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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
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