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The phase eld model is a general name for a class of diuse interface models used to study a wide variety of materials phenomena. It has several advantages over other interface tracking approaches, therefore it has been used to model general multi-phase systems with features much larger than the real interface thicknesses.

The phase-eld method, as presented here, grows out of the work of Cahn, Hilliard and Allen. It is used for two general purposes:

• to model systems in which the diuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion;

• as a front tracking technique to model general multi-phase systems.

Generally, we speak about two types of phase eld models. In the rst, called Cahn-Hilliard, the phase is uniquely determined by the value of a conserved eld variable, such as the concentration C, e.g. if C ≤ C1, then we are in one phase, if C ≥ C2 then the other. These models were rst applied to understand spin- odal decomposition, and are now used for a wide range of phenomena. In the second, called Allen-Cahn, the phase is not uniquely determined by concentra- tion, temperature, pressure, etc., so we add one or more extra eld variable(s) sometimes called the order parameter φ which determines the local phase. This class of models is widely used to study solidication and solid-state phase trans- formations in metals.

The phase-eld method is a xed-grid method; it diers from other methods in that the interface is diuse in a physical rather than numerical sense. Thus, it is also known as the diuse-interface model. More precisely, the diuse inter- face is introduced through an energetic variational procedure that results in a thermodynamic consistent coupling system. The basic idea was derived from the consideration that the two components, though nominally immiscible, does mix in reality within a narrow interfacial region. A phase-eld variable φ can be thought of as the volume fraction, to demarcate the two species and indicate the location of the interface. A mixing energy is dened based on φ which, through

a convection-diusion equation, governs the evolution of the interfacial prole. The phase-eld method can be viewed as a physically motivated level-set method. When the thickness of the interface approaches zero, the diuse-interface model becomes asymptotically identical to a sharp-interface level-set formulation. It also reduces properly to the classical sharp-interface model in general.

From the statistical (phase eld approach) point of view, the interface represents a continuous, but steep change of the properties (density, viscosity, etc.) of two uids. Within this "thin" transitional region, the uid is mixed. The mixing is determined by molecular interactions between the two species, and can be described by a stored mixing energy, which represents the balance between the competing phobic/philic relation

