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A thermodynamically consistent phase-field model for mass transport with interfacial reaction and deformation

Zhaoyang Wang, Huaxiong Huang, Ping Lin, Shixin Xu

TL;DR

The paper develops a thermodynamically consistent diffuse-interface (phase-field) model coupling mass transport, interfacial reactions, and interface deformation under flow. Using an energy variational framework, it derives a coupled Cahn–Hilliard–Navier–Stokes system with a reaction-diffusion component, ensuring mass conservation and energy dissipation at the continuous level. It then constructs a first-order, structure-preserving time discretization and a fully discrete finite element scheme that inherit these thermodynamic properties, accompanied by rigorous error analysis for a simplified setting. Numerical experiments validate accuracy, stability, and applicability to vascular problems, including straight and bifurcated microvessels that reveal microaneurysm formation driven by AGE–glucose chemistry and hemodynamics. The framework is extensible to related interfacial phenomena (e.g., surfactants, corrosion) and can be integrated with vesicle dynamics for broader biological and engineering applications.

Abstract

In this paper, a thermodynamically consistent phase-field model is proposed to describe the mass transport and reaction processes of multiple species in a fluid. A key feature of this model is that reactions between different species occur only at the interface, and may induce deformation of the interface. For the governing equations derived based on the energy variational method, we propose a structure-preserving numerical scheme that satisfies the mass conservation and energy dissipation laws at the discrete level. Furthermore, we carry out a rigorous error analysis of the time-discrete scheme for a simplified case. A series of numerical experiments are conducted to validate the effectiveness of the model as well as the accuracy and stability of the scheme. In particular, we simulate microvessels with straight and bifurcated structures to illustrate the risk of microaneurysm formation.

A thermodynamically consistent phase-field model for mass transport with interfacial reaction and deformation

TL;DR

The paper develops a thermodynamically consistent diffuse-interface (phase-field) model coupling mass transport, interfacial reactions, and interface deformation under flow. Using an energy variational framework, it derives a coupled Cahn–Hilliard–Navier–Stokes system with a reaction-diffusion component, ensuring mass conservation and energy dissipation at the continuous level. It then constructs a first-order, structure-preserving time discretization and a fully discrete finite element scheme that inherit these thermodynamic properties, accompanied by rigorous error analysis for a simplified setting. Numerical experiments validate accuracy, stability, and applicability to vascular problems, including straight and bifurcated microvessels that reveal microaneurysm formation driven by AGE–glucose chemistry and hemodynamics. The framework is extensible to related interfacial phenomena (e.g., surfactants, corrosion) and can be integrated with vesicle dynamics for broader biological and engineering applications.

Abstract

In this paper, a thermodynamically consistent phase-field model is proposed to describe the mass transport and reaction processes of multiple species in a fluid. A key feature of this model is that reactions between different species occur only at the interface, and may induce deformation of the interface. For the governing equations derived based on the energy variational method, we propose a structure-preserving numerical scheme that satisfies the mass conservation and energy dissipation laws at the discrete level. Furthermore, we carry out a rigorous error analysis of the time-discrete scheme for a simplified case. A series of numerical experiments are conducted to validate the effectiveness of the model as well as the accuracy and stability of the scheme. In particular, we simulate microvessels with straight and bifurcated structures to illustrate the risk of microaneurysm formation.

Paper Structure

This paper contains 15 sections, 3 theorems, 74 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Phase-field model with reactions at the interface
  3. Model derivation
  4. Mass conservation and energy law
  5. Structure-preserving numerical scheme
  6. Time-discrete scheme
  7. Fully discrete finite element scheme
  8. Error estimates
  9. Numerical simulations
  10. Convergence and stability tests
  11. Adsorption test under shear flow
  12. Arterial microaneurysm
  13. Straight microvessel
  14. Bifurcated microvessel
  15. Conclusions and remarks

Key Result

Theorem 2.1

\newlabeltheorem2.10 The total mass of system (the2.23) is conserved and satisfies the following energy dissipation law where

Figures (11)

  • Figure 1: Evolution of total mass and total energy with $\Delta t=10^{-2}$, $\Delta t=10^{-3}$ and $\Delta t=10^{-4}$.
  • Figure 2: Snapshots of $\phi$ at different times with $M=N=0$.
  • Figure 3: Snapshots of $Y_2$ at different times with $M=N=0$.
  • Figure 4: Snapshots of $Y_3$ at different times with $M=N=0$.
  • Figure 5: Snapshots of $Y_2$ at different times with $M=N=1$.
  • ...and 6 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Proof 1
  • Theorem 3.1
  • Proof 2
  • Remark 3.2
  • Remark 4.1
  • Theorem 4.3
  • Proof 3