Diffuse Interface Field Approach (DIFA) to Modeling and ...chen/pfm2014/talk/YuUWang_talk.pdf ·...

Materials Science and EngineeringPFM2014, State College, PA

Diffuse Interface Field Approach (DIFA) to

Modeling and Simulation of

Particle-based Materials Processes

Yu U. WangMaterials Science and Engineering Department

Michigan Technological University


Motivation

Interesting and important issues in particle processesOutline

Extend phase field method to model free-body solid particles

Moving particles of arbitrary shapes and sizes in close distance: rigid-body translations and rotations

Short-range forces: mechanical contact, friction, cohesion, steric repulsion, Stokes drag (particle shape matters)

Long-range forces: electric charge, charge heterogeneity, electric double layer, electric/magnetic dipole, van der Waals (point-charge/point-dipole approximation inaccurate)

External forces: electric/magnetic field, gravity (field-directed self-assembly)

Multi-phase liquid: fluid interface evolution, capillary force on particles (surface tension, Laplace pressure via Gibbs-Duhemrelation)


Diffuse interface field description: arbitrary particle shape, continuous motion on discrete computational grids, as desired for dynamic simulation

Model Formulation


Short-range forces: mechanical contact, steric repulsionModel Formulation

( ) ( ) ( ) ( ) ( )sr 3; ; ; ; ;d d rα α

α κ η α η α η α η α′≠

′ ′= ∇ −∇ ∑F r r r r r

( ) ( )sr sr ;V

dα α= ∫F F r

( ) ( ) ( )sr c sr ;V

dα α α = − × ∫T r r F r

soft-particle potential

action-reaction symmetry

Arbitrary particle shapes without tracking interfaces

torque


( ) ( ) ( )sr fα α α= +F F ξ

( ) ( ) ( )sr tα α α= +T T ξ

Total force and torque acting on individual particleModel Formulation

Particle dynamics in viscous liquid

thermal noise for Brownian motion

( ) ( ) ( )i ij jV M Fα α α=

( ) ( ) ( )i ij jΩ N Tα α α=small Reynolds number Re<<1,Stokes drag (friction), mobility

Equation of motion( ) ( )0

0, ; , ;t tη α η α=r r

( ) ( ) ( )0 c c0; ; ;i ij j j ir Q t r r t r tα α α = − +

( ) ( ) ( )c c; ; ;i i ir t dt r t V t dtα α α+ = +

( ) ( ) ( ); ; ;ij ik kjQ t dt R t Q tα α α+ =

( ) ( ); cos 1 cos sinij ij i j ijk kR t m m mα δ ω ω ε ω= + − −

mapping withouterror accumulation

translationrotation

incremental rotation


Particle sedimentation and stackingSimulation


Simulation Phase field model of solid-state sintering: rigid-body motions


Model Formulation Long-range force: charged particles

( ) ( ), , ;t tα

ρ ρ α=∑r r ( )( )

( )3ex

30 2

ii d k ek

ρε π

⋅= − ∫ k rkE r E n

( ) ( ) ( )el 3;V

d rα ρ α= ∫F E r r ( ) ( ) ( ) ( )el c 3;V

d rα α ρ α = − × ∫T r r E r r

( ) ( ) ( ), ; , ;t tρ α ρ α η α=r r

( ) ( ) ( ) ( ), ; , ; 1 , ;t t tρ α ρ α η α η α= − r r rbody chargesurface charge


( ) ( ) ( ) ( )el sr fα α α α= + +F F F ξ

( ) ( ) ( ) ( )el sr tα α α α= + +T T T ξ

Total force and torque acting on individual particle

Model Formulation Long-range force: charged particles

( ) ( ), , ;t tα

ρ ρ α=∑r r ( )( )

