SOC-Dependent Kinetic Parameters

• SOC-dependent kinetic parameters are continuously varying functions that model reaction rates, activation energies, and enthalpies as functions of state-of-charge in electrochemical systems.
• They are parameterized using techniques like Chebyshev polynomial expansions and neural ODE frameworks (KA-CRNN) to ensure mechanistic fidelity and accurate real-time diagnostics.
• By incorporating SOC-dependent Arrhenius laws, these models improve predictions of phenomena such as thermal runaway, phase transitions, and coupled electrolyte reactions.

SOC-dependent kinetic parameters are continuous or piecewise-continuous functions governing the rates and thermodynamics of physical or chemical processes, whose values depend explicitly on the instantaneous value of the “state of charge” (SOC) or an analogous state variable. In electrochemical or solid-state systems, SOC typically parameterizes the fractional occupancy of active species or the electron concentration, and directly modulates Arrhenius prefactors, activation barriers, reaction orders, stoichiometries, and enthalpies. Recent computational and experimental approaches enable the learning and interpretation of these parameters as smooth, data-driven functions over the operational SOC range—enabling more predictive models of phenomena such as thermal runaway and phase transformations under dynamically varying conditions (Koenig et al., 17 Dec 2025).

1. General Framework for SOC-Dependent Kinetics

State-of-charge dependence in kinetic modeling reflects the nonconstant physical environment that governs reaction energetics, accessible reaction pathways, and coupled phenomena such as phase change or gas evolution. In battery thermal runaway, the rates of oxygen-release and subsequent reactive heat release are highly nonmonotonic in SOC due to underlying structural transitions in cathode materials.

In the Kolmogorov-Arnold Chemical Reaction Neural Network (KA-CRNN) framework, each kinetic parameter $p_i$ is parameterized as a smooth, interpretable function $p_i(\mathrm{SOC})$ learned directly from differential scanning calorimetry (DSC) data, subject to mechanistic constraints. These parameters include the pre-exponential (frequency) factors $A_i$, activation energies $E_{a,i}$, temperature exponents $b_i$, reaction orders $n_i$, enthalpies $\Delta H_i$, and evolving O$_2$ stoichiometry $\nu$ (Koenig et al., 17 Dec 2025).

2. Arrhenius Rate Laws with SOC Variability

Each individual elementary reaction rate obeys an SOC-modulated Arrhenius/mass-action form

$$r_i(\mathrm{SOC},T,[c]) = A_i(\mathrm{SOC})\, T^{b_i(\mathrm{SOC})}\, [c_i]^{n_i(\mathrm{SOC})}\, \exp\left(-\frac{E_{a,i}(\mathrm{SOC})}{RT}\right),$$

where all key parameters are functions of SOC. The log-linearized rate law is

$$\ln r_i = n_i(\mathrm{SOC})\, \ln [c_i] + \ln A_i(\mathrm{SOC}) + b_i(\mathrm{SOC})\, \ln T - \frac{E_{a,i}(\mathrm{SOC})}{RT}$$

with each parameter (including enthalpy and oxygen-release stoichiometry) varying smoothly, typically parameterized as Chebyshev expansions (see below).

3. Functional Parameterization Using Chebyshev Polynomials

SOC-dependence is efficiently captured as a truncated orthogonal expansion: $$p_i(\mathrm{SOC}) = \sum_{n=0}^{N} w_{i,n}\, \psi_n(\mathrm{SOC}),\quad \psi_n(\mathrm{SOC}) = \cos\left( n \arccos(\mathrm{SOC}) \right),$$ with $N=10$ being sufficient for high-fidelity representation. Each parameter thus requires only $N+1$ coefficients, yielding a set of continuous, differentiable functions suitable for both simulation and inference (Koenig et al., 17 Dec 2025).

Parameter examples for the key decomposition step ($R_2$: spinel $\to$ rock-salt + O$_2$) in three cathode materials are provided below:

Cathode	SOC	$\ln A_2$	$E_{a,2}$ (kJ/mol)	$\Delta H_2$ (J/g)
NM	0.2	34.2	120	180
	0.8	41.7	200	245
	1.0	44.3	215	260
NMA	0.2	33.1	115	175
	0.8	39.5	185	230
	1.0	42.0	200	250
NCA	0.2	32.8	110	170
	0.8	38.7	180	225
	1.0	41.2	195	245

4. Physical Interpretation and Model Consequences

Distinct, interpretable inflections in $E_{a,2}(\mathrm{SOC})$ and $\ln A_2(\mathrm{SOC})$ appear at critical SOC (~0.8), correlating with abrupt lattice oxygen release in nickel-rich cathodes. The exothermicity $\Delta H_2(\mathrm{SOC})$ and the O$_2$ stoichiometry $\nu(\mathrm{SOC})$ both rise concomitantly, reflecting the onset of the phase transition and greater oxygen availability for subsequent electrolyte oxidation. This kinetic shift is not smooth but features a critical turning point, which cannot be captured using scalar kinetic parameters or models fit at a single SOC.

The downstream oxygen–electrolyte chemistry (R3) retains invariant kinetic parameters, but its dynamic impact is strongly SOC-modulated via its coupling to R2, resulting in amplified and narrowed calorimetric peaks at high SOC (Koenig et al., 17 Dec 2025).

5. Training Approaches and Physics Priors

Learning continuous SOC-dependent parameters is performed by minimizing an objective that combines reproduction of measured heat-flow data and penalties enforcing mechanistic fidelity:

• L_mono: Ensures monotonic increase of $\Delta H_2(\mathrm{SOC})$ and $\nu(\mathrm{SOC})$ with SOC.
• L_min, L_max: Bound reaction orders and temperature exponents to physically plausible ranges, suppressing instability from overfitting.
• Physics integration: The ODEs for mass fractions and heat release are integrated using differentiable solvers (NeuralODEs), and parameter gradients computed via algorithmic differentiation (Koenig et al., 17 Dec 2025).

6. Implications for Predictive Modeling and Real-Time Diagnostics

Continuous SOC-dependent kinetic laws enable simulations and hazard prediction at arbitrary intermediate SOC, supporting real-time inference required by battery management systems (BMS) during abuse events. The modular KA-CRNN approach further provides a framework to generalize kinetic dependencies to other control variables such as temperature, pressure, or compositional state. This paradigm supports high-resolution, interpretable, and physically constrained kinetic parameter estimation, directly improving the reliability of empirical and physics-informed simulation frameworks deployed in battery safety and lifetime modeling (Koenig et al., 17 Dec 2025).