In the relentless pursuit of sustainable transportation and green energy solutions, the development of high-performance and long-lasting energy storage systems is paramount. As a core power source for electric vehicles and portable electronics, the li ion battery faces continuous challenges related to its durability and safety. While significant research has focused on thermal management and material innovation, the mechanical stresses generated during battery operation—primarily diffusion-induced stress and thermal stress—are critical yet less explored factors influencing degradation and failure. Understanding their generation, distribution, and relative magnitudes is essential for designing more robust batteries. This study, from my perspective as a researcher, aims to dissect these mechanical phenomena within a commercial soft-pack li ion battery by developing and employing a sophisticated multiphysics simulation model.
The motivation stems from the fundamental operational mechanics of a li ion battery. During discharge, lithium ions de-intercalate from the anode, travel through the electrolyte, and intercalate into the cathode. This insertion/extraction process causes the active material particles, particularly in the graphite anode, to periodically swell and shrink. The resulting strain, due to the concentration gradient of lithium within the particles, generates significant diffusion-induced stress. Concurrently, the electrochemical reactions and internal resistance produce heat, leading to non-uniform temperature distribution and consequent thermal stress within the cell structure. In my investigation, I sought to contrast these two stress sources, quantify their scale, identify their critical locations, and evaluate their dependence on operational intensity (discharge rate). This comparative analysis provides a theoretical foundation for electrode design and structural monitoring strategies to mitigate mechanical degradation.
To achieve this, I constructed a comprehensive electrochemical-mechanical-thermal coupled model for an 18.5 Ah NCM111/Graphite soft-pack li ion battery using COMSOL Multiphysics 6.0. The model integrates three core physics: a pseudo-two-dimensional (P2D) electrochemistry model to capture lithium transport and reaction kinetics, a solid mechanics model to compute stresses, and a heat transfer model to simulate temperature distribution. The geometry of the soft-pack cell, including its essential layers and tabs, was modeled in three dimensions to accurately assess thermal stress, while the diffusion-induced stress was analyzed at the particle level derived from the 1D electrochemical model.
1. Mathematical Model Framework
The core of my simulation lies in the interplay of governing equations describing mass conservation, charge conservation, reaction kinetics, energy balance, and mechanical equilibrium within the li ion battery.
1.1 Electrochemical Model
The P2D model forms the electrochemical foundation. Lithium transport in the solid phase of the electrode particles is governed by Fick’s law of diffusion in spherical coordinates:
$$ \frac{\partial c_s}{\partial t} = \frac{D_s}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial c_s}{\partial r} \right) $$
where \( c_s \) is the solid-phase lithium concentration, \( D_s \) is the solid-phase diffusion coefficient, and \( r \) is the radial coordinate within a particle.
Conservation of lithium in the electrolyte phase (liquid) is described by:
$$ \frac{\partial (\epsilon_e c_e)}{\partial t} = \frac{\partial}{\partial x} \left( D_e^{\text{eff}} \frac{\partial c_e}{\partial x} \right) + \frac{1 – t_+^0}{F} j_{\text{Li}} $$
Here, \( c_e \) is the electrolyte concentration, \( \epsilon_e \) is the porosity, \( D_e^{\text{eff}} \) is the effective diffusivity in the electrolyte, \( t_+^0 \) is the transference number, \( F \) is Faraday’s constant, and \( j_{\text{Li}} \) is the volumetric exchange current density.
Charge conservation in the solid and electrolyte phases is given by Ohm’s law and modified Ohm’s law, respectively:
$$ \sigma^{\text{eff}} \frac{\partial^2 \phi_s}{\partial x^2} = a_s j_{\text{Li}} $$
$$ \kappa^{\text{eff}} \frac{\partial^2 \phi_e}{\partial x^2} + \kappa_D^{\text{eff}} \frac{\partial^2 (\ln c_e)}{\partial x^2} = -a_s j_{\text{Li}} $$
where \( \phi_s \) and \( \phi_e \) are solid and electrolyte potentials, \( \sigma^{\text{eff}} \) and \( \kappa^{\text{eff}} \) are effective conductivities, \( a_s \) is the specific surface area, and \( \kappa_D^{\text{eff}} \) is the diffusion conductivity.
The electrochemical reaction rate at the electrode/electrolyte interface is modeled using the Butler-Volmer equation:
$$ j_{\text{Li}} = i_0 \left[ \exp\left(\frac{\alpha_a F}{RT} \eta\right) – \exp\left(-\frac{\alpha_c F}{RT} \eta\right) \right] $$
with the overpotential \( \eta = \phi_s – \phi_e – E_{\text{eq}} \). \( i_0 \) is the exchange current density, \( \alpha_a \) and \( \alpha_c \) are anodic and cathodic transfer coefficients, \( R \) is the gas constant, \( T \) is temperature, and \( E_{\text{eq}} \) is the equilibrium potential.
