon }}^{-0.5}$
Maximum Li concentration in active materials (mol·m−3)	${c}_{S,{\rm{\max }}}$	49761	22806	20000
Initial concentration in active materials (mol·m−3)	${c}_{S,0}$	22392	228	200
Li diffusivity in active materials (m2·s−1)	${D}_{S}$	10–1436	10–1637	—
Effective electrode conductivity (S·m−1)	${\sigma }_{eff}$	1038	10a)	100
Reaction rate constant (mol·m−2·s−1·(mol·m−3)−1.5)	${k}_{0}$	3·10–1138	3·10–11a)	10−8
Charge transfer coefficient	$\alpha$	0.5
Exchange current density of Li anode (A·m−2)	${i}_{0}^{Li}$	2038
Equilibrium potential (V)	${U}_{eq}$	See noteb)
Electrolyte (1 M LiPF6 in EC/DMC 50:50 wt%) properties
Initial salt concentration (mol·m−3)	${c}_{0}$	1000
Transference number of cations	${t}_{+}$	0.3939
Ambipolar diffusivity (m2·s−1)	${D}_{amb}$	2.95·10–1039
Concentration-dependent ionic conductivity (S·m−1)	$\kappa \left(c\right)$	0.00233 $c$ c)

^a)Assumed. ^b)The equilibrium potential profiles of NMC and LFP are extracted from Fig. 2 in Ref. 40, and Fig. 2 in Ref. 41. The equilibrium potential of model cathode material is defined by Eq. 4. ^c)Calculated by $\kappa ={F}^{2}{D}_{amb}c/\left(2RT{t}_{+}\left(1-{t}_{+}\right)\right).$