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Phase Field Modeling and Simulations of Interface Problems - a Tutorial on Basic Ideas and Selected Applications

Qiang Du Department of Mathematics Pennsylvania State University

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Collaborators •  Longqing Chen, Zikui Liu, Padma Raghvan (PSU), Chris Wolverton (Ford/NWU), Steve Langer (NIST), Maria Emelianenko (GMU), Lei Zhang (PKU), Taewok Heo (LANL), Shenyang Hu (PNNL), Knuok Chung (Leuven), Sheng Guang , Jingyan Zhang, Weiming Feng, Tao Wang (Ames Lab), Materials simulations/design NSF-IUCRC, NSF-DMR,DOE •  Chun Liu, Cheng Dong, Maggie Slattery (PSU), Xiaoqiang Wang (FSU), Jian Zhang (CAS), Sovan Das (IIT), Manlin Li (Microsoft), Yanxiang Zhao (UCSD), Yanping Ma (LMU), Meghan Hoskins, Rob Kunz (ARL), Rolf Ryham (Fordham), Liyong Zhu (BUAA) Complex/biological fluids NSF-DMS, NSF-CCF, NIH-NCI

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Contributions from former PhD students •  Xiaoqiang Wang (FSU), membrane/vesicle •  Maria Emelianenko (GMU), phase diagram •  Jiakou Wang (Citi), cell aggregation •  Lei Zhang (PKU), nucleation •  Manlin Li (Microsoft), fluid-membrane •  Yanxiang Zhao (UCSD), membrane/adhesion •  Liyong Zhu (BUAA), membrane •  Yanping Ma (LMU), cell aggregation •  Jingyan Zhang (NCCM) nucleation

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Outline: basic ideas and selected applications   Motivation and Overview   Phase Field/Diffuse Interface Models

  Interfae problems   Phase field/diffuse interface models   Variational problems   Gradient flows   Coupling with external fields   Stochastic fluctuation

  Numerical Methods   Time-stepping and spatial discretizations, adaptive methods,   Spectral methods, moving mesh spectral methods

  Other Multiscale Modeling and Simulations Issues

This is not intended to be a comprehensive review of all relevant works, nor systematic studies of particular topics, we aim at presenting to beginners some basic ideas on modeling, analysis and simulation issues through selected examples

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Examples of interface:

Device

Edgerton

Wikipedia

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Complex/biological fluid •  Experiments/Analysis/Modeling/Simulations

–  membrane –  protein –  actin –  cell …

–  air –  water –  shampoo –  blood …

Courtesy of Dong’s lab

Courtesy of Pritchard lab

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Cell level: RBCs/vesicles in fluid • Experimental works

• Modeling/simulations

Tsukada et al 2001 Shelby et al 2003

Noguchi-Gompper 2005

Abkarian-Faivre-Stone 2006

Du-Liu-Ryham-Wang 2006

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Application: tumor metastasis •  Tumor cell adhesion and migration

Alberts et al., 1994

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Problems under consideration (a joint NIH project with Dong/Kunz)

Do PMNs promote metastasis of cancer cells? •  Reports on the increase in tumor cell adhesion in the

presence of leukocytes Starkey 1984,… •  Experimental works: Neeson et al. (2003) Wu et al. (2001) Pollard et al (2004) Welch et al. (1989)

Recent studies (including ours): dependence on flow conditions

Leukocytes/EC adhesion: rolling, tethering; TCs do not roll like leukocytes

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Method •  Experiments/Analysis/Modeling/Simulation(TEAMS) •  Coupling in vitro experiments

and numerical simulations

WBC

Wells for Chemoattractant

TC Flow in

Top Plate Flow out

Porous membrane Cellular Monolayer

(Penn State U, Dong Lab)

Flow Migration Chamber

Parallel flow chamber experiments show: ratio of TC/PMN population affects TC extravasation

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Multiscale aggregation process •  Initiation at nanoscale:

molecular bridging/depletion between cells •  Deformation at microscale:

shape change of individual cells •  Rheology at macroscale:

Interaction with flow, cell density statistics:

Statistical and multi-scale modeling and simulation of heterotypic cell population, coupled with CFD studies of aggregation of deformable cells, near wall cell aggregations in non-uniform shear flow, cell aggregation and adhesion to the endothelium

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Modeling multi-scales/multi-processes –  cellular level models

fluid-cell/fluid-membrane interaction phase-field Navier-Stokes equations

–  micro-macro models polymeric fluid with given interaction potential FENE dumbbell models –  statistical model cell density distribution in shear flow coagulation/population balance equations

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Interface of biology and mathematics

The biconcave shape increases their surface area, which is important in increasing the rate of diffusion as they transport O2 and CO2

Why red blood cells are biconcave in shape?