1.2 Thermal Model
The energy balance for the li ion battery accounts for heat generation and dissipation:
$$ \rho C_p \frac{\partial T}{\partial t} = \nabla \cdot (k \nabla T) + Q $$
where \( \rho \), \( C_p \), and \( k \) are the density, specific heat capacity, and thermal conductivity of the cell, respectively. The heat generation source \( Q \) comprises reversible reaction heat \( Q_{\text{rea}} \), irreversible polarization heat \( Q_{\text{rev}} \), and ohmic heat \( Q_{\text{ohm}} \):
$$ Q = j_{\text{Li}} \eta + j_{\text{Li}} T \frac{\partial E_{\text{eq}}}{\partial T} + \sigma^{\text{eff}} \left( \frac{\partial \phi_s}{\partial x} \right)^2 + \kappa^{\text{eff}} \left( \frac{\partial \phi_e}{\partial x} \right)^2 + \frac{2 \kappa^{\text{eff}} R T}{F} (1 – t_+^0) \frac{\partial \ln c_e}{\partial x} \frac{\partial \phi_e}{\partial x} $$
Heat dissipation to the environment via convection and radiation is modeled using Newton’s law of cooling and the Stefan-Boltzmann law.
1.3 Mechanical Model
The mechanical analysis is separated into two scales. For diffusion-induced stress in anode particles, the stress-strain constitutive relation, considering lithium insertion-induced swelling, is:
$$ \varepsilon_{ij} = \frac{1}{E} \left[ (1+\nu)\sigma_{ij} – \nu \sigma_{kk} \delta_{ij} \right] + \frac{1}{3} \Omega c_s \delta_{ij} $$
where \( E \) and \( \nu \) are Young’s modulus and Poisson’s ratio of the active material, \( \Omega \) is the partial molar volume of lithium, and \( \delta_{ij} \) is the Kronecker delta. Assuming spherical particles and spherically symmetric diffusion, the radial \( \sigma_r \) and tangential \( \sigma_\theta \) stress components can be derived analytically. The von Mises stress, used to evaluate the stress magnitude, is calculated as:
$$ \sigma_{vM} = | \sigma_r – \sigma_\theta | $$
For thermal stress at the cell level, the constitutive relation incorporates thermal expansion:
$$ \varepsilon_{ij}^T = \frac{1}{E_T} \left[ (1+\nu_T)\sigma_{ij}^T – \nu_T \sigma_{kk}^T \delta_{ij} \right] + \alpha_T \Delta T \delta_{ij} $$
where \( E_T \), \( \nu_T \) are the cell’s effective mechanical properties, \( \alpha_T \) is the coefficient of thermal expansion, and \( \Delta T \) is the temperature change.
1.4 Model Parameters, Geometry, and Validation
The model parameters for the NCM111 cathode, graphite anode, separator, and electrolyte were sourced from literature and calibrated to match the specific 18.5 Ah cell. Key cell parameters are summarized below:
| Parameter | Value | Unit |
|---|---|---|
| Nominal Capacity | 18.5 | Ah |
| Charge Cut-off Voltage | 4.2 | V |
| Discharge Cut-off Voltage | 2.75 | V |
| Cell Dimensions (L x W x H) | 142 x 12 x 73 | mm |
| Anode Thickness | 65 | μm |
| Cathode Thickness | 55 | μm |
| Separator Thickness | 10 | μm |
Critical electrochemical parameters are listed in the following table:
| Parameter | Anode (Graphite) | Separator | Cathode (NCM111) |
|---|---|---|---|
| Particle Radius, \( R_s \) | 5 μm | — | 3.5 μm |
| Porosity, \( \epsilon_e \) | 0.23 | 0.54 | 0.332 |
| Solid Phase Diffusivity, \( D_s \) | \( D_{s,\text{ref}} \exp\left[\frac{E_a}{R}\left(\frac{1}{T_{\text{ref}}}-\frac{1}{T}\right)\right] \) | — | 3×10⁻¹³ m²/s |
| Max. Li Concentration, \( c_{s,\text{max}} \) | 25000 mol/m³ | — | 36220 mol/m³ |
The mechanical property of the cell stack, Young’s modulus, was determined experimentally from a stress-strain test, yielding a value of 33.04 MPa for use in the thermal stress model.
I validated the coupled model by simulating discharge at 0.5C, 1C, 1.5C, and 2C rates and comparing the voltage and temperature profiles with experimental data. The simulation results showed excellent agreement, with average relative errors well below 1% for voltage and 0.3% for temperature, confirming the model’s reliability for the subsequent stress analysis.
2. Results and Discussion: A Deep Dive into Stress Generation
Using the validated model, I systematically analyzed the behavior of the li ion battery under different discharge rates (0.5C to 2C). The focus was on the evolution of lithium concentration, diffusion-induced stress in anode particles, and cell-level thermal stress and displacement.
2.1 Lithium Concentration Gradients and Their Implications
The diffusion-induced stress originates directly from the lithium concentration gradient within the active material particles. My simulation tracked the normalized lithium concentration \( c_s / c_{s,\text{max}} \) across the radius of an anode particle located near the separator. As expected, the concentration decreased during discharge as lithium de-intercalated. A significant radial gradient developed, with the surface concentration lower than the center concentration due to finite solid-state diffusion. This center-surface concentration difference, \( \Delta c_s \), is a key driver for stress.
