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Linear and Unconditionally Energy Stable Schemes for the binary Fluid-Surfactant Phase Field Model
Xiaofeng Yang, Lili Ju
TL;DR
The paper addresses efficient numerical solution of a binary fluid-surfactant phase-field model with coupled nonlinear Cahn-Hilliard equations and multiple nonlinear free-energy terms. It applies Invariant Energy Quadratization to construct first- and second-order linear schemes with semi-explicit nonlinear treatment. The schemes are proved unconditionally energy stable, yield symmetric positive definite systems, and are evaluated in two- and three-dimensional experiments.
Problem
The model combines thin-interface stiffness with strong nonlinear coupling among phase fields, making efficient unconditionally energy-stable discretization difficult.
Method
Invariant Energy Quadratization transforms the system using auxiliary variables so nonlinear terms can be treated semi-explicitly in first- and second-order linear time-marching schemes.
Results
The schemes are linear, unconditionally energy stable, and produce positive definite systems that can be solved efficiently.
Takeaways & Limitations
The approach provides accurate and efficient linear schemes for the targeted nonlinearly coupled binary fluid-surfactant phase-field model.
Takeaways & Limitations
The paper does not address hydrodynamics-coupled schemes, and error analysis for the multi-variable BFS-PF model is left for future work.
Abstract
from arXiv · showhide
In this paper, we consider the numerical solution of a binary fluid-surfactant phase field model, in which the free energy contains a nonlinear coupling entropy, a Ginzburg-Landau double well potential, and a logarithmic Flory-Huggins potential. The resulting system consists of two coupled, nonlinear Cahn-Hilliard type equations. We develop a set of first and second order time marching schemes for this system using the "Invariant Energy Quadratization" approach, in particular, the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi-explicitly. Both schemes are linear and lead to symmetric positive definite systems at each time step, thus they can be efficiently solved. We further prove that these schemes are unconditionally energy stable in the discrete sense. Various 2D and 3D numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes.
1. Introduction
The paper targets efficient, accurate, and unconditionally energy-stable schemes for a strongly coupled binary fluid-surfactant phase-field model. It uses IEQ to obtain linear time-stepping schemes whose systems are efficiently solvable and whose accuracy and stability are tested numerically.
- Background: Binary fluid-surfactant models use phase fields for fluid densities and surfactant concentration, with Ginzburg-Landau and nonlinear coupling energy terms.The model also includes a logarithmic Flory-Huggins potential in the paper’s formulation.
- Challenges: Thin-interface stiffness makes simple implicit or explicit discretizations conditionally energy stable and imposes severe time-step constraints.The strong nonlinear coupling among multiple phase-field variables further complicates algorithm development.
- Contribution: The paper adopts Invariant Energy Quadratization to develop first- and second-order schemes that are accurate, unconditionally energy stable, linear, and efficient to implement.The schemes are designed for a representative nonlinear coupled multivariate fluid-surfactant model.
- Contribution: The proposed schemes produce fully linear systems at each time step, while the paper reports symmetric positive definiteness and unconditional energy stability.The paper presents these properties as targets of the scheme design and discusses their rigorous analysis.
- Validation: Two- and three-dimensional numerical experiments validate the accuracy and energy stability of the proposed schemes.The numerical experiments are presented as part of the paper’s validation strategy.
2. The BFS-PF model and its energy law
The BFS-PF model describes coupled fluid and surfactant phase fields through a free energy combining interface, surfactant, and coupling contributions. Its variational formulation yields coupled Cahn-Hilliard equations with a continuous energy law that the numerical schemes aim to reproduce discretely.
- Phase-field variables: The model uses φ for fluid densities and ρ for local surfactant concentration in a binary fluid-surfactant phase-field description.The interface is represented by the zero level set of φ, with a transition layer of width O(ϵ).
- Free energy: The mixing free energy uses a Ginzburg-Landau double-well potential to represent competing mixing and separation interactions.Their competition produces a diffuse interface at equilibrium.
- Free energy: The surfactant contribution uses the logarithmic Flory-Huggins potential G(ρ) = ρlnρ + (1 −ρ)ln(1 −ρ), restricting ρ to (0, 1).The potential reflects surfactant concentration approaching saturation at the interface.
- Free energy: A nonlinear coupling entropy term penalizes the system so surfactant concentration accumulates near the fluid interface.The coupling is introduced because surfactants alter interfacial tension and preferentially remain at the interface.
- Energy law: The variational formulation produces a nonlinear coupled Cahn-Hilliard system, and summing the relevant L2 inner products yields its PDE energy law.The schemes are designed to satisfy a discrete version of this continuous law under periodic boundary conditions.
