Probing the Role of Electrode Microstructure in the Lithium-Ion Battery Thermal Behavior

Active material particles are considered to have spherical geometry and Fickian diffusion is used for solving lithium transport inside the particles.

Diffusion of lithium ions inside the electrolyte is also modeled using Fick's law with D^eff_e giving the effective diffusion rate of lithium ions in the electrolyte phase. Volumetric current production/destruction is given by j.

Electric potential in the solid phase ϕ_s can be evaluated using Ohm's law.

Charge transport in the electrolyte phase is driven by gradients of electrolyte potential, ϕ_e and ionic concentration, c_e.

Temperature evolution in a typical Li-ion cell sandwich can be studied based on a lumped thermal model under typical external cooling conditions owing to low Biot number characteristics.

The heat generation rate consists of reaction/irreversible heat, reversible/entropic heat and joule/ohmic heat. Joule heating is the predominant component with contributions from electronic resistance (σ^eff_s∇ϕ_s · ∇ϕ_s), ionic resistance under potential gradient, (κ^eff∇ϕ_e · ∇ϕ_e) and concentration gradient, (κ^eff_D∇ln c_e · ∇ϕ_e). Reaction heat is given by(j(ϕ_s − ϕ_e − U_eq) and entropic heat is given by . The total rate is calculated by integrating the source terms over the entire length of the cell and multiplying with the electrode area, A. The open circuit potential, U_eq and entropic coefficient, dU_eq/dT for both electrodes are provided in Table II. The electrolyte properties, listed in Table III, are adopted from Valoen and Reimers,³¹ which show functional dependence on Li⁺ concentration in the electrolyte and cell temperature. The boundary conditions for the electrochemical-thermal model are given in Table IV. A Dirichlet BC is prescribed at the cathode – current collector interface for the electrolyte phase potential, ϕ_e to enable the solution of the Poisson charge conservation equations which otherwise have all Neumann BCs. The initial conditions correspond to the SOC values in each electrode at the start of discharge.

Table II. Electrode Properties.

	Values
Parameters	Anode	Cathode
k (m2.5 mol−0.5 s−1)	2.3238 × 10− 10	2.4177 × 10− 11
csmax (mol/m3)	30900	49500
Eact (i0) (J/mol)	68000	50000
Ds (m2/s)	1.6 × 10− 14	3.0 × 10− 14
Eact(Ds)(J/mol)	30000	30000
Ueq(V)
(mV/K)
Rc (Ohm-m2)	6 × 10− 4

Table III. Electrolyte Properties.

Parameters	Value
De(m2/s)
κ(S/m)
V(−)	0.601 − 0.24(10− 3c)0.5
	+ 0.982(1 − 0.0052(T − 294))(10− 3c)1.5
t+(−)	0.363

Table IV. Boundary Conditions.

Parameters	CC - anode x = 0	Anode–separator x = La	Separator-cathode x = La + Ls	Cathode-CC x = La + Ls + Lc
ce
ϕe
ϕs

In this work, several realizations of 3D electrode microstructures are stochastically reconstructed with varying composition of the active material, conductive additive, and binder.^19,21,32,33 Active material, representative of nearly spherical shape of NMC (Ni-Mn-Co oxide) based cathode active particles, are randomly distributed with the desired volume %. The particles are prohibited from overlapping when the vol. % of the active material is smaller than 50 vol. %. The conductive additives are represented as platelets and also randomly distributed. The conductive additives are prohibited to overlap with active materials, but allowed to overlap with each other. The binder is randomly added to a structure in the shape of a concave meniscus in locations where material surfaces get close together. In the cathode, such as (Li(Ni_1/3Mn_1/3Co_1/3)O₂) cathode in this work, the wt % of the active material carbon-based conductive additive, and binder (PVDF) is maintained at 90:5:5. In the anode, wt % of the active material and the binder is maintained at 95:5. Figure 1a shows representative cathode microstructures with different active material loadings based on stochastic reconstruction.