( )3ex

30 2

ii d k ek

ρε π

⋅= − ∫ k rkE r E n

( ) ( ) ( )el 3;V

d rα ρ α= ∫F E r r ( ) ( ) ( ) ( )el c 3;V

d rα α ρ α = − × ∫T r r E r r

( ) ( ) ( ), ; , ;t tρ α ρ α η α=r r

( ) ( ) ( ) ( ), ; , ; 1 , ;t t tρ α ρ α η α η α= − r r rbody chargesurface charge


Particles of same charge: repulsionSimulation


Particles of opposite charges: attractive self-assemblySimulation

dipolar stable

mutually induced dipoles


Self-Assembly Mechanisms

(1) neutral & symmetric (2) induced dipole (3) attraction (4) repeated growth & dipolar

ρsurf = -2.0

ρsurf = +1.0

N-:N+ = 1:2

neutral chain formation

growth stopper


(1) charged & dipole (2) alignment (3) attraction (4) repeated growth & charged

charged chain formation, mutually repulsive,

as straight as possible

repel at long distanceattract at short distance

ρsurf = -2.0

ρsurf = +1.0

N-:N+ = 1:1

Self-Assembly Mechanisms


Particles of opposite charges: non-spherical shapesSimulation


Stacking of charged particles under external fieldsSimulation

g


Model Formulation Long-range force: dipolar particles

( ) ( ), , ;t tα

α=∑P r P r ( )( )

( )3

ex3

0

12

id k eε π

⋅ = − ⋅ ∫ k rE r E n P k n

( ) ( ) ( ) ( )el 3;V

d rα α η α= ⋅∇ ∫F P E r r

( ) ( ) ( ) ( )

( ) ( ) ( ){ } ( )

el 3

c 3

;

;V

V

d r

d r

α α η α

α α η α

= ×

+ − × ⋅∇

∫∫

T P E r r

r r P E r r

( ) ( ) ( ), ; ; , ;t t tα α η α=P r P r


Model Formulation Long-range force: dipolar particles

( ) ( ), , ;t tα

α=∑P r P r ( )( )

( )3

ex3

0

12

id k eε π

⋅ = − ⋅ ∫ k rE r E n P k n

( ) ( ) ( ) ( )el 3;V

d rα α η α= ⋅∇ ∫F P E r r

( ) ( ) ( ) ( )

( ) ( ) ( ){ } ( )

el 3

c 3

;

;V

V

d r

d r

α α η α

α α η α

= ×

+ − × ⋅∇

∫∫

T P E r r

r r P E r r

( ) ( ) ( ), ; ; , ;t t tα α η α=P r P r

Long-range force: magnetic particles( ) ( ) ( ), ; , ;t t t

α

α η α=∑M r M r ( )( )

( )3

ex32

id k eπ

⋅ = − ⋅ ∫ k rH r H n M k n

( ) ( ) ( ) ( )mag 30 ;

Vd rα µ α η α= ⋅∇ ∫F M H r r

( ) ( ) ( ) ( ) ( ) ( ) ( ){ } ( )mag 3 c 30 0; ;

V Vd r d rα µ α η α µ α α η α = × + − × ⋅∇ ∫ ∫T M H r r r r M H r r


Simulation Dipolar particles: agglomeration


Self-Assembly of Arbitrary-Shaped Dipolar Particles


Simulation Dipolar particles: field-directed self-assembly


Self-Assembly of Arbitrary-Shaped Dipolar Particles

E


Processing-Microstructure Relationship Mechanisms of Filler Particle Self-Assembly

attraction repulsion

Strongly anisotropic force that can be tuned by external field

Rigid-body motion (translation and rotation) of colloidal particles in liquids (water, organic solvent, polymer melt, etc.)


Simulation Phase field model of dielectric/magnetic composites


Dielectric: PZT fillers Electro-Optic: PbTiO3 nanoparticlesMagnetostrictive: Terfenol-D particles

Alignment of irregular-shaped functional filler particles

Or et al, J. Appl. Phys., 93, 8510, 2003.Duenas et al, J. Appl. Phys., 90, 2433, 2001.

