A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

1. Problem Statement

The paper addresses the mathematical modeling and numerical simulation of thermally driven phase separation in viscous, incompressible binary mixtures, specifically motivated by water condensation inside electric vehicle battery housings.

• Physical Context: Moisture entering battery enclosures can condense on colder internal surfaces, leading to corrosion and insulation degradation. Experimental testing across wide geometric and operating ranges is costly, necessitating robust numerical tools.
• The Gap: Existing models often focus on isothermal settings or generic boiling/condensation configurations. There is a lack of frameworks that couple phase-field dynamics with variable-density Stokes flow and heat transport under realistic, mixed boundary conditions relevant to confined geometries (like battery casings).
• Core Challenge: The system involves strong coupling between three distinct physical mechanisms:
  1. Phase Separation: Governed by the Cahn–Hilliard equation (CHE).
  2. Fluid Flow: Governed by the Stokes equations with variable density and viscosity.
  3. Heat Transport: Governed by the heat equation with convection.
     The coupling is bidirectional: temperature affects the free energy landscape (driving phase change), while the phase field (concentration) affects fluid density and viscosity (driving flow).

2. Methodology

A. Mathematical Modeling

The authors propose a diffuse-interface model consisting of three coupled partial differential equations (PDEs) in a bounded domain $\Omega \subset \mathbb{R}^d$ ($d=2,3$):

1. Convective Cahn–Hilliard Equation:
   Describes the evolution of the order parameter $c$ (representing volume fraction/concentration).
   $$\frac{\partial c}{\partial t} + (u \cdot \nabla)c = \frac{1}{Pe} \Delta \mu$$
   where $\mu$ is the chemical potential derived from a temperature-dependent Landau free energy.

2. Stokes Equations (Quasi-static):
   Describes the velocity $u$ and pressure $p$ for creeping flow ($Re \ll 1$) with variable coefficients.
   $$-\nabla p + \nabla \cdot (2\eta(c) E(u)) + G \rho(c) \hat{g} = 0, \quad \nabla \cdot u = 0$$
   Here, density $\rho(c)$ and viscosity $\eta(c)$ are affine functions of $c$. A buoyancy term $G \rho(c) \hat{g}$ accounts for gravity.

3. Heat Equation:
   Describes the temperature field $\Theta$.
   $$\frac{\partial \Theta}{\partial t} + u \cdot \nabla \Theta = \frac{1}{Pe_\Theta} \Delta \Theta$$

Key Innovation in Free Energy:
The bulk free energy density $f(c, \Theta)$ is modified to be temperature-dependent:
$$f(c, \Theta) = 2c^4 - (1 - \beta(\Theta))c^2$$
where $\beta(\Theta) = \Theta / \Theta_S$ is the ratio of local temperature to saturation temperature.

• If $\Theta > \Theta_S$ ($\beta > 1$): The potential is a single-well, favoring a homogeneous phase.
• If $\Theta < \Theta_S$ ($\beta < 1$): The potential becomes a double-well, driving phase separation.

B. Boundary Conditions

The model utilizes mixed boundary conditions to reflect realistic battery geometries:

• Velocity: No-slip on some boundaries ($\Gamma_D$) and traction-free on others ($\Gamma_N$).
• Temperature: Dirichlet (fixed temperature) on some boundaries and Neumann (insulated) on others.
• Phase Field: Homogeneous Neumann (no-flux) for both $c$ and $\mu$.

C. Numerical Scheme

A fully discrete finite element scheme is proposed using a sequential splitting strategy:

1. Time Discretization: First-order semi-implicit.
   • Heat Equation: Backward Euler.
   • Stokes: Implicit solve for velocity/pressure with updated coefficients.
   • Cahn–Hilliard: Convex-splitting (Eyre-type) discretization. The free energy is split into convex (treated implicitly) and concave (treated explicitly) parts to ensure unconditional energy stability for the diffusive substep.
2. Spatial Discretization:
   • Velocity/Pressure: Taylor–Hood elements ($P_2/P_1$).
   • Scalars ($c, \mu, \Theta$): Linear elements ($P_1$).
3. Solver: Nonlinear algebraic systems for the Cahn–Hilliard step are solved via damped Newton iteration.

3. Key Contributions

1. Thermodynamically Motivated Potential: Introduction of a Landau-type potential where the phase behavior (homogeneous vs. separated) is directly controlled by the local temperature ratio $\beta(\Theta)$, rather than ad-hoc source terms.
2. Rigorous Existence Theory:
   • Proved local-in-time existence of weak solutions for the coupled system in 2D and 3D.
   • Addressed the lack of global $L^\infty$ bounds on the order parameter (due to the polynomial potential) by working in the functional framework $H^2 \to L^2 \to H^{-2}$.
   • Used a Schauder fixed-point argument on a composite operator $T(u) = S(C(u, H(u)))$, analyzing the continuity and compactness of the Heat, Cahn–Hilliard, and Stokes operators separately.
   • Handled mixed boundary conditions (traction-free and thermal Neumann) which complicate standard energy estimates.
3. Numerical Stability: Developed a scheme that conserves mass (under impermeable boundaries) and is unconditionally energy-stable for the Cahn–Hilliard substep in the isothermal diffusive limit.
4. Application to Battery Safety: Provided a specific framework for simulating condensation in confined, battery-relevant geometries with realistic boundary conditions.

4. Results

The authors validated the model through four numerical experiments:

1. Benchmark (Lid-Driven Cavity):

   • Compared against existing literature (Khatavkar et al.) for two-phase Stokes flow.
   • Result: The scheme accurately reproduced velocity profiles and interface deformation trends, validating the hydrodynamic discretization.

2. Isothermal Phase Separation:

   • Tested the potential's response to temperature.
   • Result:
     • At $\Theta > \Theta_S$: The system relaxed to a homogeneous state (single-well behavior).
     • At $\Theta < \Theta_S$: The system underwent rapid spinodal decomposition and coarsening (double-well behavior).

3. Gravity-Driven Phase Separation:

   • Activated buoyancy with variable density ($\lambda_\rho \approx 0.001$) and viscosity.
   • Result: Dense liquid domains formed and migrated downward due to gravity, accumulating at the bottom boundary and merging into a heavier layer, demonstrating the coupling between phase separation and buoyancy.

4. Thermally Induced Phase Change (Heated/Cooled Boundaries):

   • Simulated a thermal gradient across a domain.
   • Heated Boundary: Local heating suppressed phase separation, causing liquid domains near the hot wall to dissolve into a homogeneous vapor phase.
   • Cooled Boundary: Local cooling triggered nucleation of liquid droplets near the cold wall, which subsequently grew and coarsened.
   • Significance: This successfully mimics the physical process of condensation on cold surfaces within a battery housing.

5. Significance

• Theoretical Advancement: The paper bridges a gap in the mathematical analysis of diffuse-interface models by handling the complex coupling of temperature, variable density/viscosity, and mixed boundary conditions, which are often omitted in simpler isothermal models.
• Engineering Utility: It provides a robust, mathematically grounded tool for predicting condensation patterns in electric vehicle battery packs. This is critical for designing thermal management systems that prevent moisture accumulation, thereby extending battery life and safety.
• Computational Robustness: The proposed convex-splitting scheme allows for stable simulations of complex phase transitions without the severe time-step restrictions often associated with explicit methods or the stiffness of singular potentials.

In summary, the work presents a comprehensive framework that unifies thermodynamics, fluid mechanics, and heat transfer to model thermally driven phase separation, offering both rigorous mathematical proofs and practical numerical simulations for real-world engineering applications.