Effective electrode properties are estimated based on direct numerical simulation (DNS) calculations on reconstructed electrode microstructures. The electrolyte transport properties depend on the porosity and tortuosity of the pore space. Similarly, the presence of multiple components in the solid phase (AM+CA+B) necessitates estimation of the effective electronic conductivity. The DNS calculations involve the solution of diffusion equation on 3-D electrode microstructures with Dirichlet boundary conditions along the transport direction and periodic boundary conditions span-wise directions. Stochastic electrode microstructure reconstruction and effective transport property calculations were performed using GeoDict.³⁴

It is worth mentioning that the diffusion mechanism is dependent on the Knudsen number, which is characterized by the pore size relative to the mean free path of the diffusing ions. In the case of typical porous electrodes for Li-ion batteries, the Knudsen number is very small (Kn<<1), which means that the pore diameters are large compared to the mean free path length of the diffusing ions. Thus, diffusion can be modeled based on the Laplace equation,

along with the following boundary conditions.

Periodic boundary conditions are applied on the remaining four faces for the diffusivity calculations. At the electrode-electrolyte interface, is satisfied. The effective diffusivity can now be computed by equating species flux obtained from the DNS calculation and effective diffusivity based flux formulation:

Similar diffusivity calculations are performed in y, and z directions which are then used to determine the tortuosity in the principal directions using following relation^25,35

where D is the intrinsic diffusivity of void phase, ɛ is the porosity (i.e. volume fraction of pore space) of the microstructure and τ is the tortuosity of the microstructure. The arithmetic mean of the tortuosity calculated in the individual direction is used to estimate the tortuosity value, τ = (τ_x + τ_y + τ_z)/3.0 which is then used for computation of effective ionic diffusivity, ionic conductivity and diffusional conductivity for input into the electrochemical performance model.

Similar to the evaluation of effective diffusivity, the effective electrical conductivity can also be calculated by solving the Laplace's equation for electrical potential ∇.(σ∇ϕ) = 0 and Eq. 21.

The effective electrical conductivity is calculated according to the material electrical conductivity σ_j provided in Table V. The calculated effective transport properties (i.e. tortuosity, electrical conductivity) of different electrode microstructures and the corresponding theoretical capacity values are furnished in Table VI. One should note that, in this study, we focus on the influence of cathode microstructures on the Li-ion cell thermal behavior. Therefore, the anode microstructure is kept fixed for all the simulations.

The macrohomogeneous thermo-electrochemical model is validated for varying C-rates and ambient temperature (25°C) operation with the experimental data for a Li-ion cell, with 2.2 Ah nominal capacity, from Ji and Wang et al.¹² (see, Figure 2a). The simulation results e.g., discharge performance and average temperature rise at 1 C and 3 C, show good agreement with the experimental data. The effect of tortuosity is further investigated. In the macrohomogeneous, porous electrode theory, the tortuosity is calculated based on the idealized Bruggeman relation (Eq. 22).

However, the Bruggeman relation is based on the assumption that the particles are spherical and uniformly distributed.²⁴ Since typical Li-ion battery electrodes include multiple phases (e.g., active particle, conductive additive, binder) and dense solid phase packing, the Bruggeman relation is not sufficient to represent the pore space tortuosity.³⁰ Figure 2b shows the comparison of the tortuosity estimation based on stochastically reconstructed 3-D electrode microstructures and the idealized Bruggeman relation. The larger deviation in the tortuosity estimation with decreasing porosity will further affect the discharge performance, heat generation, and temperature evolution as shown in Figures 2c, 2d, and 2e, respectively. An increase in pore space tortuosity also amplified the internal resistance, which increases voltage drop, joule heating rate, and consequently the cell temperature rise. It is important to note that tortuosity significantly affects joule heating rate, while its influence on reaction and reversible heat rates is quite negligible.

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The Li-ion cell temperature needs to be maintained below a safety limit (e.g. < 80°C) with as little compromise on the achievable capacity as possible. The objective, therefore, is to study the existence of a potential electrode design window within which the capacity can be maximized while maintaining the maximum temperature rise within the safety limit. In this regard, the spatio-temporal variations of different heat sources for varying cathode microstructures are analyzed based on the cell sandwich model. The trade-off between achievable capacity and maximum temperature rise is investigated in order to develop a thermo-electrochemical safety window based on the electrode microstructure analysis.

