The development of sustainable energy storage technologies is paramount for the future of electric mobility and renewable energy integration. Within this landscape, the lithium-ion battery remains the dominant technology due to its high energy density and established manufacturing base. However, the reliance on cobalt (Co) in prevalent cathode chemistries like Nickel-Cobalt-Manganese (NCM) oxides presents significant challenges, including price volatility, geopolitical supply chain risks, and ethical sourcing concerns. Transitioning to cobalt-free cathode materials, particularly high-nickel layered oxides, is therefore a critical research frontier aimed at reducing costs while maintaining, or even enhancing, the energy density of lithium-ion batteries.
Cobalt plays a specific role in traditional NCM cathodes; it is known to mitigate cation mixing (where Ni2+ ions occupy Li+ sites) and stabilize the layered structure during cycling, especially at high states of charge. Its removal, while economically and ethically attractive, introduces technical hurdles. Cobalt-free LiNixMn1-xO2 (LNMO) cathodes often exhibit inferior rate capability and faster capacity fade, particularly at low temperatures. These performance limitations are intrinsically linked to altered electrochemical kinetics and thermodynamics. Therefore, a deep understanding of the internal electrochemical processes governing cobalt-free lithium-ion battery performance across a wide operational temperature range (-5°C to 50°C) is essential for their optimization and reliable deployment.
Electrochemical models, particularly the pseudo-two-dimensional (P2D) model pioneered by Newman and colleagues, serve as a vital bridge between the internal physicochemical parameters of a lithium-ion battery and its external electrical behavior. Unlike equivalent circuit models (ECMs), the P2D model is physics-based, explicitly simulating lithium-ion diffusion in solid particles and electrolyte, charge transfer kinetics at electrode/electrolyte interfaces, and ionic/electronic conduction. This makes it an invaluable tool for analyzing performance limitations, diagnosing aging mechanisms, and designing advanced battery management systems (BMS). However, the model’s accuracy hinges on the correct identification of numerous, often interdependent, parameters. This task becomes exceptionally difficult for novel chemistries like cobalt-free cathodes and under wide temperature swings, where parameter values and their relative sensitivities change dramatically.
Experimental Performance Evaluation of Cobalt-Free Batteries
To establish a baseline, the performance of a commercial 3.35 Ah cobalt-free LiNi0.5Mn0.5O2/graphite pouch cell was characterized under various thermal and electrical stresses. Tests were conducted in an environmental chamber at temperatures of -5°C, 5°C, 25°C, and 45°C, with discharge rates ranging from 0.2C to 1C.
Analysis of Incremental Capacity (IC) curves, derived from low-rate open-circuit voltage (OCV) tests, reveals the phase transition behavior of the electrode materials. For the cobalt-free cathode, characteristic peaks associated with hexagonal (H1, H2, H3) and monoclinic (M) phase transitions are observed. A key finding is the significant dampening or shift of specific peaks (e.g., Peak A and Peak D) at low temperatures (-5°C). Peak A, related to the H2→H3 transition at high SOC, nearly disappears. This phase transition involves a contraction of the c-axis lattice parameter, inducing mechanical stress. At low temperatures, sluggish solid-state diffusion in the anode and increased polarization at the cathode exacerbate the kinetic limitations in this high-voltage region, effectively creating a “kinetically hindered zone.” This underscores the strong temperature dependence of the reaction and transport processes within this lithium-ion battery chemistry.
From the galvanostatic tests, key electrical performance metrics were extracted: capacity (Q), energy (E), Coulombic efficiency (CE), energy efficiency (EE), and direct current internal resistance (DCIR at 1s and 10s). Their dependencies are summarized below:
| Performance Metric | Rate Dependence | Temperature Dependence |
|---|---|---|
| Charge Capacity (Q) | High (++++), decreases with rate | Medium (+), increases with temperature |
| Charge Energy (E) | High (++++), decreases with rate | Medium (+), increases with temperature |
| Coulombic Efficiency (CE) | Low (+), ~99.5% stable | Medium (++) |
| Energy Efficiency (EE) | High (+++) | Medium (++) |
| 1s DCIR (RΩ) | Low (+) | Very High (++++) |
| 10s DCIR (RΩ) | Low (+) | Very High (++++) |
This analysis shows that capacity and energy are predominantly limited by high C-rates due to increased polarization, while internal resistance and efficiencies are more severely impacted by low temperatures. The stable CE suggests minimal irreversible side reactions under these test conditions, a positive sign for the cobalt-free chemistry’s coulombic reversibility.
P2D Model Fundamentals and Sensitivity Analysis Methodology
The P2D model conceptualizes the lithium-ion battery as a one-dimensional sandwich of porous positive electrode, separator, and porous negative electrode. Active material particles in each electrode are modeled as uniform spheres. The model couples several conservation laws and kinetic equations:
1. Solid-Phase Diffusion (Fick’s second law in spherical particles):
$$ \frac{\partial c_s}{\partial t} = \frac{D_s}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial c_s}{\partial r} \right) $$
where $$c_s$$ is the Li concentration in the solid particle and $$D_s$$ is the solid-phase diffusion coefficient.