Per unit-volume, given a fixed surface area, what is the optimal shape of a cell? “Optimal”? energetic considerations (bending energy) lead to a minimal surface problem

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Diffuse Interface Description of Surfaces/Interfaces

•  A popular approach for free/moving interface problems

•  Sharp interfaces diffuse interfaces characterized by some order parameters (phase field functions)

Eg: phase field simulations of microstructure evolution (Yu-Hu-Chen-Du, JCP 2005)

•  Idea goes back to van de Waals Ginzburg-Landau, Cahn-Hilliard, Halperin-Hohenberg,…

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Diffuse Interface/Phase Field To describe an interface Γ, a smooth phase field function φ is used to label the two sides, with nearly constant values except in a thin (diffuse) layer

Γ

φ ~ 1 φ ~ -1 ε φ

ε Γ

+1

-1

•  Interface Γ: zero level set of φ diffuse interfacial layer

An implicit surface representation

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Multiscale Modeling and Simulations

Materials Computation And Simulation Environment

Liu-Chen-Raghavan-Du-Sofo-Langer-Wolverton, 2004: An integrated Framework for multi-scale materials simulation and design, J. Computer Aided Materials Design

Microstructure evolution

Atomic structure

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Nanoscale Grain/Domain Structures in Ferroelectrics

(from L.Q. Chen)

Atomic

Macroscale/device level

ARRAY PERIPHERY

Bit Line

DriveLine

PZT

WordLineW

Domain

Grain

Device

Geometry and topology f microstructure control material property

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Implicit interface representation - any advantage?

A single set of equations to be solved throughout the domain, no need to track interface

φ=0 φ>0

φ<0

Interface with different topology is described by a single level set function

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Diffuse interface/Phase field: a geometric view

ε Γ

φ~1

φ~-1

Ω

Γ

•  How to describe the geometric features of Γ by φ ?

Volume (difference) :

Computational domain

Area:

(Cahn-Hilliard, Modica, Fonseca-Tartar, Rubinstein-Sternberg-Keller, Kohn, Gurtin, X.F. Chen, Elliott, Nochetto-Paulini-Verdi, Evans-Souganidis-Soner, …)

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Diffuse interface/Phase field: a geometric view

φ~1

φ~-1

Ω

Γ

•  A phase field description of isoparametric problem

•  Minimize surface area Subject to given volume

Min:

Subject to:

Phase field/Diffuse interface models

Why “phase field” ?

•  Concentration in a mixture: volume fraction, mass fraction •  State of matter (phase): like gas, liquid, solid •  Order parameter (measure of the degree of order in a system), eg:

crystal lattice configuration Why diffuse interface? •  Materials interface may not be sharp •  Numerically more difficult with sharp interface (such as formation of singularity, topological changes)

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Deckelnick-Dziuk-Elliott

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Phase Field/Diffuse Interface Model Idea goes back to van de Waals, Ginzburg-Landau, Cahn-Hilliard, Halperin-Hohenberg, ….

•  V. D. Waals, (1893). Verhandel. Konink. Akad. Weten. Amsterdam 1(8); Rowlinson, J. S. (1979). Translation of J. D. van der Waals' thermodynamic theory of capillarity under the hypothesis of a continuous variation of density.J. Stat. Phys. 20: 197.

•  The phase field variable labels different states of a material. •  A diffuse interface between stable phases of a material is more

natural than a sharp interface with a discontinuity

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Phase field/Diffuse interface •  Landau, L.D., 1937, On the theory of phase transitions,

an order parameter characterizes the phase change

•  Ginzburg & Landau 1950, On the theory of superconductivity. (Nobel prize 2003)

complex order parameter (wave function) Ψ= ρ eiθ ρ2: density of superconducting carriers

For more mathematical and computational studies of the G-L models, see Du Tutorials at IMA 2004, IMS 2007 Du-Gunzburger-Peterson 1992 SIAM Review

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Phase-field method for phase transition

•  J. W. Cahn (1961). Acta Metallurgica 9: 795-801; J. W. Cahn and J. E. Hilliard (1958). J. Chem. Phys. 28: 258-267; Allen, S. M. and J. W. Cahn (1977). Journal de Physique C7: C7-51. •  G. J. Fix (1983). Free Boundary Problems: Theory

and Applications. Boston, Piman: 580.