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- 1. Oleksiy Varfolomiyev Oleksiy Varfolomiyev | Dortmund Introduction to the Phase Field Method Allen-Cahn vs. Cahn-Hilliard Model Supervisor : Prof. S.Turek LSIII, TU Dortmund
- 2. What for? <ul><li>The phase-field method (PFM), as presented here, grows out of the work of Cahn, Hilliard and Allen </li></ul><ul><li>It is used for two general purposes: </li></ul><ul><li>to model systems in which the diffuse nature of interfaces is essential to the problem, such as spinodal decomposition and solute trapping during rapid phase boundary motion; </li></ul><ul><li>as a front tracking technique to model general multi-phase systems . </li></ul>
- 3. PFM Applications Multiphase Systems Spinodal Decomposition
- 4. PFM Applications Atomization
- 5. PFM Applications Dynamics of drop formation from a capillary tube: inkjet printing
- 6. Two types of phase field models <ul><li>Cahn Hillard </li></ul><ul><li>Phase is uniquely determined by the value of a conserved field variable , e.g. concentration </li></ul><ul><li>C < C1 we are in one phase </li></ul><ul><li>C > C2 we are in the other </li></ul><ul><li>Allen -Cahn </li></ul><ul><li>Phase is not uniquely determined by concentration, temperature, </li></ul><ul><li>pressure, etc. </li></ul><ul><li>We define the order parameter field variable to determine the phase, φ </li></ul>Oleksiy Varfolomiyev | Dortmund
- 7. Models Cahn-Hilliard Free Energy Allen-Cahn
- 8. Models Cahn-Hilliard Free Energy Allen-Cahn Because C is locally conserved, according to Fick‘s second law Double-well potential Because is not conserved
- 9. Models Cahn-Hilliard Free Energy Allen-Cahn Because C is locally conserved, according to Fick‘s second law Define potential Constitutive equation Denote Double-well potential Because is not conserved
- 10. Allen-Cahn Equation Cahn-Hilliard Equation Lagrange multiplier
- 11. Allen-Cahn Equation Cahn-Hilliard Equation Lagrange multiplier Momentum equation with continuity condition
- 12. Allen-Cahn Equation Cahn-Hilliard Equation Lagrange multiplier Momentum equation with continuity condition +IC & BC
- 13. Allen-Cahn-Hilliard-Navier-Stokes Problems Initial conditions Allen-Cahn Problem Cahn-Hilliard Problem Initial conditions Boundary conditions Boundary conditions
- 14. Solver for the CHNS Problem Step 0: Step1: A projection method on a fixed half-staggered mesh Half-staggered mesh
- 15. Solver for the CHNS Problem Step2: (the projection step) Project the intermediate velocity field onto the divergence-free vector space Step 0: Step1: A projection method on a fixed half-staggered mesh Update the pressure Half-staggered mesh
- 16. Solver for the CHNS Problem Step2: (the projection step) Project the intermediate velocity field onto the divergence-free vector space Step 0: Step1: A projection method on a fixed half-staggered mesh Update the pressure Half-staggered mesh Pressure-Poisson Equation
- 17. Step 3 (the phase evolution step): Compute the phase field by
- 18. Solver for the CHNS Problem Step 3 (the phase evolution step): Compute the phase field by Simulation Analysis
- 19. Simulation – Surface Tension
- 20. Simulation – 2 Kissing Bubbles
- 21. Adaptive vs fixed mesh method
- 22. Inference <ul><li>Conclusion </li></ul><ul><li>The phase-field method is a very versatile and robust method for studying interfacial motion in multi-component flows. It casts geometric evolution in Lagrangian coordinates into an Eulerian formulation, and provides a way to represent surface effects as bulk effects. The whole process allows us to use an energetic variational formulation that makes it possible to ensure the stability of corresponding numerical algorithms. The elastic relaxation built into the phase-field dynamics prevents the interfacial mixing layer from spreading out. Moreover, being a physically motivated approximation based on the competition between different parts of the energy functionals, the phase-field model can be adapted easily to incorporate more complex physical phenomena such as Marangoni effect and non-Newtonian rheology. </li></ul>
- 23. The End Vielen Dank für Ihre Aufmerksamkeit!
- 24. References <ul><li>Literature </li></ul><ul><li>1 Adam Powell, Introduction to Phase Field Method, Group Seminar, September 5, 2002 </li></ul><ul><li>2 Xiaofeng Yang, James J. Feng, Chun Liu, Jue Shen, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, Journal of Computational Physics 218 (2006) pp.417-428 </li></ul><ul><li>3 Chun Liu, Jie Shen, A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method, Physica D 179 (2003) pp.211-228 </li></ul><ul><li>4 James J. Feng, Chun Liu, Jie Shen, Pengtao Yue, An energetic variational formulation with phase field methods for interfacial dynamics of complex fluids - advantages and challenges, In Modeling of soft matter, vol. 141 of IMA Vol. Math Appl., pp.1-26, Springer, New York, 2005 </li></ul><ul><li>5 Yana Di, Ruo Li, Tao Tang, A General Moving Mesh Framework in 3D and its Application for Simulating the Mixture of Multi-Phase Flows, Communications in Computational Physics, Vol. 3, No.3, pp.582-602 </li></ul><ul><li>6 C.M. Elliott, D.A. French, and F.A. Milner, A Second Order Splitting Method for the Cahn-Hillard Equation, Numer Math. 54, 575-590 (1989) </li></ul><ul><li>7 David Kay, Richard Welford, A Multigrid Finite Element Solver for the Cahn-Hilliard Equation, Journal of Computational Physics, Volume 212, Issue 1, (2006), pp.288-304 </li></ul><ul><li>8 David Kay, Richard Welford, Efficient Numerical Solution of Cahn-Hillard-Navier-Stokes Fluids in 2D, SIAM J. Sci. Comput. Vol 29, No. 6, pp. 2241-2257 </li></ul><ul><li>9 C.M. Elliott, The Cahn-Hillard model for the kinetics of phase separation, in Mathematical Models for Phase Problems, Internat, Ser. Numer. Math. 88, Birkhäuser-Verlag, Basel, Swi tzerland, 1989, pp. 35-73 </li></ul><ul><li>10 Zhengru Zhang, Huazhong Tang, An adaptive phase field method fort he mixture of two incompressible fluids, Computers & Fluids 36, (2007), pp.1307-1318 </li></ul>

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