Pore-scale modeling of capillary-driven binder migration during battery electrode drying

1. Problem Statement

The manufacturing of battery electrodes (specifically for Sodium-ion batteries using hard carbon) involves a critical drying step where solvent evaporation leads to the formation of the final solid microstructure. A major challenge in this process is binder migration.

• The Issue: During the "pore-emptying" phase of drying, capillary forces drive the solvent (and dissolved polymer binder) through the porous microstructure. This often results in an inhomogeneous binder distribution, characterized by accumulation at the electrode surface and depletion near the current collector.
• Consequences: Surface accumulation reduces adhesion to the current collector, while local depletion compromises particle cohesion, leading to mechanical failure and poor electrical performance.
• Limitations of Existing Models: Previous modeling approaches largely relied on:
  • 1D Homogenized Continuum Models (HCM): These treat drying as a film shrinkage problem using convection-diffusion equations but neglect pore-scale capillary effects.
  • Coarse-Grained Molecular Dynamics (CGMD): These capture microstructure but often implicitly handle capillary forces or cannot model the explicit pore-emptying physics.
  • Gap: There is a lack of physically consistent, spatially resolved models that explicitly link capillary-driven solvent flow to binder transport in realistic 3D microstructures.

2. Methodology

The authors developed an extended spatially resolved pore-scale continuum model using the multiphase-field method within the PACE3D simulation framework.

A. Mathematical Framework

The model couples three physical phenomena:

1. Phase Evolution (Fluid-Fluid Interface): Uses an Allen-Cahn formulation to track the liquid-air interface. The evolution equation includes terms for surface tension minimization and explicitly models solvent evaporation at the interface.
2. Fluid Dynamics: Solves the incompressible Navier-Stokes equations for the two fluid phases (solvent and air), incorporating capillary forces (via the phase-field variable) and rigid-body coupling with solid particles.
3. Binder Transport (Novel Contribution):
   • Introduces a convection-diffusion equation for mobile binder concentration ($c$) dissolved in the solvent.
   • Introduces a separate field for deposited binder ($c_a$) in the air phase.
   • Deposition Mechanism: Uses a source term ($q_d$) to model irreversible binder deposition from the solvent to the air phase (or solid surface). The deposition is weighted by a separation parameter ($f$) to control whether binder deposits at the fluid-fluid interface or preferentially at triple-phase boundaries (solid-liquid-gas).
   • Mass Conservation: Employs a Lagrange multiplier ($\lambda$) to enforce global binder mass conservation, compensating for the continuous reduction of solvent volume due to evaporation.

B. Simulation Setup

• Microstructures: Realistic 2D microstructures of Hard Carbon (HC) electrodes were generated from SEM images (labeled HC-B and HC-C), differing in mean particle diameter and pore size distribution.
• Parameters: The study systematically varied:
  • Solvent viscosity ($\mu$)
  • Evaporation rate ($\dot{r}$)
  • Surface tension ($\sigma$)
  • Contact angle (wettability: $60^\circ, 90^\circ, 120^\circ$)
• Validation: The model was validated against:
  1. A bubble rise benchmark (conservation of mass in moving fluid without evaporation).
  2. A drying capillary benchmark (comparing results against literature for different Péclet numbers).

3. Key Contributions

1. First Explicit Capillary-Driven Binder Model: This work presents the first spatially resolved continuum framework that consistently captures binder migration driven specifically by capillary transport during pore emptying, rather than just film shrinkage.
2. Coupled Transport-Deposition System: The introduction of a dual-field formulation (mobile vs. deposited binder) coupled with a mass-conserving Lagrange multiplier allows for accurate tracking of binder redistribution in evaporating porous media.
3. Microstructure-Process Linkage: The model successfully links specific microstructural features (particle size, pore network) and process parameters (drying rate, wettability) to the final binder distribution.

4. Key Results

• Microstructure Influence:
  • Particle Size: Smaller particle sizes (HC-B) lead to a more homogeneous binder distribution. Larger particles (HC-C) result in significant binder accumulation at the top surface.
  • Mechanism: Narrower capillary channels in smaller-particle microstructures promote more uniform solvent flow, whereas larger pores drain first, creating strong gradients.
• Evaporation Rate:
  • Higher evaporation rates lead to steeper binder gradients (more inhomogeneity). This is due to stronger capillary pressure gradients driving faster solvent flow toward the drying front.
• Surface Tension:
  • Increased surface tension promotes stronger binder gradients.
  • Lower surface tension weakens capillary forces, leading to a more homogeneous drying front and reduced binder migration.
  • A sharp transition in binder gradient was observed for HC-C at specific surface tension ratios, linked to "pinning" effects of the drying front.
• Viscosity:
  • Surprisingly, solvent viscosity showed only a minor influence on binder migration in the Newtonian regime.
  • Caveat: The authors note that real slurries are often shear-thinning (non-Newtonian), which the current model does not capture. This may explain discrepancies with experimental data where viscosity plays a larger role.
• Wettability (Contact Angle):
  • Deviations from a neutral contact angle ($90^\circ$) significantly amplify the sensitivity of binder migration to viscosity.
  • Hydrophilic conditions ($120^\circ$) showed a stronger increase in binder gradient with viscosity compared to hydrophobic conditions.

5. Significance and Outlook

• Process Optimization: The study demonstrates that binder homogeneity can be effectively optimized by tuning surface tension and contact angle (wettability), rather than just drying rates. This provides a theoretical basis for designing better electrode slurries and drying profiles.
• Modeling Paradigm Shift: The results confirm that 1D convection-diffusion models are insufficient for predicting binder migration because they ignore the dominant capillary mechanisms at the pore scale. Accurate prediction requires explicit resolution of the microstructure and capillary forces.
• Future Work: The authors plan to extend the model to:
  • 3D microstructures to capture realistic pore connectivity.
  • Non-Newtonian rheology (shear-thinning) to better match experimental slurry behavior.
  • Gas-phase mass transport resistance to account for saturation effects in thick electrodes.

In conclusion, this paper provides a robust, physics-based tool for understanding and mitigating binder migration, a critical factor in the reliability and performance of next-generation battery electrodes.