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

Qiang Du Department of Mathematics Pennsylvania State University

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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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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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Strategy: Phase Field Calculus Geometric calculus: variation of surface area/surface tension leads to mean curvature

Phase Field Calculus

Consistency in energy (functional)

and variation (force)

Γ-limit

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Diffuse Interface Bending Energy Model

•  Extension to other geometric features, curvatures … Du-Liu-Wang JCP 2004, 2006, Du-Liu-Ryham-Wang CPAA 2005

•  Geometric calculus translates into phase field calculus for mean curvature square energy

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Sharp interface limit •  In Du-Liu-Ryham-Wang (Nonlinearity 2005), and

later Wang (SIAM Math Ana 2008), by taking a general ansatz

it was shown

in addition, the variation Willmore stress

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Phase Field Calculus for Bending Elasticity

Geometric calculus Phase field calculus

and the variation Willmore stress

Again: a consistent phase field model

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Bending energy/Willmore functional Some relevant works Image impaintng: use of Eulers elastica Chan-Kang-Shen, Esedoglu-Shen, March-Dozio Amphiphilic layer, vesicle membrane Grompper-Schick,Misbah et al, Campelo et al Willmore problem: given surface boundary Dziuk et al, Clarenz

Action minimization: a time dependent variation Kohn et al

Gamma convergence: on least squares DeGiorgi, Roger-Schätzle

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Simulations: equilibrium shapes

Configurations controlled by excess area:     σ = 1 sphere   σ > 1 ?

discocytes

stomatocytes Du-Zhang 2009

Du-Liu-Wang 2004

Michalet-Bensimon

Theory of Ou-Yang

Du-Liu-Wang 2006 JCP

¾ = ( A = 4 ¼ ) 1 = 2

( 4 V ¼ = 3 ) 1 = 3

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Simulations vs. Experiments

Berndl et al 1990: vesicle deformation in response to changing temperature

cross sections of 3d vesicles (Du-Liu-Wang JCP 2004)

43.8-44.1oC

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Numerical simulation

energy diagram

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Modeling multi-component membranes

•  Diffuse interface simulations (Wang- D. 2007 J Math Bio)

Comparing our simulation with experiment (Baumgart et al, Nature 2003)

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Multi-component vesicles •  Recent experimental studies of multicomponent membranes revealed intricate domain structures which are important for biological functions

•  Phase field description of deformable multicomponent membrane: multiple phase field functions for both the vesicle (co-dimension 1 surface) and phase boundary (co-dimension 2 curve) of individual components

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Phase Field Modeling •  Wang- D. 2007 J. Math. Bio. : Phase field model for Bending elastic energy + line tension

subject to fixed total volume and fixed individual phase areas

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Diffuse Interface Modeling •  Diffuse interface model for line tension

Adelman-Kawasaki-Kawakatsu

Ji-Lookman-Saxena

Double well Φ

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Similar extension: membrane with free edges

•  An open membrane is taken as a closed membrane with two components, with zero bending rigidity in one of them

(Wang-Du 2007)

Saitoh-Takiguchi-Tanaka-Hotani Capovilla-Guven-Santiago Umeda-Suezaki-Takiguchi-Hotani Ouyang-Tu

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Phase field models of vesicle membranes

So far, we have discussed:

Models and simulations of equilibrium configurations of the single and multi-component vesicles

Next, we discuss various extensions: • Gradient dynamic models • Interaction with fluid and hydrodynamics models • Interactions with substrate

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Dynamic phase field bending elasticity models

Dynamics via gradient flow:

Allen Cahn type: 4-th order in space

Conservative Cahn Hilliard : 6-th order in space

Compare with Cahn Hilliard dynamics for surface tension: different energy law

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Extension: vesicle membranes in fluid • Experimental works

• Modeling/simulations Lipowsky, Grompper, Noguchi, Misbah, Nochetto…

Tsukada et al 2001 Shelby et al 2003

Noguchi-Gompper 2005 Abkarian-Faivre-Stone 2006

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Phase-Field Navier-Stokes Model Du-Liu-Ryham-Wang 2009

Du-Liu-Wang 2006 Du-Li-Liu 2006

Membrane Elastic bending

Fluid Incompressible

viscous

transport

Willmore stress

Hydrodynamics interaction with flow:

Navier-Stokes

Willmore stress

Phase field

Allen-Cahn/Cahn-Hilliard fluid transport

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PFBENS

Lemma: Energy law

Du-Li-Liu 2007 DCDS

For positive ε, µ and ν Theorem: Existence of weak solution (via Galerkin) Theorem: Uniqueness is shown (for 3d case under extra

regularity condition on the velocity profile)

ν can be 0, I or -Δ

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PFNS: sharp interface limit interface conditions (no slip ) (γ = 0)

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Comments on PFNS model:

• Among many studies of membrane/fluid interactions, only a few models incorporate bending elastic energy (Lipowski, Grompper, Noguchi, Brown, Misbah, Nochetto, Atzberger)