Figure 3 shows the influence of microstructure on the cell capacity, and temperature evolution. Figure 3a shows the reconstructed 3-D cathode microstructures with varying active material loading. Since the wt % of active material, carbon additive, and binder is maintained at 90:5:5, the amount of binder and carbon additives increases with increase in the amount of active material. Consequently, the porosity of the electrode decreases, which results in an increase in tortuosity as shown in Figure 2a. Figure 3b shows the improvement of capacity when the amount of active material is increased. The corresponding 2-D microstructural slices are also included in Figure 3b. The temporal evolution of temperature is shown in Figure 3c. It can be observed that the temperature increases faster in the cell with low porosity cathode. This can be further explained by analyzing the effect of porosity, and hence tortuosity, variation on the heat generation source terms which is presented in the following section.

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For the sake of completeness, the terminology and salient parameters used in the simulations are reviewed briefly. The state of charge (SOC) has been defined both at the cell level and individual electrode level for better explaining the electrode microstructure implications on the cell capacity and temperature rise. At the cell level, SOC is defined as the ratio of the discharge capacity at a particular C-rate to the discharge capacity obtained at 1 C and 25°C. Depth of discharge (DOD) is defined as (1 – SOC) at the cell level. The electrode level SOC is defined as the average fractional occupancy of lithium in the electrode. The anode SOC can range from 0 to 1 as graphite shows fully reversible intercalation/deintercalation over the entire SOC range. The Li(Ni_1/3Mn_1/3Co_1/3)O₂ cathode shows reversible intercalation/deintercalation in the SOC range of 0.3–1. At the beginning of discharge, the anode SOC is fixed at 0.95 and the cathode SOC is close to 0.3 which gives an initial voltage close to 4.2 V. The discharge simulations are carried out till the voltage goes below 3 V (or the cathode is fully lithiated).

Joule/ohmic heat

Figures 4a and 5a–5c show the temporal and spatial distributions of joule heat rates, respectively for different electrode microstructural variation. From Figure 4a, it can beconcluded that the lowest porosity cathode (ε = 0.2) is associated with the highest Joule heat rate. Lower porosity results in higher tortuosity of the cathode, which impedes Li⁺ ion transport in the electrolyte, thereby increasing the charge transport resistance. This high internal resistance increases the polarization of Li⁺ concentration and electrolyte potential. Consequently, according to Eq. 13, steep concentration and potential gradients ensue, leading to increased ohmic heat. Figures 5a–5c show the spatial distribution of the Joule heating rates for different cathode microstructures with varying porosities. It is evident that the maximum ohmic heating rate occurs in the vicinity of the cathode-separator interface. This is due to the existence of higher concentration and potential gradients close to the separator-electrode interface. Since the anode microstructure is unchanged for all the scenarios studied here, the heating rate inside the anode shows little variation, except near the end of discharge. It can be concluded that higher concentration and potential gradients in the region near cathode-electrolyte interface contribute significantly to the Joule heating rate in a Li-ion cell with a low-porosity cathode. It is also worth noticing that, toward the end of discharge, the joule heating rate as well as the extent of the heating region in the anode diminishes for such a low-porosity cathode configuration. This phenomenon can be explained by the temperature dependence of electrolyte transport properties. The steeper temperature rise accompanying the low porosity cathode improves the ionic conductivity and diffusivity. Hence, the electrolyte concentration and potential gradients become less steep in the anode (with a fixed porosity), thereby leading to lower ohmic heat rate in anode as the cathode porosity decreases from 0.4 to 0.2. This temperature increase during discharge also alters the heating zone in the cathode. However, in the cathode, there are two competing factors, decreased porosity leading to higher tortuosity, and higher temperature leading to improved transport properties. The tortuosity increase emerges as the dominant factor and the deleterious effect thereof cannot be compensated by the betterment of the electrolyte transport properties. Consequently, higher ohmic heat rates can be observed with lower porosity in the cathode during the discharge. At the end of discharge, the disparity in the heat rates for the three different porosity cathodes becomes smaller which can be attributed to the temperature rise, which can improve electrolyte transport properties, as compared to the negative effect from enhanced tortuosity for lower porosity cathode microstructure.

