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Toward a Physical Interpretation of Phase Field Models with Dynamic Boundary Conditions

Xiaobo Jing, Qi Wang

TL;DR

This work clarifies the physical interpretation of dynamic boundary conditions in phase-field models by formulating bulk–surface exchange within a closed system and grounding it in nanothermodynamics. Using the generalized Onsager principle, it derives four thermodynamically consistent models that couple bulk and surface dynamics through a mobility operator, showing how mass/volume conservation and energy dissipation constrain both the mobility and the free energy. A key insight is the introduction of the dimensionless parameter $\beta$, linked to a characteristic length, which governs material exchange between the bulk and surface and recovers classical limits when $\beta \to 0$ or $\infty$. Numerical simulations demonstrate that reversible exchanges (encoded by $\mathbf{M}^{(a)}$) and the value of $\beta$ markedly influence pattern formation and energy dissipation, providing a framework for analyzing finite-size effects and guiding extensions to hydrodynamic systems with dynamic boundaries.

Abstract

In recent decades, considerable research has been devoted to partial differential equations (PDEs) with dynamic boundary conditions. However, the physical interpretation of the parameters involved often remains unclear, which in turn limits both theoretical analysis and numerical computation. For instance, the Robin boundary condition used in thermodynamically consistent models with dynamic boundary conditions has been misinterpreted as representing a chemical reaction, or has been generalized in an unjustified manner in numerous works. In this paper, we treat the bulk and surface as a closed system and develop thermodynamically consistent phase field models to clarify the physical meaning of parameters in governing equations and boundary conditions, with particular focus on material and energy exchange between the bulk and surface by connecting it with the nanothermodynamics. Firstly, we commence with the mass and volume conservation law in the close system and elucidate the physical interpretation of the Robin boundary condition, demonstrating that the relevant parameters are connected to the system's characteristic length scale and play a crucial role on the exchanging of material and energy. Furthermore, our analysis justifies the physical necessity for the phase variable in the bulk to differ from that on the surface. Secondly, we construct four more general models capable of describing both irreversible and irreversible-reversible coupling processes using the generalized Onsager principle. Thirdly, we reveal that both conservation and dissipation laws simultaneously determine the mobility operator and free energy, which are two dual variables. Finally, we perform structure-preserving numerical simulations to systematically investigate how reversible processes and characteristic length affect pattern formation.

Toward a Physical Interpretation of Phase Field Models with Dynamic Boundary Conditions

TL;DR

This work clarifies the physical interpretation of dynamic boundary conditions in phase-field models by formulating bulk–surface exchange within a closed system and grounding it in nanothermodynamics. Using the generalized Onsager principle, it derives four thermodynamically consistent models that couple bulk and surface dynamics through a mobility operator, showing how mass/volume conservation and energy dissipation constrain both the mobility and the free energy. A key insight is the introduction of the dimensionless parameter , linked to a characteristic length, which governs material exchange between the bulk and surface and recovers classical limits when or . Numerical simulations demonstrate that reversible exchanges (encoded by ) and the value of markedly influence pattern formation and energy dissipation, providing a framework for analyzing finite-size effects and guiding extensions to hydrodynamic systems with dynamic boundaries.

Abstract

In recent decades, considerable research has been devoted to partial differential equations (PDEs) with dynamic boundary conditions. However, the physical interpretation of the parameters involved often remains unclear, which in turn limits both theoretical analysis and numerical computation. For instance, the Robin boundary condition used in thermodynamically consistent models with dynamic boundary conditions has been misinterpreted as representing a chemical reaction, or has been generalized in an unjustified manner in numerous works. In this paper, we treat the bulk and surface as a closed system and develop thermodynamically consistent phase field models to clarify the physical meaning of parameters in governing equations and boundary conditions, with particular focus on material and energy exchange between the bulk and surface by connecting it with the nanothermodynamics. Firstly, we commence with the mass and volume conservation law in the close system and elucidate the physical interpretation of the Robin boundary condition, demonstrating that the relevant parameters are connected to the system's characteristic length scale and play a crucial role on the exchanging of material and energy. Furthermore, our analysis justifies the physical necessity for the phase variable in the bulk to differ from that on the surface. Secondly, we construct four more general models capable of describing both irreversible and irreversible-reversible coupling processes using the generalized Onsager principle. Thirdly, we reveal that both conservation and dissipation laws simultaneously determine the mobility operator and free energy, which are two dual variables. Finally, we perform structure-preserving numerical simulations to systematically investigate how reversible processes and characteristic length affect pattern formation.

Paper Structure

This paper contains 16 sections, 41 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The exchange of material, conservation law and dissipation law
  3. Mass conservation law with the exchange of material between bulk and surface
  4. Volume conservation law with the exchange of material between the bulk and surface
  5. Energy dissipation law with the exchange of energy between bulk and surface
  6. Generalized Onsager principle, dynamics and energy dissipation
  7. Generalized Onsager principle
  8. Dynamics in the bulk
  9. Dynamics on the boundary
  10. Thermodynamically consistent phase filed models with dynamic boundary conditions
  11. Model A for irreversible transport system
  12. Model B for irreversible-reversible transport system
  13. Model C for irreversible reactive transport system
  14. Model D for irreversible-reversible reactive transport system
  15. Numerical simulation
  16. ...and 1 more sections

Figures (6)

  • Figure 5.1: Time evolution of the spinodal decomposition of the binary fluid system with $M_{23}=0$ (a-d) and $M_{23}=1\times 10^{-5}$ (e-h), respectively. Snapshots of numerical solutions $\phi$ are taken at $t=0,20,50,100$ for the two cases, respectively. Dynamic boundary effects on the bulk solution near $\Gamma$ are observed in both cases and $M_{23}^{(s)}$ can affect the phase separation significantly.
  • Figure 5.2: Time evolutions of the free energy in total, free energy in bulk and free energy on surface, the free energy in total dissipates with time, however, the free energy in bulk is not monotonic decreasing with time. The numerical results are coincident with our theory.
  • Figure 5.3: Time evolutions of the mass in total, bulk mass and surface mass, the mass in total is conserved, in meanwhile, the mass in bulk and on the surface are not monotonic with time. The numerical results are coincident with our theory, this is attribute to the structure-preserving numerical algorithms are used.
  • Figure 5.4: Time evolutions of the spinodal decomposition of the binary fluid system with $\beta =1$ in (a-c), $\beta =1\times 10^{-3}$ in (d-f) and $\beta =10$ in (g-i) at $t=7,50,100$, respectively. The effects of characteristic length on the patterns are observed in these three cases.
  • Figure 5.5: Time evolutions of the free energy in total, free energy in bulk and free energy on surface with $\beta=1, \beta=1\times 10^{-3}, 10$. The total energy is dissipated in three cases, the bulk enengy and surface energy are not always monotonic with time (for example, the bulk energy with $\beta=10$ increases at the first stage and then decreases with time). The numerical results are coincident with our theory.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9