The PFNS model • follows from the kinetic/elastic energy competition • has an energy law consistent to sharp interface • enforces incompressibility everywhere • can incorporate inhomogeneities, fluactuations

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Numerical Examples

•  Membrane/fluid interactions (competition of kinetic energy and bending elastic energy)

Du-Liu-Wang 2006 Spectral code developed by Dr. Xiaoqiang Wang (IMA and FSU)

cutting view 3d view

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Some examples

•  Membrane/fluid interactions (single vesicle)

Du-Liu-Wang 2006 cutting view 3d view

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Some examples

•  Membrane/fluid interactions (single vesicle)

Noguchi-Gompper cutting view 3d view

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Stochastic Implicit Interface Method

•  Developed for –  Fluid-Microstructure (Interface) interaction. – External fluctuation effects.

•  Motivated by stochastic immersed boundary method (Kramer-Peskin/Atzberger-Kramer-Peskin)

•  Based on –  Implicit interface formulation. –  Fluctuation-dissipation theorem.

(Du – M. Li 2010 DCDS and 2010 Ach. Rat. Mech )

(Manlin Li Ph.D thesis 2010 Penn State)

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Stochastic Implicit Interface Method •  Reformulation

nice structure if G=0, but need to estimate the effect due to the stochastic integral G

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Stochastic Implicit Interface Method A special case: quasi-steady Stokes flow

•  Formally

•  Multiplicative noise (very little analytic theory)

•  Comparison with Langevin gradient dynamics

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Stochastic Implicit Interface Method •  Quasi-steady Stokes flow •  Some insight from spatial discretization

•  Let

•  Stratonovich (instead of Ito) Boltzmann

,

L: semi-infinite

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Stochastic Immersed Boundary Method

•  Quasi-steady Stokes flow, finite # of particle

Stratonovich

•  In Ito form:

Periodic BC/Fourier basis: Stratonovich reduces to Ito, allowing further reduction of the multiplicative noise to a finite dimensional representation

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Diffuse interface  Consistency with sharp interface description  Diffuse interface:

 Single set of equations with smooth solutions

 Easier to handle topological changes (interface recovered from the phase field function);

 Need to use a parameter ε (artificial or physical)

 Computational domain has a higher dimension;

 Adaptivity helps to reduce computational cost

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Spatial discretization: FEM •  Mixed formulation with C0 linear element + mass lumping ( FVM in general mesh or FDM for

Cartesian mesh) : lower differentiation order via integration by parts

•  Theorem: as h 0 , a subsequence of approximate solutions converges to a solution of the continuous problem (via uniform bounds and compactness)

(Du-Wang, IJNAM 2006)

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Spatial discretization: FEM •  Mixed formulation : TZ=F(Z) with T of the form

Linear part: nested saddle point problem (framework developed in Chen-Du-Zou SINUM 1999)

Nonlinear part: as compact perturbation (BRR)

– Theorem: as h 0 , we have the convergence and optimal order error H 1 estimates (Du-Zhu, JCM 2006)

•  Extension of Aubin-Nitsche to nonlinear case

– Theorem: as h 0 , we have the convergence and optimal order error L 2 estimates (L. Zhu, MMAS)

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Adaptive 3d FEM

mesh density

Total #s of freedom N ~ O(ε-2)

code developed by Jian Zhang, anisotropic adaptivity?

•  Phase field functions are computed in 3d domain mainly to resolve the 2d interface, adaptive FEM improves the efficiency (Provatas-Goldenfeld-Dantzig, Tonhardt-Amberg, Feng-Wu, Bänsch-Haußer-Lakkis-Li-Voigt, Braun-Murray, … ) •  Adaptive FEM for vesicle deformation based on residual type a posteriori error estimates (Du-Zhang SISC 2008) •  3D Computation gets reduced to 2D complexity

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Spatial discretization: Fourier spectral

Application to phase field models (Chen-Shen 92):   Phase field functions are smooth with given interfacial width   Superior accuracy, fast implementation via FFT  Rigid Cartesian grid structure  Accuracy degradation if the interface is under resolved

To better represent interfacial region  Moving mesh Fourier spectral (Yu-Du-Chen 2006,

Feng-Yu-Hu-Liu-Du-Chen 2006): following works of Winslow, Nakashashi-Deiwert, Bayliss-Kuske-Matkowsky, Huang-Ren-Russell, Ceniceros-Hou,… )

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Moving Mesh/Fourier Spectral (MMFS) (Feng-Yu-Hu-Liu-Du-Chen, JCP 2006)

•  Large gradient in physical domain requires high resolution, thus necessitates fine uniform mesh

•  To make grids clustered near physical interface, a map is used between the computational and physical domain so that the grids still remain uniform in the computational domain

•  MMFS (Moving Mesh Fourier spectral method): mesh evolves with solution to provide maximum resolution in the physical domain

•  Overcome the complication of inhomogeneous coefficients due to mesh motion to allow FFT, reduce overall system size and improve accuracy

•  Performance gain despite of the overhead

Computational

mesh

Physical

mesh

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Moving Mesh/Fourier Spectral (MMFS)

•  Overcome the complication of inhomogeneous coefficients due to mesh motion to allow FFT

e.g. Allen-Cahn

•  Reduce system size and improve accuracy •  Mesh update may be less frequent •  Performance gain despite of the overhead

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Phase field/diffuse interface model •  Phase field type methods are insensitive

to topological changes as interfaces are recovered implicitly from the order parameters. Large scale simulations are often able to reveal very complex morphological patterns

•  It is important to perform quantitative statistical analysis (data mining) for large scale simulations!

