Phase Field Modeling and Simulations of Interface Problems

Transcription

1 Phase Field Modeling and Simulations of Interface Problems - a Tutorial on Basic Ideas and Selected Applications Qiang Du Department of Mathematics Pennsylvania State University 1

2 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 2

3 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 3

4 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 4

5 Examples of interface: Wikipedia Edgerton Device 5

6 Complex/biological fluid Experiments/Analysis/Modeling/Simulations Courtesy of Pritchard lab Courtesy of Dong s lab membrane protein actin cell air water shampoo blood 6

7 Cell level: RBCs/vesicles in fluid Experimental works Tsukada et al 2001 Shelby et al 2003 Abkarian-Faivre-Stone 2006 Modeling/simulations Noguchi-Gompper 2005 Du-Liu-Ryham-Wang

8 Application: tumor metastasis Tumor cell adhesion and migration Alberts et al.,

9 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 9

10 Method Experiments/Analysis/Modeling/Simulation(TEAMS) Coupling in vitro experiments and numerical simulations Flow Migration Chamber Top Plate Flow in Cellular Monolayer Porous membrane WBC TC Flow out Wells for Chemoattractant (Penn State U, Dong Lab) Parallel flow chamber experiments show: ratio of TC/PMN population affects TC extravasation 10

11 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 11

12 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 12

13 Interface of biology and mathematics Why red blood cells are biconcave in shape? The biconcave shape increases their surface area, which is important in increasing the rate of diffusion as they transport O2 and CO2 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 13

14 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, 14

15 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 -1 Γ +1 φ ε ε Γ φ ~ 1 φ ~ -1 Interface Γ: zero level set of φ diffuse interfacial layer An implicit surface representation 15

16 Multiscale Modeling and Simulations Materials Computation And Simulation Environment Atomic structure Microstructure evolution Liu-Chen-Raghavan-Du-Sofo-Langer-Wolverton, 2004: An integrated Framework for multi-scale materials simulation and design, J. Computer Aided Materials Design 16

17 Nanoscale Grain/Domain Structures in Ferroelectrics (from L.Q. Chen) Atomic Domain Geometry and topology f microstructure control material property Grain W Drive Line Bit Line PZT Macroscale/device level Word Line ARRAY PERIPHERY Device 17

18 Implicit interface representation - any advantage? φ=0 φ>0 φ<0 A single set of equations to be solved throughout the domain, no need to track interface Interface with different topology is described by a single level set function 18

19 Diffuse interface/phase field: a geometric view How to describe the geometric features of Γ by φ? Γ ε Γ φ~1 Ω φ~-1 Computational domain Volume (difference) : Area: (Cahn-Hilliard, Modica, Fonseca-Tartar, Rubinstein-Sternberg-Keller, Kohn, Gurtin, X.F. Chen, Elliott, Nochetto-Paulini-Verdi, Evans-Souganidis-Soner, ) 19

20 Diffuse interface/phase field: a geometric view A phase field description of isoparametric problem Minimize surface area Subject to given volume Min: Γ φ~1 Ω φ~-1 Subject to: 20

21 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) Deckelnick-Dziuk-Elliott 21

22 Phase Field/Diffuse Interface Model Idea goes back to van de Waals, GinzburgLandau, 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 22

23 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 e iθ ρ 2 : 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 23

24 Phase-field method for phase transition J. W. Cahn (1961). Acta Metallurgica 9: ; J. W. Cahn and J. E. Hilliard (1958). J. Chem. Phys. 28: ; 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,

25 Phase-field Method Reviews: Boettinger W J, Warren J A, Beckermann C and Karma A Phase-field simulation of solidification, Annu. Rev. Mater. Res Chen L-Q Phase-field models for microstructure evolution Annu. Rev. Mater. Res Steinbach I. 2009, Phase-field models in materials science, Modelling Simul. Mater. Sci. Eng. 17,

26 Phase field via DFT Classical density functional theory for inhomogeneous fluid, ρ(r) atomic number density, attraction potential U(r)= - k δ (r) Solution of Euler-Lagrange: (nonlocal) Slowly varying: Landau expansion 26