“A phase field model is derived for free boundary problems where the effects of supercooling and surface tension are present. A scheme for obtaining numerical approximations is derived, and sample numerical results are presented. “

•  G. Caginalp (1986) “An analysis of a phase field model of a free boundary” Archive for Rational Mechanics and Analysis 92, 205-245.

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Phase-field Method

Reviews:

•  Boettinger W J, Warren J A, Beckermann C and Karma A 2002. Phase-field simulation of solidification, Annu. Rev. Mater. Res. 32 163–94

•  Chen L-Q 2002. Phase-field models for microstructure

evolution Annu. Rev. Mater. Res. 32 113–40

•  Steinbach I. 2009, Phase-field models in materials science, Modelling Simul. Mater. Sci. Eng. 17, 073001

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Phase field via DFT

Classical density functional theory for inhomogeneous fluid, ρ(r) atomic number density, attraction potential U(r)= - kδ(r)

Solution of Euler-Lagrange: (nonlocal)

Slowly varying:

“Landau expansion”

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Diffuse interface: a thermodynamic description

A rescaled double well potential

W (c) = [ 14ε (1− c2 )2 + ε2 |∇c |

2 ]dx∫

420)( ccfcf βα +−≈

20 )( cfcf α+≈ Single well

Double well

0cc =

21 , ccc =

1 ,1 21 −== cc

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Diffuse interface: a thermodynamic description

A rescaled double well potential

420)( ccfcf βα +−≈

Double well

21 , ccc =

1 ,1 21 −== cc

Γ c~1

c~-1

Ω

Γ

W (c) = (ε2∇c 2 + 1

4ε(c2 −1)2 )dx

Ω

∫

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Diffuse interface: a thermodynamic description

Variational problem: minimizing the total energy with various given constraints.

A one-dimensional profile:

A multidimensional profile:

0)(1 322

2=−+− ccc

dxd

ε

W (c) = [ 14ε (1− c2 )2 + ε2 |∇c |

2 ]dx∫

ε

( ) 222

2)1(

21 −= cdx

dcε

)2

tanh()(εxxc =

)2),(tanh()(

εΓ

≈xdxc

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Dynamic phase field equations

Dynamics via gradient flow for energy W=W(φ ):

Allen Cahn type:

Conservative Cahn Hilliard : 4-th order in space H-1 gradient flow

Cahn Hilliard with non-constant mobility

∂φ∂t = −

δWδφ

∂φ∂t = Δ

δWδφ

∂φ∂t = div(M∇ δW

δφ )

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Diffuse interface: a thermodynamic description

Given a composition variable c , the total free energy

•  temperature dependent bulk free energy density •  composition gradient energy coefficient

W (c) = [kc2

|∇c |2 + f (c)] dx∫

)(cf

ck

20 )( cfcf α+≈ Single well

0cc =

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Phase-Field Simulation of Microstructure Evolution

Thermodynamic & kinetic parameters

Input or generate initial microstructure

Calculate driving forces

Integrate microstructure evolution equations

Microstructure & statistics output

http://matcase.psu.edu

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Applications of Phase-field Method •  Solidification microstructures •  Domain/phase microstructures in solid state phase

transformation in bulk systems and thin films Order-disorder transformations, phase separation, martensitic transformations, ferroelectric transitions, ferromagnetic domains, precipitate nucleation and growth

•  Microstructure coarsening •  Defect microstructures

–  Dislocation microstructures and evolution –  Interactions between dislocation and precipitate microstructures –  Crack propagation, void formation in electromigration

•  Film deposition, morphological instability of thin films and quantum dot formation

(L. Q. Chen, Annual Review of Materials Research 32, 113 (2002))

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Diffuse interface/Phase field: a geometric view

ε Γ

φ~1

φ~-1

Ω

Γ

•  How to describe the geometric features of Γ by φ ?

How about other geometric features, interfacial physics?

Volume (difference) :

Computational domain

Area:

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Another exmaple: complex morphological patterns in cells and membranes

Red Blood Cells

Multi- Component GUV

mitochondria

Pictures from various sources

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Cells and Biomembranes

wikivisual

Each compartment is surrounded by a biomembrane

Cells are composed of compartments (organelles) with specific functions

•  Maintains cellular stability/integrity •  Is a protective and selective barrier •  Controls and directs cellular activity

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Cell Membranes

8µm

5 nm

•  Red blood cells and lipid bilayer

Biomembrane as a composite shell (E. Sackman)

Ongoing Budding Fission Fusion

Sackamn

Some important aspects,

I. Cells are composed of compartments (organelles) with specific functions

2. Each compartment is surrounded by a biomembrane: a soft elastic shell,, which fulfills many functional proteins.

3. There is a bidirectional material flow from the endoplasmatic reticulum (ER) to the extracelluare space.

4. It is mediated by the ongoing budding and fission of vesicles from one organelle (say the ER) and their fusion with target organelles (say Golgi or plasma membrane)

