The transition towards sustainable energy systems has placed lithium-ion batteries at the forefront of technological innovation. As the demand for electric vehicles and grid-scale energy storage escalates, optimizing the manufacturing processes of these batteries becomes paramount. Among the critical steps in electrode production, calendering—the process of compacting the coated electrode layer—plays a decisive role in determining the final electrode’s microstructure, mechanical integrity, and electrochemical performance. The pressure applied during calendering directly influences the electrode’s porosity, tortuosity, and adhesion to the current collector, all of which are crucial for battery capacity, rate capability, and cycle life.
Despite its importance, predicting the calendering pressure for lithium-ion battery electrodes remains a significant challenge. The electrode is a complex composite structure, comprising a porous coating of active material, conductive agent, and binder on a thin metal foil current collector. This porous, compressible nature means that traditional metal-forming theories, such as those based on the von Mises yield criterion, are not directly applicable. Existing approaches often rely on empirical data fitting, which lacks generalizability and mechanistic insight. Therefore, developing a physics-based model to accurately predict pressure distribution and total roll force is essential for process optimization, equipment design, and ultimately, for producing higher-performance and more consistent lithium-ion batteries.
Current research on lithium-ion battery calendering has extensively focused on the final electrode properties. Studies have examined how calendering pressure affects porosity, ionic/electronic conductivity, adhesion strength, and electrochemical performance. Numerical simulations, particularly using the Discrete Element Method (DEM), have provided valuable insights into the microstructural evolution of the coating during compaction. However, a comprehensive analytical model for predicting the roll force and pressure distribution within the nip region—the gap between the two calender rolls—is still lacking. Such a model would bridge the gap between process parameters (like roll diameter and reduction rate) and the resulting mechanical loads on the equipment.
This work addresses this gap by introducing a novel analytical model for the calendering process of lithium-ion battery electrodes. Recognizing the porous, powder-like behavior of the electrode coating, we adopt the Kuhn yield criterion from the field of powder compaction. This criterion effectively accounts for the influence of hydrostatic pressure on the yielding of porous materials, making it far more suitable for modeling electrode compaction than classical metal plasticity theories. We derive the equilibrium differential equations governing stress within the electrode coating as it passes through the roll gap. The model is then solved to obtain the distribution of unit pressure along the contact arc. Furthermore, we analyze the distribution of frictional stress and calculate the total force and torque per unit width. The model’s predictions are rigorously validated against experimental calendering data for both cathode and anode electrodes. Finally, we employ the validated model to systematically investigate the effects of key process parameters—specifically, the reduction rate and roll diameter—on the calendering pressure and torque, providing crucial insights for industrial process design.
Theoretical Model Development
Fundamental Assumptions and the Kuhn Yield Criterion
The calendering process involves the electrode being drawn into the gap between two counter-rotating rolls. To develop a tractable analytical model, several simplifying assumptions are made, consistent with established rolling theory and the specific nature of the electrode:
- The process is treated as a plane strain problem, neglecting strain in the width direction due to the large width-to-thickness ratio.
- The electrode coating and the current collector foil are modeled as rigid-plastic materials, neglecting elastic recovery and work hardening for simplification.
- Vertical cross-sections of the electrode remain plane during deformation.
- The thickness change of the thin metal current collector during calendering is negligible compared to that of the porous coating.
- The horizontal normal stress is uniformly distributed across the height of both the coating and the current collector.
- The thickness and length directions are the principal stress directions.
- The relative density of the coating is uniform through its thickness but varies along the length of the deformation zone.
