Phase-field dynamics with transfer of materials: The Cahn-Hillard equation with reaction rate dependent dynamic boundary conditions

The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.

[1]  Songmu Zheng,et al.  The Cahn-Hilliard equation with dynamic boundary conditions , 2003, Advances in Differential Equations.

[2]  Bernd Rinn,et al.  Phase separation in confined geometries: Solving the Cahn–Hilliard equation with generic boundary conditions , 2001 .

[3]  Pierluigi Colli,et al.  Equation and dynamic boundary condition of Cahn-Hilliard type with singular potentials , 2015, 1502.05159.

[4]  Matthias Liero,et al.  Passing from bulk to bulk-surface evolution in the Allen–Cahn equation , 2013 .

[5]  Stefan Metzger,et al.  On convergent schemes for two-phase flow of dilute polymeric solutions , 2018, ESAIM: Mathematical Modelling and Numerical Analysis.

[6]  Paula Harder,et al.  Error estimates for the Cahn-Hilliard equation with dynamic boundary conditions , 2020, ArXiv.

[7]  Hao Wu,et al.  Asymptotic behavior of a Cahn-Hilliard equation with Wentzell boundary conditions and mass conservation , 2008 .

[8]  Zheng Songmu,et al.  Asymptotic behavior of solution to the Cahn-Hillard equation , 1986 .

[9]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[10]  Taishi Motoda,et al.  Time periodic solutions of Cahn-Hilliard systems with dynamic boundary conditions , 2018 .

[11]  Philipp Maass,et al.  Novel Surface Modes in Spinodal Decomposition , 1997 .

[12]  A. Buffa,et al.  On traces for H(curl,Ω) in Lipschitz domains , 2002 .

[13]  Alain Miranville,et al.  Higher-order Cahn–Hilliard equations with dynamic boundary conditions , 2017 .

[14]  Helmut Abels,et al.  Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy , 2007 .

[15]  Pierluigi Colli,et al.  Cahn-Hilliard equation with dynamic boundary conditions and mass constraint on the boundary , 2014, 1412.1932.

[16]  Hao-qing Wu,et al.  Convergence to Equilibrium for the Cahn-Hilliard Equation with Wentzell Boundary Condition , 2004, 0705.3362.

[17]  Takeshi Fukao,et al.  Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn–Hilliard type with singular potential , 2019, Asymptotic Analysis.

[18]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[19]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[20]  Günther Grün,et al.  On Modeling and Simulation of Electrokinetic Phenomena in Two-Phase Flow with General Mass Densities , 2012, SIAM J. Appl. Math..

[21]  Alain Miranville,et al.  The Cahn–Hilliard Equation: Recent Advances and Applications , 2019 .

[22]  Sergey Zelik,et al.  The Cahn-Hilliard Equation with Logarithmic Potentials , 2011 .

[23]  Robert Nürnberg,et al.  Linear Functional Analysis: An Application-Oriented Introduction , 2016 .

[24]  Piotr Rybka,et al.  Convergence of solutions to cahn-hilliard equation , 1999 .

[25]  Chun Liu,et al.  An Energetic Variational Approach for the Cahn–Hilliard Equation with Dynamic Boundary Condition: Model Derivation and Mathematical Analysis , 2017, Archive for Rational Mechanics and Analysis.

[26]  L. Ambrosio,et al.  Chapter 1 – Gradient Flows of Probability Measures , 2007 .

[27]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[28]  Philipp Maass,et al.  Time-dependent density functional theory and the kinetics of lattice gas systems in contact with a wall , 1998 .

[29]  C. M. Elliott,et al.  Finite element analysis for a coupled bulk-surface partial differential equation , 2013 .

[30]  Kei Fong Lam,et al.  Convergence of a Robin boundary approximation for a Cahn–Hilliard system with dynamic boundary conditions , 2019, Nonlinearity.

[31]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[32]  Hao Wu,et al.  Convergence to equilibrium for a Cahn-Hilliard model with the Wentzell boundary condition , 2007, Asymptot. Anal..

[33]  Sergey Zelik,et al.  THE CAHN-HILLIARD EQUATION WITH SINGULAR POTENTIALS AND DYNAMIC BOUNDARY CONDITIONS , 2009, 0904.4023.

[34]  Stefan Metzger.  On stable, dissipation reducing splitting schemes for two-phase flow of electrolyte solutions , 2018, Numerical Algorithms.

[35]  Craig Cowan,et al.  The Cahn-Hilliard Equation as a Gradient Flow , 2005 .

[36]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[37]  Stefan Metzger,et al.  On Fully Decoupled, Convergent Schemes for Diffuse Interface Models for Two-Phase Flow with General Mass Densities , 2016 .

[38]  Patrik Knopf,et al.  On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems , 2021, Interfaces and Free Boundaries.

[39]  Pierluigi Colli,et al.  On the Cahn–Hilliard equation with dynamic boundary conditions and a dominating boundary potential , 2014, 1403.1663.

[40]  Charles M. Elliott,et al.  Finite element methods for surface PDEs* , 2013, Acta Numerica.

[41]  Robbins,et al.  Simulations of contact-line motion: Slip and the dynamic contact angle. , 1989, Physical review letters.

[42]  Robert L. Pego,et al.  Front migration in the nonlinear Cahn-Hilliard equation , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[43]  Peter W. Bates,et al.  The Dynamics of Nucleation for the Cahn-Hilliard Equation , 1993, SIAM J. Appl. Math..

[44]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[45]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[46]  Stefan Metzger,et al.  An efficient and convergent finite element scheme for Cahn-Hilliard equations with dynamic boundary conditions , 2019, SIAM J. Numer. Anal..

[47]  Ciprian G. Gal,et al.  A Cahn–Hilliard model in bounded domains with permeable walls , 2006 .

[48]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[49]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[50]  C. M. Elliott,et al.  On the Cahn-Hilliard equation with degenerate mobility , 1996 .

[51]  P. Colli,et al.  A class of quasi-linear Allen-Cahn type equations with dynamic boundary conditions , 2016, 1610.08687.

[52]  Alain Miranville,et al.  A Cahn-Hilliard model in a domain with non-permeable walls , 2011 .

[53]  Harald Garcke,et al.  On Cahn—Hilliard systems with elasticity , 2003, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[54]  Harald Garcke,et al.  Weak Solutions of the Cahn-Hilliard System with Dynamic Boundary Conditions: A Gradient Flow Approach , 2018, SIAM J. Math. Anal..

[55]  Charles M. Elliott,et al.  On the Cahn-Hilliard equation , 1986 .

[56]  Günther Grün,et al.  Two-phase flow with mass density contrast: Stable schemes for a thermodynamic consistent and frame-indifferent diffuse-interface model , 2012, J. Comput. Phys..