2. Liquid-Phase Mass Conservation (in electrolyte):
$$ \epsilon_e \frac{\partial c_e}{\partial t} = \frac{\partial}{\partial x} \left( D_e^{eff} \frac{\partial c_e}{\partial x} \right) + \frac{1 – t_+^0}{F} a_s j $$
where $$c_e$$ is the electrolyte Li concentration, $$D_e^{eff}$$ is the effective electrolyte diffusivity, $$t_+^0$$ is the Li+ transference number, and $$j$$ is the pore wall flux.
3. Electrode Kinetics (Butler-Volmer equation):
$$ j = i_0 \left[ \exp\left(\frac{\alpha_a F \eta}{RT}\right) – \exp\left(-\frac{\alpha_c F \eta}{RT}\right) \right] $$
$$ i_0 = k (c_e)^\alpha (c_{s,max} – c_{s,surf})^\alpha (c_{s,surf})^\alpha $$
where $$i_0$$ is the exchange current density, $$k$$ is the reaction rate constant, and $$\eta$$ is the surface overpotential.
4. Charge Conservation (Ohm’s law in solid and electrolyte phases).
The terminal voltage $$V_{cell}$$ is calculated as:
$$ V_{cell} = \phi_s|_{x=L} – \phi_s|_{x=0} – R_{SEI} \cdot I $$
where $$\phi_s$$ is the solid-phase potential and $$R_{SEI}$$ represents the impedance of the Solid Electrolyte Interphase layer on the anode.
The model contains over 20 parameters, categorized as structural (e.g., electrode thickness $$L$$, particle radius $$R_p$$, $$R_n$$, volume fractions $$\epsilon_s$$, $$\epsilon_e$$), thermodynamic (e.g., maximum solid concentration $$c_{s,max}$$, initial lithiation $$SOC_{ini}$$, OCV curves), and kinetic (e.g., $$D_s$$, $$k$$, $$D_e$$, $$R_{SEI}$$). Identifying these parameters for a new cell chemistry is non-trivial. To streamline this process and understand which parameters most critically affect model output under different conditions, a multi-dimensional sensitivity analysis is indispensable.
We employ a One-at-a-Time (OAT) local sensitivity analysis across a matrix of conditions: Temperature T = {-5, 25, 50}°C, C-rate = {0.2, 0.5, 0.8, 1}C, and SOC interval = {0-10%, 10-40%, 40-60%, 60-80%, 80-100%}. For each parameter $$b_i$$, its sensitivity index SA for a given condition is calculated as the normalized root-mean-square error (RMSE) between the voltage output of the baseline model (with nominal parameters) and the model where only $$b_i$$ is perturbed within its plausible physical range.
$$ SA(SOC, C_{rate}, T, b_i) = \frac{1}{n} \sqrt{ \sum_{i=1}^{n} \left( V_{model}(b_i) – V_{model}(\bar{b}) \right)^2 } $$
This analysis reveals that the sensitivity of parameters is highly dependent on the operating condition. Key findings are:
| Parameter Category | High Sensitivity Parameters | Primary Dependence |
|---|---|---|
| Kinetic | Reaction rate constant (kp, kn) | Strongly on C-rate and SOC |
| Transport | Solid diffusion coeff. (Ds,p, Ds,n) | Strongly on SOC, moderately on C-rate |
| Liquid diffusion coeff. (De) | Strongly on C-rate | |
| Structural | Particle radius (Rp, Rn) | Significant across all conditions |
A critical, overarching result is the dramatic increase in sensitivity for almost all parameters at low temperatures (-5°C). This “sensitivity explosion” explains why conventional parameter identification routines, often calibrated at room temperature, fail for low-temperature modeling. The model output becomes excessively sensitive to minute changes in many parameters simultaneously, making unique identification from limited voltage data nearly impossible.
Wide-Temperature-Range Modeling and Parameter Identification Strategy
The sensitivity analysis dictates a pragmatic, tiered parameter identification strategy:
1. Structural and Thermodynamic Parameters: Structural parameters are obtained from direct measurement (e.g., SEM for particle size) or cell specifications. Thermodynamic parameters ($$c_{s,max}$$, $$SOC_{ini}$$, OCV curves) are obtained by fitting half-cell or full-cell OCV data at multiple temperatures using methods like Particle Swarm Optimization (PSO). These parameters show clear temperature trends, as fitted below:
$$ c_{s,max}(T) = A – B \cdot T $$
$$ SOC_{ini}(T) = C + D \cdot T $$
2. Kinetic Parameters at Elevated Temperatures (T ≥ 15°C): For temperatures where sensitivity is manageable, a multi-stage experimental identification is feasible. Fast dynamics parameters (e.g., $$k$$, $$R_{SEI}$$) are identified from short current pulse responses. Slow dynamics parameters (e.g., $$D_s$$) are identified from long relaxation or low-rate discharge data.