The core of our model is the application of the Kuhn yield criterion. For porous materials under plane strain conditions, the Kuhn criterion relates the principal stresses ($\sigma_1$, $\sigma_3$) to the material’s yield stress ($\sigma_{sc}$) and Poisson’s ratio ($\nu$):
$$ (\sigma_1 – \sigma_3)^2 + \frac{1-2\nu}{1-\nu}(\sigma_1 + \sigma_3)^2 = \frac{4}{1+\nu}\sigma_{sc}^2 $$
The yield locus described by this equation is an ellipse. The porous material’s Poisson’s ratio ($\nu$) and yield stress ($\sigma_{sc}$) are not constants but functions of its instantaneous relative density ($\rho$), which changes during compaction:
$$ \nu = \frac{1}{2} – \frac{1}{2}e^{-m(1-\rho)} $$
$$ \sigma_{sc} = \sigma_0 \rho^n $$
Here, $m$ and $n$ are material constants. $\sigma_0$ is the yield stress of the fully dense base material (the solid matrix), which itself may be strain-hardened:
$$ \sigma_0 = K \bar{\epsilon}_0^{\, s} $$
where $K$ is a strength coefficient, $s$ is a strain-hardening exponent, and $\bar{\epsilon}_0$ is the accumulated effective plastic strain in the base material. The relationship between the strain in the porous body ($d\bar{\epsilon}_c$) and the dense base material ($d\bar{\epsilon}_0$) is given by:
$$ d\bar{\epsilon}_0 = \rho \, d\bar{\epsilon}_c $$
Model Derivation: Force Balance in the Roll Gap
Similar to metal rolling, the deformation zone in calendering is divided into a backward slip zone (where the electrode surface speed is less than the roll surface speed) and a forward slip zone (where it is greater). The boundary is the neutral point. We analyze a differential element of the electrode within the roll gap.
The element is subject to horizontal normal stresses in the coating ($\sigma_c$) and the current collector ($\sigma_f$), the roll pressure on the coating ($p$), the friction stress at the roll-coating interface ($\tau_{Rc}$), and the interfacial pressure ($p_{cf}$) and shear stress ($\tau_{cf}$) between the coating and the current collector. Assuming a small contact angle, $p_{cf} \approx p$.
Establishing force equilibrium in the horizontal direction for both the coating and the current collector in the backward and forward slip zones, and combining these with the yield condition for the current collector (modeled as a dense material obeying the von Mises criterion: $p – \sigma_f = \frac{2}{\sqrt{3}}\sigma_{sf}$), we derive the governing differential equation for the coating:
$$ \frac{d}{dx}(h_c \sigma_c) + \frac{dh_c}{dx}p = 2(\sigma_f \pm \tau_{Rc}) $$
where $h_c$ is the coating thickness, $x$ is the horizontal coordinate, and the $\pm$ sign corresponds to the backward ($-$) and forward ($+$) slip zones, respectively.
To solve this equation, the relationship between the roll pressure $p$ and the coating’s horizontal stress $\sigma_c$ is needed. Under our assumption that the thickness and length directions are principal directions, we have $\sigma_3 \approx -p$ and $\sigma_1 \approx \sigma_c$. Applying the Kuhn yield criterion (Equation 1) provides this link:
$$ \sigma_c = \frac{1-\nu}{1+\nu}p – \frac{1}{1+\nu}\sqrt{4\sigma_{sc}^2 – \frac{1-2\nu}{1-\nu}(2p)^2} $$
The friction stress is modeled using Coulomb’s law: $\tau_{Rc} = \mu p$, where $\mu$ is the coefficient of friction.
The evolution of the base material’s accumulated strain $\bar{\epsilon}_0$ is calculated by integrating the incremental strain in the porous coating. For plane strain, the effective strain increment in the coating is:
$$ d\bar{\epsilon}_c = \sqrt{ \frac{4}{3(1+\nu+\nu^2)} \left( d\epsilon_x^2 + d\epsilon_z^2 + \nu \, d\epsilon_x d\epsilon_z \right) } $$
where $d\epsilon_z = -dh_c / h_c$ and $d\epsilon_x$ is related to the distribution of elongation along the roll gap.
Elongation and Relative Density Distribution
The model requires the distribution of the electrode’s relative density $\rho(x)$ within the roll gap. This is determined by the geometry of compaction and the elongation (stretching) of the electrode. Experimental evidence suggests a linear relationship between the overall reduction rate ($C$) and elongation ($\lambda$) for a given roll diameter ($D$):
$$ \lambda = a C + b $$
We assume the local elongation distribution $d\lambda$ is linear with the instantaneous thickness. The local strain in the x-direction is then $d\epsilon_x = d\lambda / (1+\lambda)$. The relative density at any point $x$ can be found from mass conservation:
$$ \rho(x) = \frac{h_{c0} \rho_0}{h_c(x) (1 + \lambda(x))} $$
where $h_{c0}$ and $\rho_0$ are the initial coating thickness and relative density. The relationship between elongation and roll diameter for the same reduction is given by: $\lambda_2 / \lambda_1 = \sqrt{D_2 / D_1}$.