I found that \( \Delta c_s \) increased substantially with the discharge rate. At the end of discharge (100% Depth of Discharge, DOD), the normalized \( \Delta c_s \) was 0.0234 for 0.5C but surged to 0.0886 for 2C. This indicates that high-rate operation forces a steeper concentration profile, potentially leading to higher mechanical strain.
More intriguingly, analyzing this difference along the anode thickness (from the current collector to the separator) revealed a dynamic shift. During early and mid-discharge (up to ~60% DOD), the particles near the separator exhibited a larger \( \Delta c_s \) than those near the current collector. This aligns with the “proximity principle,” where ions de-intercalate first from regions closer to the lithium source (the separator). However, after a turning point between 60-70% DOD, this trend reversed. In the late discharge stage (80-100% DOD), the particles near the current collector experienced a larger \( \Delta c_s \). This shift has direct consequences for the spatial distribution of diffusion-induced stress within the electrode.
2.2 Diffusion-Induced Stress in Anode Particles
The tangential stress on the surface of the anode particles, a critical component of the von Mises stress, was calculated. The results powerfully demonstrate the impact of discharge rate and the spatial shift observed in the concentration analysis.
- Rate Dependency: The maximum tangential stress increased dramatically with the discharge rate, from 6.97 MPa at 0.5C to 30.3 MPa at 2C. This underscores that milder discharge protocols can significantly reduce this intrinsic mechanical degradation factor in a li ion battery.
- Spatial-Temporal Evolution: Mirroring the lithium concentration shift, the location of the highest stress migrated during discharge. Before the ~70% DOD turning point, particles near the separator endured higher stress. After this point, the particles near the current collector became the most stressed region. Therefore, the risk of particle cracking or structural degradation is not static but moves within the electrode during a single discharge cycle.
The magnitude of diffusion-induced stress was consistently in the MPa range, which is critically relevant for the micro-scale integrity of the active material particles.
2.3 Cell-Level Thermal Stress and Expansion
In contrast to the particle-level stress, thermal stress operates at the macro scale of the entire cell pouch. The simulation solved the 3D thermal and structural fields to map the von Mises stress and displacement. The key findings are:
- Rate Dependency: Both the maximum thermal stress and the maximum displacement (swelling) of the cell increased with higher discharge rates, correlating directly with the larger temperature gradients generated.
- Magnitude Comparison: Crucially, the thermal stress was in the kPa range, approximately three orders of magnitude smaller than the diffusion-induced stress within the particles. For instance, at the end of a 1C discharge, diffusion-induced stress accounted for about 99.97% of the combined stress magnitude from both sources, with thermal stress contributing only 0.03%. This highlights that for this li ion battery under normal operating conditions, diffusion-induced stress is the dominant mechanical degradation mechanism at the material level.
- Critical Location: A distinctive feature of the soft-pack design emerged. The highest thermal stress concentrations occurred at the junction between the tabs and the cell body. This is a potential weak point for mechanical fatigue, differing from cylindrical cell designs where stress is distributed differently. The aluminum (positive) and copper (negative) tabs, having different mechanical properties, also showed different stress levels, with the copper tab generally experiencing higher stress.
- Linear Relationship: A clear linear correlation was established between the cell’s maximum thermal stress (and displacement) and its internal temperature difference (\( \Delta T \)). This provides a simple predictive relationship: managing the cell’s temperature uniformity is directly proportional to managing its thermal-mechanical strain.
3. Conclusion
Through the development and application of a high-fidelity multiphysics model, this study provides a clear comparative analysis of diffusion-induced and thermal stress in a commercial ternary soft-pack li ion battery during discharge. The conclusions drawn from my analysis are as follows:
First, the discharge rate is a paramount factor influencing both stress mechanisms. Lowering the discharge rate effectively reduces the magnitude of both diffusion-induced stress in anode particles and thermal stress in the cell, suggesting a straightforward pathway to enhance mechanical longevity.
Second, diffusion-induced stress, which is on the order of Megapascals (MPa), is the dominant mechanical stressor at the particle level. Its distribution is dynamic, shifting from separator-side particles in the early discharge phase to current-collector-side particles in the late phase, with a transition around 60-70% DOD. This spatial-temporal shift is critical for targeted electrode design and degradation monitoring.
Third, thermal stress, while critical for pack design and tab integrity, is orders of magnitude smaller (on the order of Kilopascals (kPa)) under normal operation. However, it exhibits a strong linear dependence on the cell’s internal temperature difference. The tab-to-cell junction in soft-pack batteries is identified as a critical hotspot for this type of stress.
In summary, while thermal management remains crucial for safety and uniformity, mitigating diffusion-induced stress at the particle level through material engineering, electrode architecture optimization, and moderate operational protocols appears to be the more significant challenge for improving the mechanical durability and cycle life of the modern li ion battery. This work delineates the scale and nature of these challenges, offering a quantitative framework for future design and diagnostic advancements.