3. Extrapolation to Low Temperatures using Temperature-Dependent Functions: Due to the identification challenge at low T, we utilize the successfully identified kinetic parameters from higher temperatures (15°C, 25°C, 35°C, 50°C) to establish their temperature dependence. The Arrhenius equation is used for thermally activated processes:
$$ \theta(T) = \theta_{ref} \cdot \exp \left[ \frac{E_a}{R} \left( \frac{1}{T_{ref}} – \frac{1}{T} \right) \right] $$
where $$\theta$$ represents parameters like $$k$$ and $$D_s$$, $$E_a$$ is the activation energy, and $$R$$ is the gas constant. The electrolyte diffusion coefficient $$D_e$$ often follows a linear or power-law relationship with temperature. Fitted equations for a cobalt-free cell are:
| Parameter | Temperature Correlation | R² |
|---|---|---|
| kn, kp | Arrhenius Eqn. (Ea,k ~ 12-16 kJ/mol) | 0.86 – 0.96 |
| Ds,n, Ds,p | Arrhenius Eqn. (Ea,D ~ 33 kJ/mol) | 0.83 – 0.86 |
| De | Linear: De = a + b·T | 0.95 |
This approach allows for the calculation of credible low-temperature parameter values without direct, futile identification attempts at those conditions.
The finalized P2D model, incorporating temperature-dependent parameters, is implemented and solved using the finite element method in COMSOL Multiphysics®. Its validation across a wide temperature range (-5°C to 50°C) and multiple C-rates shows significant improvement over a model using room-temperature parameters exclusively. The voltage prediction RMSE remains within 20 mV across the entire range, a marked achievement for modeling a cobalt-free lithium-ion battery under such varied conditions.
Linking Electrochemical Parameters to Macroscopic Performance
With a validated model, we can now probe the causal relationship between internal electrochemical parameters and the macroscopic electrical performance metrics. By varying high-sensitivity kinetic parameters ($$k$$, $$D_s$$, $$D_e$$) within ±40% of their nominal values and simulating constant-current discharges at different rates, we can quantify their influence.
The results indicate a clear decoupling of influence:
Diffusion-Limited Performance (Capacity & Energy): The solid-state ($$D_s$$) and electrolyte ($$D_e$$) diffusion coefficients are the primary governors of accessible capacity and energy, especially at medium to high C-rates (≥0.5C). When these parameters are low, concentration gradients build rapidly, leading to early voltage cut-off and reduced usable capacity. This correlation is crucial for State-of-Health (SOH) diagnosis, as degradation mechanisms like particle cracking or electrolyte depletion directly impair these diffusion coefficients.
Reaction Kinetics & Internal Resistance: The reaction rate constant $$k$$ has a more pronounced effect on the initial voltage drop and DCIR, particularly at very high rates or low temperatures. However, its impact on total capacity is secondary compared to diffusion. The SEI resistance $$R_{SEI}$$ directly adds to the ohmic voltage drop.
This analysis provides a foundational map for linking non-destructively measurable performance fade (loss of capacity, increase in resistance) to underlying degradation modes in cobalt-free lithium-ion batteries. For instance, a capacity fade dominated by loss of active lithium might correlate more with changes in $$SOC_{ini}$$, while rate capability fade would point towards degradation in $$D_s$$ or $$k$$.
Conclusion
This work presents a comprehensive framework for the electrochemical modeling and analysis of cobalt-free lithium-ion batteries across wide temperature ranges. The multi-dimensional sensitivity analysis of the P2D model parameters is instrumental, revealing that key kinetic and transport parameters ($$k$$, $$D_s$$, $$D_e$$, $$R_p$$, $$R_n$$) are highly sensitive to C-rate and SOC, and that an “explosion” in sensitivity occurs at low temperatures, confounding traditional identification methods. To address this, we proposed and demonstrated a pragmatic identification strategy: obtaining parameters at tractable higher temperatures and extrapolating to low temperatures using established physical correlations like the Arrhenius equation. The resulting temperature-dependent model achieves high-fidelity voltage prediction (RMSE < 20 mV) from -5°C to 50°C.
Furthermore, using the validated model as a virtual testbed, we elucidated the distinct roles of different parameter classes on external performance. We established that diffusion-related parameters ($$D_s$$, $$D_e$$) are the primary determinants of capacity and energy output at practical rates, while charge-transfer parameters ($$k$$) more strongly influence polarization and DCIR. These findings provide critical theoretical support for developing advanced BMS algorithms for cobalt-free lithium-ion batteries, including accurate state estimation (SOC, SOH) and diagnostic routines that connect measurable electrical signatures to specific internal aging mechanisms. This work thus advances the fundamental understanding and practical management of this promising, cobalt-free energy storage technology.