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Reaction/irreversible heat

Reversible heat

Total heat rate

Since the heat generation rate is also affected by the discharge rate and ambient temperature, analyzing the influence of electrode microstructure on the cell thermal behavior under different operation conditions is important. Figure 6 presents the maximum temperature and cell capacity for varying cathode porosities during discharge for different C-rates. Figures 6a, 6c show the maximum temperature and capacity when the cell operates at ambient temperature of 25°C, and Figs. 6b, 6d show the show the same for cell operation at an ambient temperature of 40°C. In Figs. 6a and 6b, based on the maximum temperature, there distinct regimes can be identified. The "Unsafe" region refers to T >80°C, where the cell runs high risk of going into potential thermal runaway. The region of "Potential Risk" refers to a temperature range of 60°C <T <80°C, where the cell has the potential for being driven into thermal runaway if a sudden heat generation rate spike is observed,for example due to an external short. The "Safe" region refers to T < 60°C, where the cell can operate within a thermally safe window. Figure 6a suggests that when the discharge current is below 7A, the maximum cell temperature rise within a cathode porosity range of ɛ ∼ 0.4–0.18 can remain outside the unsafe zone. However, the cell can reach the unsafe temperature region, for the same cathode porosity range, when the discharge current is higher than 7A. An interesting thermo-electrochemical behavior can be seen in the low porosity region. The maximum temperature reached in this region does not cross the unsafe barrier even at higher C-rates. The lower maximum temperature for the cell with cathode porosity of ɛ ∼0.18–0.20 is due to the self-shutdown of the cell. At high discharge rate, because of large internal resistance of a low-porosity cathode, the cell shuts down before the full discharge in the operating window of 3.0 - 4.2 V. Figure 6c also shows the capacity lost due to the self-shut down for a low-porosity cathode. Under high discharge currents (e.g., I > 5A), there is significant capacity lost for a low-porosity electrode compared to the capacity obtained under low discharge current. Due to the self-shutdown, although the heat generation rate is large, the reduction in time for temperature rise decreases the maximum temperature obtained for a low-porosity cathode. The dashed lines in Figs. 6c and 6d correspond to the current at a fixed C-rate value for different porosity cathodes. As the porosity decreases, the nominal cathode capacity increases, hence an increase in the current corresponding to a particular C-rate can be observed with lower porosity. For porosity greater than 0.35, for 1 C operation, the current goes below 2A as evident by the dashed line intersecting the porosity axis.

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From Fig. 6b, it is evident that the unsafe region expands when the cell operates at a higher ambient temperatures (e.g., T_amb = 40°C). With the exception of the cathode porosity of 40%, cells with other cathode porosities run the risk of reaching the unsafe temperatures, and hence the maximum allowable discharge current decreases with decrease in the cathode porosity. Similar to Fig. 6a, for the cathode porosity of ∼18%, if the discharge current is larger than 7A, the induced self-shutdown limits the maximum temperature below 80°C. The self-shutdown for T_amb = 40°C also can be observed from the capacity contour plot shown in Fig. 6d. It is worth noticing that a loss in the achievable capacity happens at higher discharge current and lower porosity cathode when T_amb = 40°C as compared to that for T_amb = 25°C. This is because an increase in the ambient temperature enhances the ionic conductivity and diffusivity, which further decreases the internal resistance. The decrease in internal resistance shrinks the range of discharge current and porosity, which can trigger self-shutdown.