Boundary Conditions and Solution
Assuming no front or back tension, the horizontal stress $\sigma_c$ is zero at both the entry ($x=0$) and exit ($x=l$) of the deformation zone. Using Equation (5), these translate into boundary conditions for the roll pressure $p$:
At exit ($x=l$): $$ p(l) = \frac{\sigma_{sc}(l)}{1-\nu(l)} $$
At entry ($x=0$): $$ p(0) = \frac{\sigma_{sc}(0)}{1-\nu(0)} $$
The deformation zone length $l$ and the coating thickness $h_c(x)$ are geometrically related to the roll radius $R$ and the entry/exit thicknesses. The Hitchcock formula is used to account for roll elastic flattening, where the effective roll radius $R’$ increases with pressure:
$$ R’ = R \left[ 1 + \frac{16(1-\nu_R^2)}{\pi E_R} \frac{p_0}{h_1 – h_0} \right] $$
where $\nu_R$ and $E_R$ are the roll’s Poisson’s ratio and Young’s modulus, and $p_0$ is the total force per unit width.
The system of equations is solved numerically using a shooting method, integrating from both entry and exit towards the interior until the pressure profiles match at the neutral point. The total calendering force and torque per unit width are then obtained by integration:
$$ p_0 = \int_{0}^{l} p \, dx $$
$$ T = R \int_{0}^{\beta} \tau_{Rc} \, d\beta $$
where $\beta$ is the contact angle.
Model Validation with Experimental Data
To validate the proposed model, calendering experiments were performed on both NMC (Lithium Nickel Manganese Cobalt Oxide) cathodes and graphite anodes. The initial electrode thicknesses and current collector foils are listed below.
| Electrode Type | Initial Total Thickness | Current Collector (Thickness) | Initial Coating Relative Density ($\rho_0$) |
|---|---|---|---|
| NMC Cathode | 145 µm | Aluminum Foil (12 µm) | 0.581 |
| Graphite Anode | 196 µm | Copper Foil (7 µm) | 0.461 |
Experiments were conducted using two different roll diameters (110 mm and 130 mm) at various reduction rates. The roll force was measured directly during the process. The material parameters for the coating, determined through a combination of material property estimation and calibration against experimental force data, are summarized in the following table.
| Parameter | Symbol | NMC Cathode | Graphite Anode |
|---|---|---|---|
| Base Material Yield Strength | $\sigma_0$ (MPa) | 410 | 153 |
| Density Exponent | $n$ | 3.5 | 2.5 |
| Strength Coefficient | $K$ (MPa) | 4200 | 700 |
| Strain Hardening Exponent | $s$ | 1 | 1 |
| Poisson’s Ratio Constant | $m$ | 4 | 6 |
| Friction Coefficient | $\mu$ | 0.15 | 0.10 |
The elongation-reduction relationship for the 110 mm rolls was found to be: $\lambda_{Ca} = 0.116C_{Ca} – 0.007$ for the cathode and $\lambda_{An} = 0.025C_{An} – 0.002$ for the anode.
The model predictions for the total force per unit width ($p_0$) are compared with the experimental measurements in the figures below. For comparison, results from an empirical Kawakita equation fit ($p_0 = A C p_0 / (p_0 + B)$) are also shown.
The agreement between the model calculations and the experimental data is excellent. For the NMC cathode, the average prediction error was 1.9% for the 110 mm rolls and 3.2% for the 130 mm rolls. For the graphite anode, the average errors were 6.9% and 6.6%, respectively. All errors remained below 10%, demonstrating the high accuracy and robustness of the proposed mechanistic model based on the Kuhn yield criterion for lithium-ion battery electrode calendering.