27 Diffuse interface: a thermodynamic description f ( c) f + α c 0 2 Single well c = c 0 f 2 4 ( c) f0 α c + βc Double well c = c 1, c 2 A rescaled double well potential c = 1, c = W (c) = [ 1 4ε (1 c2 ) 2 + ε 2 c 2 ]dx 27

28 Diffuse interface: a thermodynamic description f 2 4 ( c) f0 α c + βc Double well c = c 1, c 2 A rescaled double well potential c = 1, c = W (c) = Ω ( ε 2 c ε (c2 1) 2 )dx Γ Γ c~1 Ω c~-1 28

29 Diffuse interface: a thermodynamic description Variational problem: minimizing the total energy W (c) = with various given constraints. A one-dimensional profile: [ 1 4ε (1 c2 ) 2 + ε 2 c 2 ]dx ε A multidimensional profile: d( x, Γ) c( x) tanh( ) 2ε d dx 2 2 c + 1 ( c 3 2 ε c) = 0 c ( x) = tanh( ( ) 2 dc 2 2 = 1 ( c 1) dx x ) 2ε 2 2ε 29

30 Dynamic phase field equations Dynamics via gradient flow for energy W=W(φ ): Allen Cahn type: φ t = δw δφ Conservative Cahn Hilliard : 4-th order in space H -1 gradient flow φ t = Δ δw δφ Cahn Hilliard with non-constant mobility φ t = div(m δw δφ ) 30

31 Diffuse interface: a thermodynamic description Given a composition variable c, the total free energy W (c) = [ k c 2 c 2 + f (c)] dx f (c) temperature dependent bulk free energy density composition gradient energy coefficient k c f ( c) f + α c 0 2 Single well c = c 0 31

32 Phase-Field Simulation of Microstructure Evolution Thermodynamic & kinetic parameters Input or generate initial microstructure Calculate driving forces Integrate microstructure evolution equations Microstructure & statistics output 32

33 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)) 33

34 Diffuse interface/phase field: a geometric view How to describe the geometric features of Γ by φ? Γ ε Γ φ~1 Ω φ~-1 Computational domain Volume (difference) : Area: How about other geometric features, interfacial physics? 34

35 Another exmaple: complex morphological patterns in cells and membranes Multi- Component GUV Red Blood Cells mitochondria Pictures from various sources 35

36 Cells and Biomembranes Cells are composed of compartments (organelles) with specific functions wikivisual Each compartment is surrounded by a biomembrane Maintains cellular stability/integrity Is a protective and selective barrier Controls and directs cellular activity 36

37 Cell Membranes Red blood cells and lipid bilayer 8µm 5 nm 37

38 Biomembrane as a composite shell (E. Sackman)

39 Sackamn Ongoing Budding Fission Fusion

40 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

41 Bilayer Vesicle Biomimetic cell membrane: lipid vesicle fluid-like bilayer membrane formed by lipids (mostly amphiphilic lipids and sterols) simple models of membranes 41

42 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 42

43 Vesicle Membrane Models/Simulations Atomistic simulations: Lindahl and Edholm Biophys J., 2000 All-atom lipid bilayer 20nm x 20nm 1024 lipids, 10ns Roark and Feller Langmuir

44 Vesicle Membrane Models/Simulations Atomistic simulations: supported membrane Water Lipids lipid Upper leaflet Lower leaflet Bilayer water Water substrate Heine et al. Molecular Simulations, 2007, 33(4-5), pp Substrate 44

45 Vesicle Membrane Models/Simulations Atomistic simulations 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. Alternatives: Coarse-grained models Continuum models 45

46 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 46

47 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) dv m dt = F DISSIPATIVE + F + F RANDOM CONSERVATIVE Deserno 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 47

48 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) 48

49 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 min k 1 k 2 subj. to volume/area constraints Related to the Willmore problem Special case of Helfrich energy mean curvature 49

50 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 * 50

51 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 2 nd derivatives of coordinates Feng-Klug C1 element Bonito-Nocheto-Pauletti 51

52 Continuum theory Bending elasticity model Diffuse interface formulation Numerical methods Multiphase vesicle, hydrodynamic interaction, adhesion 52

53 Phase Field Bending Elasticity Model min k 1 k 2 subj. to volume/area constraints Γ φ~1 Ω φ~-1 mean curvature A new problem: how to describe the curvature and bending energy in phase field form? phase field calculus 53