- Scope: The paper excludes hydrodynamic coupling from the present numerical treatment because coupling velocity and phase-variable computations presents further challenges.Hydrodynamics are identified as a subject for future work.
3. Numerical schemes
The paper uses IEQ to transform the coupled BFS-PF model into an equivalent quadratic-energy system, enabling linear first- and second-order schemes with unconditional discrete energy stability.
- Challenges: The schemes address cubic Ginzburg-Landau, logarithmic Flory-Huggins, and nonlinear coupling-entropy terms in the BFS-PF model.These are identified as the principal discretization challenges for constructing linear, unconditionally energy-stable methods.
- Linear schemes: All nonlinear coefficients of the auxiliary variables are treated explicitly, producing linear time-stepping systems rather than nonlinear solves.The schemes are designed for temporal discretization of the transformed system and use semi-explicit treatment of nonlinear terms.
- Invariant Energy Quadratization: The regularized Flory-Huggins potential replaces the logarithmic functional with a C2 continuous, convex, piecewise function, with the approximation error controlled by bϵ.The regularized formulation is used for the numerical model, with bϵ tending to zero recovering the original potential.
- Invariant Energy Quadratization: IEQ introduces auxiliary variables for the nonlinear potentials, transforming the free energy and PDE system into an equivalent quadratic form.The auxiliary variables represent the square roots of F(φ), (ρ − |∇φ|)^2, and G(ρ)+B, while preserving equivalence with the original system.
- First-order scheme: The first-order scheme has a symmetric positive definite linear system and satisfies an unconditional discrete energy dissipation law.The resulting system admits a unique weak solution, and the unconditional stability theorem applies without a time-step restriction.
- Second-order scheme: The second-order scheme is unconditionally energy stable, and its discrete energy law approximates the continuous dissipation law to second order.The auxiliary variables remain first-order approximations in the first-order construction, while the second-order energy relation is justified through a second-order temporal expansion.
- Analysis scope: The analysis notes that optimal error estimates for the multi-variable BFS-PF model are more complicated than for the classical Cahn-Hilliard equation.The stability derivation is formal, with the additional coupling and multiple variables complicating error analysis.
4. Numerical experiments
The numerical experiments test convergence, energy stability, and phase-separation dynamics in two and three dimensions. The schemes achieve their stated temporal orders, while simulations show energy decay and composition-dependent coarsening behavior.
- Experimental setup: The Fourier-spectral discretization uses 129^d modes, making spatial errors negligible relative to time-discretization errors.
- Accuracy test: LS1 and LS2 are first- and second-order accurate, respectively, with LS2 more accurate than LS1 at the same time step.Errors are measured at t = 0.5 against an LS2 solution with δt = 7.8125×10^-5 used as an approximate benchmark.
- 2D spinodal decomposition: All five tested time steps produce decaying free-energy curves, supporting unconditional stability, although δt = 0.01 gives noticeably different accuracy.The tested values are δt = 0.00005, 0.0001, 0.0005, 0.001, and 0.01.
- 2D spinodal decomposition: For ¯φ0 = 0, initially mixed fluids decompose and form a banded equilibrium by t = 1500, with relatively high ρ at the interface.
- 2D spinodal decomposition: For ¯φ0 = 0.3, the less abundant fluid forms satellite drops that collide and merge into one large bubble by t = 1500.
- 3D spinodal decomposition: Three-dimensional simulations show coarsening from mixed or separated states, while energy decreases monotonically in both cases.For ¯φ0 = 0.3, the fluids decompose into small drops and evolve toward accumulation; the energy curves decay monotonically.
- Surfactant distribution: With a uniformly distributed initial interface and surfactant, the surfactant concentrates near interfaces and the final steady shape is a large drop.
- Surfactant distribution: When surfactant is initially concentrated at the center, diffusion away from that region takes longer, but it is absorbed into the binary-fluid interfaces by t = 500.
5. Concluding remarks
The paper develops efficient first- and second-order linear schemes for the binary fluid-surfactant phase-field model using a novel IEQ approach. The schemes are accurate, unconditionally energy stable, and produce positive-definite linear systems suitable for efficient solution.
- The proposed first- and second-order schemes are linear, accurate, and unconditionally energy stable.
- The schemes generate positive-definite linear systems that can be solved efficiently with Krylov subspace methods and mass-lumping preconditioners.
- 3D simulations examine spinodal decomposition for initial values ¯φ0 = 0 and ¯φ0 = 0.3 using phase-field isosurfaces over time.
- The equilibrium isosurfaces are compared for the two initial values ¯φ0 = 0 and ¯φ0 = 0.3.
- The free-energy curves decay for all time steps in the 3D spinodal-decomposition cases, confirming unconditional stability of the algorithm.