From Figure 6, it can be concluded that the maximum cell temperature rise is linked to electrode porosity and discharge current. To limit the maximum temperature of the cell, two routes may be adopted. One way is external shut down, which shuts down the cell when it reaches the maximum temperature limit with external control or mechanism. However, with this shutdown mechanism, the cells are forced to stop before full discharge as shown in Fig. 7a. Figure 7a shows an example of a scenario if we want to limit the temperature below 60°C. Figure 7b presents the obtained capacities if the temperature is limited to 60°C and 70°C, respectively. If there is no temperature control, the cell with cathode porosity of 20% has the highest capacity when the discharge current is 6A. The decrease in capacity for a cathode porosity of ∼18% is due to the self-shutdown. When the temperature is limited to 70°C, a cell with the cathode porosity of 20% can no longer provide the maximum capacity, while a cell with the cathode porosity of 25% becomes the best case scenario. If we further limit the temperature at 60°C, the cathode porosity, which can provide the maximum capacity, shifts to 35%.

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An alternate way to control the cell temperature is by controlling the electrode (cathode) microstructure. From Figs. 6a and 6b, it can be concluded that by controlling the porosity (i.e. amount of active material), the maximum temperature can be controlled. To keep the maximum temperature away from the unsafe region, shown in Figs. 6a and 6b, the cathode porosity can be an internal control, for example increase the cathode porosity slightly to lower the temperature rise or decrease the porosity to a threshold value (such as, ∼20%) to induce self-shutdown of the cell. Figures 8a and 8b expand on this point further and show the optimum window of cathode microstructure design when the discharge current is 8A. In Figure 8a, if the desired temperature safety limit is 80°C, the cell should have cathode porosity larger than 26% or smaller than 18%. Similarly, if the desired temperature limit is 70°C, the cell should have cathode porosity larger than 36% or smaller than 16%. The lower the desired maximum temperature limit, the lesser the available options are for the cathode microstructure design. Similar to the external temperature control mechanism, when the electrode microstructure is designed to control the temperature, there is some variation in the obtained cell capacity as well. The corresponding capacity that the designed electrode microstructure can obtain is also shown in Fig. 8a. As is evident, the choice of increasing porosity (i.e. decreasing the amount of active material) seems to be a preferred route as it can result in a higher capacity of the cell (e.g. 2.2 Ah for 26% porosity as compared to 1.7 Ah for 18% porosity for T_max = 80°C). A similar analysis was also performed for obtaining the optimum microstructure design window when the ambient temperature is 40°C, as shown in Fig. 8b. The design window of the cathode microstructure shrinks further for a cell operating at T_amb = 25°C, as compared to 40°C operation.

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This study highlights that for cell temperature control via electrode microstructure, porosity increase is more beneficial especially for operation at high C-rates. However, this is only true if the high discharge current is "intentional". There can be extenuating circumstances, where the high current is "not intentional", such as the failure of the control system and external short. To account for these conditions, decreasing the porosity to such limits so as to initiate self shutdown is a better choice. As shown in Fig. 8a, for the discharge current range of 0A–10A, both low-porosity electrodes (ɛ = 0.15–0.18) and high-porosity electrodes (ɛ = 0.32–0.4) maintain the maximum temperature below 80°C. An additional advantage of using the low-porosity electrode is the higher capacity under low C-rate discharge. Generally, for normal operation the load on the cell draws less than 1C current, and consequently low porosity electrode (ɛ = 0.15–0.18) will allow higher capacity to achieve due to the larger amount of active material without an appreciable temperature rise. Therefore, low-porosity electrodes (ɛ = 0.15–0.18) have the advantage of: (i) high capacity under low discharge current, (ii) self shutdown mechanism to prevent the temperature rise to an extreme and avoid thermal runaway at high discharge currents. Such cells can be useful for constant load operations where the current drawn from the cell is not very high (< 2C) and does not show significant fluctuations. The usage of these low-porosity electrodes for electric vehicle batteries will, however, be a bit tricky as vehicular operation involves C-rate swings, which may sometime require currents in excess of 5C. It can be inferred that that a battery pack consisting of both low- and high-porosity (electrode) cells may be a promising option for electric vehicle (EV) batteries. A battery management system devised such that low currents are drawn from the low porosity cells and high current loads invoke the usage of higher porosity cells will go a long way into increasing the capacity as well as power capability of EV batteries without allowing the temperature to exceed the critical (safety) limit. This study points to the existence of an optimum electrode microstructure design, based on the cell operation conditions and electrode microstructure dependent thermal behavior, to ensure the safety of a Li-ion cell.