Analysis and Discussion of Results
Distribution of Unit Pressure and Frictional Stress
The validated model allows us to examine the detailed mechanics within the roll gap, which are difficult to measure experimentally. The figure below shows the calculated distribution of unit pressure $p(x)$ for different reduction rates using 110 mm diameter rolls.
The pressure distribution exhibits a characteristic shape: it is relatively low at the entry and exit points, rises to a maximum value inside the deformation zone, and this peak is located closer to the exit (near the neutral point). The exit pressure is higher than the entry pressure because the coating’s yield strength $\sigma_{sc}$ is highest at the exit due to maximum densification and accumulated strain. As the reduction rate increases, the pressure distribution becomes steeper and the peak pressure rises significantly. The length of the deformation zone also increases with higher reduction due to greater roll flattening.
The corresponding distribution of frictional stress $\tau_{Rc}(x)$ is directly proportional to the unit pressure ($\tau_{Rc} = \mu p$) but changes direction at the neutral point. This leads to a sharp discontinuity or sign change in the friction stress profile at that location, which is a direct consequence of the Coulomb friction model.
To further validate the non-uniform pressure distribution predicted by the model, we compared the average unit pressure from calendering with that from a simple flat-plate compression test at the same final density. The average calendering pressure was consistently lower than the flat-plate pressure. Since flat-plate compression approximates a uniform pressure state, this discrepancy confirms that the pressure in calendering is indeed non-uniform, with significant regions of lower pressure, as predicted by our model.
Influence of Process Parameters on Force and Torque
Using the model, we can systematically analyze the impact of key calendering parameters. The first critical parameter is the reduction rate ($C$). The following figure illustrates the relationship between reduction rate and the total force per unit width ($p_0$) for a fixed roll diameter.
The force increases monotonically with reduction rate. Notably, the slope of the curve also increases, meaning that achieving additional densification becomes progressively more difficult at higher reductions. This trend is consistent for both cathode and anode materials, though the absolute force is higher for the NMC cathode due to its greater intrinsic strength. A similar monotonic and increasingly steep relationship is observed between the reduction rate and the driving torque per unit width ($T$).
The second key parameter is the roll diameter ($D$). The figures below show how the force and torque per unit width vary with roll diameter for different reduction rates.
Both the force ($p_0$) and torque ($T$) increase approximately linearly with roll diameter. This is because a larger roll diameter increases the length of the contact arc, over which the pressure integrates to give the total force. The slope of these linear relationships becomes steeper at higher reduction rates, indicating that the sensitivity of the calendering load to roll diameter is more pronounced when compressing the electrode to a higher density. These insights are invaluable for the design and scaling of calendering equipment for lithium-ion battery manufacturing, allowing engineers to predict the mechanical loads for different roll sizes and process targets.
Conclusion
In this work, we have developed and validated a comprehensive analytical model for predicting the calendering pressure in the manufacturing of lithium-ion battery electrodes. The model’s foundation is the Kuhn yield criterion, which accurately captures the plastic behavior of the porous electrode coating—a critical advancement over traditional metal-forming theories. By solving the force equilibrium equations within the roll gap, the model successfully predicts the distribution of unit pressure and frictional stress, as well as the total force and torque per unit width.
Experimental validation on both NMC cathodes and graphite anodes confirmed the model’s accuracy, with prediction errors for total force consistently below 10%. The analysis revealed that pressure within the nip is non-uniform, peaking near the neutral point, and becomes increasingly non-uniform at higher reduction rates. The study further quantified the effects of key process variables: both the reduction rate and the roll diameter have a significant and monotonic positive influence on the required calendering force and torque. The relationship with roll diameter is approximately linear, and the sensitivity to diameter increases with higher reduction.
This model provides a powerful tool for the lithium-ion battery industry. It moves beyond empirical fitting, offering a physics-based approach to optimize calendering parameters, design robust calendering machinery, and ensure consistent electrode quality. By enabling accurate prediction of mechanical loads, it contributes to more efficient, reliable, and scalable manufacturing processes for next-generation lithium-ion batteries. Future work could integrate this mechanical model with microstructural simulations (e.g., DEM) to form a complete digital twin of the calendering process, linking process parameters directly to final electrode performance.