•  Sometimes it may be desirable to detect/control the topology of implicitly defined interfaces

LQ Chen et al

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Phase field/diffuse interface model

•  How to retrieve interesting microstructure statistics from the order parameter?

•  Connect geometry with topology, allow effective detection and control of topological events

Du-Liu-Wang SIAM Appl Math 2005

•  Formulae are developed to retrieve certain topological quantities such as surface genus (Euler number) from the order parameter

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Phase field/diffuse interface model •  Phase field type methods are insensitive

to topological changes as interfaces are recovered implicitly from the order parameters. Large scale simulations are often able to reveal very complex morphological patterns

•  It is important to perform quantitative statistical analysis (data mining) for large scale simulations!

•  Sometimes it may be desirable to detect/control the topology of implicitly defined interfaces

LQ Chen et al

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Phase field/diffuse interface modeling

•  Eg: how to retrieve/detect topological info of implicitly defined structure?

(Note: level-set/phase field methods are insensitive to topological changes: advantage or deficiency?)

Du -Liu-Wang SIAM Appl Math 2005

•  Formulae are developed to detect /control topology change in shape transformation via Euler number

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Numerical simulation •  Shapes with same elastic bending energy

but different topology (Euler number)

elastic bending energy

Euler number

Key 1: Euler numbers may be recovered from curvature via Gauss-Bonnet formula Key 2: phase field models can compute curvature related energy effectively via suitable integration

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Euler-Poincare index with diffuse interface

•  In 3d:

•  In 2d:

Diffuse-Interface Based Euler number !

D. -Liu-Wang 2004 SIAP

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Euler-Poincare index •  Further simplification in some special ansatz

D. -Liu-Ryham-Wang 2007 CMS Regularization + Renormalization

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Automated topological information retrieval

•  Gradient flow of bending energy for vesicles subject to volume and surface area

•  A spectral method based implementation

D.-Liu-Wang SIAM Appl Math 2005

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Topology of self-intersecting surfaces •  Nature often leads to piecewise smooth interfaces (3d cylindrically symmetric surfaces)

(2d curves) •  Phase field solutions allow self-intersecting (piecewise smooth) interfaces •  The phase field based Euler number provides detection of singularities/self-intersections

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Extension: study of adhesion and vesicle fusion

 Adhesion: critical to biological processes such as exocytosis, endocytosis,... (Yeagle, The membrane of cells, 1993)

 Fusion: important for lipid transport, pathogen entry,…

www.yellowtang.org/cells.php wikipedia

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Vesicle substrate interaction via diffuse interface Zhang-Das-Du JCP 2009

with Additional contribution due to adhesion

Diffuse interface formulation

No need to specify contact region

− w(x)P(d(x,Γ

∫ ΓS ))dΓ

H 2

Γ

∫ dΓ

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Multicomponent Vesicle substrate interaction Parthasarathy-Yu-Groves, Rozycki-Weikl-Lipowsky, ….

Sohn-Tseng-Li-Voigt-Lowengrub,

Funkhouser-Solis-Thorton,

Adelman-Kawasaki-Kawakatsu

Ji-Lookman-Saxena

+ Z

¡ ¾

£ » 2 ¯ ¯ r ¡ ´

¯ ¯ 2 + 1 4 »

© ( ́ ) ¤ d ¡ ¡

Z

¡ w ( c 0 + c 2 ́ ) P ( d ( x ) ) d ¡ ;

E ( ́ ) = Z

¡ ( c 0 + c 1 ́ )

£ H ¡ a ( c 0 + c 3 ́ )

¤ 2 d ¡

sharp-interface diffuse interface ( η : new phase field function)

Zhang-Das-Du PRE 2010

Phase change on Γ:

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Multi-component Vesicle substrate interaction Zhang-Das-Du PRE 2010

Adhesion induced phase separation, substantiating claims made in Gordon-Deserno-Andrew-Egelhaaf-Poon, 2008 EPL

With

Adhesion

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Multi-component Vesicle Fusion

Zhang-Du PRE 2011

Motivation: adhesion-induced fusion – Lipowsky 1991 Nature

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Multicomponent Vesicle substrate interaction Zhang-Das-Du PRE 2010

Adhesion induced phase separation