Post on 16-Mar-2018
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
Materials Science and EngineeringPFM2014, State College, PA
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)
Materials Science and EngineeringPFM2014, State College, PA
Diffuse interface field description: arbitrary particle shape, continuous motion on discrete computational grids, as desired for dynamic simulation
Model Formulation
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
( ) ( ) ( )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
Materials Science and EngineeringPFM2014, State College, PA
Particle sedimentation and stackingSimulation
Materials Science and EngineeringPFM2014, State College, PA
Particle sedimentation and stackingSimulation
Materials Science and EngineeringPFM2014, State College, PA
Simulation Phase field model of solid-state sintering: rigid-body motions
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
( ) ( ) ( ) ( )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
Materials Science and EngineeringPFM2014, State College, PA
Particles of same charge: repulsionSimulation
Materials Science and EngineeringPFM2014, State College, PA
Particles of opposite charges: attractive self-assemblySimulation
dipolar stable
mutually induced dipoles
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
(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
Materials Science and EngineeringPFM2014, State College, PA
Particles of opposite charges: non-spherical shapesSimulation
Materials Science and EngineeringPFM2014, State College, PA
Stacking of charged particles under external fieldsSimulation
g
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
Simulation Dipolar particles: agglomeration
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
Materials Science and EngineeringPFM2014, State College, PA
Simulation Dipolar particles: field-directed self-assembly
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
Self-Assembly of Arbitrary-Shaped Dipolar Particles
E
Materials Science and EngineeringPFM2014, State College, PA
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.)
Materials Science and EngineeringPFM2014, State College, PA
Simulation Phase field model of dielectric/magnetic composites
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
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 = ∂ ∂
Materials Science and EngineeringPFM2014, State College, PA
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
Materials Science and EngineeringPFM2014, State College, PA
Irregular-shaped particle at curved fluid interfaceSimulation
Materials Science and EngineeringPFM2014, State College, PA
Particle self-assembly directed by fluid interface: encapsulationSimulation
negative pressure
positive pressure
zero pressure
Materials Science and EngineeringPFM2014, State College, PA
Bijel: bicontinuous interfacially jammed emulsion gelsSimulation
Materials Science and EngineeringPFM2014, State College, PA
Bijel: bicontinuous interfacially jammed emulsion gelsSimulation
Materials Science and EngineeringPFM2014, State College, PA
Capillary bridges for in-situ firming of colloidal crystalsSimulation
100 nm, 10,000 Pa
Materials Science and EngineeringPFM2014, State College, PA
Capillary bridges for in-situ firming of colloidal crystalsSimulation
Materials Science and EngineeringPFM2014, State College, PA
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