Shanmugham et al, J. Mater. Res., 19, 795, 2004.Or et al, J. Magn. Magn. Mater., 262, L181, 2003.

equiaxedirregular

needle-shapedirregular

random aligned

Particle-Filled Polymer-Matrix Composites


Particles in multi-phase liquid: capillary forcesModel Formulation

{ } { }( ) 21,2

F f c c dVα β α αα

η κ = + ∇ ∑∫

{ } { }( ) ( ) ( )2 2

4 3 4 3 2 2 2 21 2

1 1, 3 4 3 4 6f c A c c c c cα β α α β β α α β

α β β α

η η η χ λ η= =

= − + − + +

∑ ∑ ∑∑

c FMt cα

αα

δδ

∂= ∇ ⋅ ∇ ∂

A A B Bdp c d c dµ µ= +0

1 1 2 2p p c cµ µ− = +

Cahn-Hilliard

Landau polynomial

Gibbs-Duhem

Young-LaplacepRγ

∆ =

f cα αµ = ∂ ∂


Particles in multi-phase liquid: capillary forcesModel Formulation

( )1 2 1 2c c c c c∇ = ∇ −∇

LPP( , ) ( , ) ( )d p dVβ κ η β= ∇F r r r

ITT

2T

( ) [ ( )]

[( ) ]

d c c dV

c c c dV

β

β β

β κ η

κ η η

= ∇ × ∇ ×∇

= ∇ ⋅∇ ∇ − ∇ ∇

F

Laplace pressure

interfacial tension


Irregular-shaped particle at curved fluid interfaceSimulation


Particle self-assembly directed by fluid interface: encapsulationSimulation

negative pressure

positive pressure

zero pressure


Bijel: bicontinuous interfacially jammed emulsion gelsSimulation


Capillary bridges for in-situ firming of colloidal crystalsSimulation

100 nm, 10,000 Pa


Capillary bridges for in-situ firming of colloidal crystalsSimulation


Y.U. Wang, “Phase Field Model of Dielectric and Magnetic Composites,” Appl. Phys.Lett., 96, 232901-1-3, 2010.

Y.U. Wang, “Computer Modeling and Simulation of Solid-State Sintering: A PhaseField Approach,” Acta Mater., 54, 953-961, 2006.

Y.U. Wang, “Modeling and Simulation of Self-Assembly of Arbitrary-Shaped Ferro-Colloidal Particles in External Field: A Diffuse Interface Field Approach,” Acta Mater.,55, 3835-3844, 2007.

P.C. Millett, Y.U. Wang, “Diffuse Interface Field Approach to Modeling and Simulationof Self-Assembly of Charged Colloidal Particles of Various Shapes and Sizes,” ActaMater., 57, 3101-3109, 2009.

P.C. Millett, Y.U. Wang, “Diffuse-Interface Field Approach to Modeling Arbitrarily-Shaped Particles at Fluid-Fluid Interfaces,” J. Colloid Interface Sci., 353, 46-51, 2011.

T.L. Cheng, Y.U. Wang, “Spontaneous Formation of Stable Capillary Bridges forFirming Compact Colloidal Microstructures in Phase Separating Liquids: AComputational Study,” Langmuir, 28, 2696-2703, 2012.

T.L. Cheng, Y.U. Wang, “Shape-Anisotropic Particles at Curved Fluid Interfaces andRole of Laplace Pressure: A Computational Study,” J. Colloid Interface Sci., 402, 267-278, 2013.

NSF DMR-0968792; TeraGrid supercomputers.Acknowledgement


Diffuse Interface Field Approach (DIFA) to

Modeling and Simulation of

Particle-based Materials Processes

Yu U. WangMaterials Science and Engineering Department

Michigan Technological University

Diffuse Interface Field Approach (DIFA) to Modeling and ...chen/pfm2014/talk/YuUWang_talk.pdf ·...

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