5.The inner space of the organelles (the lumen) does not mix with the cytoplasmatic space

Sackamn

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Bilayer Vesicle

Biomimetic cell membrane: lipid vesicle fluid-like bilayer membrane formed by lipids (mostly amphiphilic lipids and sterols)

simple models of membranes

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Models/Simulations

  Atomistic models: ab initio, MD

  Coarse-grained models: effective particle, triangulated networks, Browning dynamics, DPD

  Continnum mechanics: bending elasticity model, diffuse interface formulation

  Multiscale models

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Vesicle Membrane Models/Simulations   Atomistic simulations:

Roark and Feller Langmuir 2008 Lindahl and Edholm

Biophys J., 2000 All-atom lipid bilayer 20nm x 20nm 1024 lipids, 10ns

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Vesicle Membrane Models/Simulations   Atomistic simulations: supported membrane

Substrate

Water

Bila

yer

Water

Lipids

Upper leaflet

Lower leaflet

Heine et al. Molecular Simulations, 2007, 33(4-5), pp.391-397.

lipid

water

substrate

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Vesicle Membrane Models/Simulations   Atomistic simulations

Alternatives:   Coarse-grained models   Continuum models

20 nm to 200 nm: 1,000,000 times more the cost Benchmark of Lindahl and Edholm ~ 40 years of simulation (Moore’s Law) - M. Deserno Full atomistic simulation: 46 years - G. Brannigan et. al.

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Vesicle Membrane Models/Simulations   Coarse grained models:

Coarse-grained modeling of lipids, Bennun-Hoopes-Xing-Faller, Chemistry and Physics of Lipids 159(2009)

Mesoscopic models of biological membranes, Venturoli-Maddalen-Sperotto-Kranenburg-Smit, Phy. Rep. 437(2006)

Top-down: particles represents a number of atoms (a few to a few dozen)

Bottom-up: aggregates, patches, discretization of continuum

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Vesicle Membrane Models/Simulations   Coarse grained models:

  Systematic CG vs Empirical CG   Explicit-solvent vs. implicit-solvent   Pair-wise interaction vs. multi-body interaction   Molecular dynamics vs. Monte Carlo or DPD (Hoogerbrugge-Koelman 1992, Espanol-Warren 1995)

∑+

+=

VECONSERVATI

RANDOMEDISSIPATIV

F

FFdtvdm

Suitable choices of weights in the dissipative and noise forces can lead to an equilibrium distribution depending only on the conservative part of the force

Deserno

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Vesicle Membrane Models/Simulations   Atomistic models: ab initio, MD

  Coarse-grained models: MC, effective particle, triangulated networks, DPD

  Continnum mechanics: bending elasticity model for lipid bilayer (our starting point)

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Continuum Theory: Bending Elasticity Model •  Earlier studies: Canhem 70, Helfrich 73, Evans 79, Fung, …

•  Hypothesis: vesicle Γ minimizes bending elasticity energy, subject to volume/area constraints

Related to the Willmore problem Special case of Helfrich energy

k1 k2

mean curvature

min subj. to volume/area constraints

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Solution techniques •  Analytical/geometrical constructions:

Jenkins, Lipowsky, Seifert, Ouyang, Guven, …

•  Numerical simulations: solving Euler-Lagrange (axis-symmetric),

triangulated networks * FEM boundary integrals * surface evolver moving Least-Squares, lattice Boltzmann, particle dynamics * advected field * Diffuse Interface / Phase Field *

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Solution techniques •  Analytical/geometrical constructions:

Jenkins, Lipowsky, Seifert, Ouyang, Guven, …

•  Numerical simulations: solving Euler-Lagrange (axis-symmetric),

triangulated networks FEM boundary integrals * surface evolver moving Least-Squares, lattice Boltzmann, particle dynamics * advected field * Diffuse Interface / Phase Field Model *

Energy involving 2nd derivatives of coordinates Feng-Klug C1 element

Bonito-Nocheto-Pauletti

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Continuum theory

  Bending elasticity model   Diffuse interface formulation   Numerical methods   Multiphase vesicle, hydrodynamic

interaction, adhesion

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Phase Field Bending Elasticity Model

k1 k2

mean curvature

min subj. to volume/area constraints

φ~1

φ~-1

Ω

Γ

A new problem: how to describe the curvature and bending energy in phase field form?

phase field calculus