Diffuse Interface Field Approach (DIFA) to Modeling and Simulation of Particle-based Materials Processes Yu U. Wang Department Michigan Technological University
Motivation Extend phase field method to model free-body solid particles Outline Interesting and important issues in particle processes 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-Duhem relation)
Model Formulation 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 repulsion ( ; α) κ η( ; α) η( ; α ) η( ; α) η( ; α ) sr 3 df r = r r r r dr α α action-reaction symmetry ( α sr ) ( ; α ) sr F d F r V = soft-particle potential ( α) = ( α) d ( ; α) T sr c sr V r r F r torque Arbitrary particle shapes without tracking interfaces
Total force and torque acting on individual particle ( α) = sr ( α) + f ( α) ( α) = sr ( α) + t ( α) F F ξ T T ξ Model Formulation Particle dynamics in viscous liquid ( α) = ( α) ( α) ( α) = ( α) ( α) V M F i ij j Ω N T i ij j Equation of motion ( r,; t ) = ( r 0, t0; ) η α η α ( ; α) 0 c ( ) 0; α ( ; α) ( + ; α) = c( ; α) + ( ; α) ( + ; α) = ( ; α) ( ; α) ( ) ( ) r = Q t r r t + r t i ij j j i r t dt r t V t dt c i i i Q t dt R t Q t ij ik kj thermal noise for Brownian motion small Reynolds number Re<<1, Stokes drag (friction), mobility R t; α = δ cosω+ mm 1 cosω ε m sinω ij ij i j ijk k mapping without error accumulation translation rotation incremental rotation
Simulation Particle sedimentation and stacking
Simulation Particle sedimentation and stacking
Simulation Phase field model of solid-state sintering: rigid-body motions
Model Formulation Long-range force: charged particles ( r,; t ) = ( ) ( r,; t ) ( r,; t ) ( ) ( r,; t ) 1 ( r,; t ) ρ α ρ α η α ρ α = ρ α η α η α 3 ρ( r, t) = ρ( r,; t α ex ) i dk ρ ( k) ikr Er ( ) = E e 3 α ε n 0 ( 2π ) k el 3 F ( α) Er ( ) ρ( r ; α) dr T ( α) = r r ( α) Er ( ) ρ( r ; α) = V el c 3 V body charge surface charge dr
Total force and torque acting on individual particle ( α) = el ( α) + sr ( α) + f ( α) F F F ξ ( α) = el ( α) + sr ( α) + t ( α) T T T ξ Model Formulation Long-range force: charged particles ( r,; t ) = ( ) ( r,; t ) ( r,; t ) ( ) ( r,; t ) 1 ( r,; t ) ρ α ρ α η α ρ α = ρ α η α η α 3 ρ( r, t) = ρ( r,; t α ex ) i dk ρ ( k) ikr Er ( ) = E e 3 α ε n 0 ( 2π ) k el 3 F ( α) Er ( ) ρ( r ; α) dr T ( α) = r r ( α) Er ( ) ρ( r ; α) = V el c 3 V body charge surface charge dr
Simulation Particles of same charge: repulsion
Simulation Particles of opposite charges: attractive self-assembly dipolar stable mutually induced dipoles
Self-Assembly Mechanisms ρ surf = -2.0 ρ surf = +1.0 N - :N + = 1:2 growth stopper neutral chain formation (1) neutral & symmetric (2) induced dipole (3) attraction (4) repeated growth & dipolar
Self-Assembly Mechanisms ρ surf = -2.0 ρ surf = +1.0 N - :N + = 1:1 repel at long distance attract at short distance charged chain formation, mutually repulsive, as straight as possible (1) charged & dipole (2) alignment (3) attraction (4) repeated growth & charged
Simulation Particles of opposite charges: non-spherical shapes
Simulation Stacking of charged particles under external fields g
Model Formulation Long-range force: dipolar particles Pr (,; t α) = P( t; α) η( r,; t α) Pr (, t) Pr (,; t α ) ( ) = α ( α) = ( α) ( ) η( ; α) F P Er r el 3 V ( α) = ( α) ( ) η( ; α) T P Er r el 3 V V 1 Er E npk ( ) ne dr 3 ex dk = 3 ε 0 dr ( ) ( ) ( ) { } ( ; ) + r r α Pα Er η rα ( 2π ) dr c 3 ikr
Model Formulation Long-range force: dipolar particles Pr (,; t α) = P( t; α) η( r,; t α) Pr (, t) Pr (,; t α ) ( ) = α ( α) = ( α) ( ) η( ; α) F P Er r el 3 V ( α) = ( α) ( ) η( ; α) T P Er r el 3 V V 1 Er E npk ( ) ne dr 3 ex dk = 3 ε 0 dr ( ) ( ) ( ) { } ( ; ) + r r α Pα Er η rα ( 2π ) dr c 3 ikr Long-range force: magnetic particles M( r, t) M( t; α) η( r,; t α) ( ) = α ( α ) = µ ( α) ( ) η( α) F M Hr r mag 3 0 ; V 3 ex dk = 3 ( 2π ) ( ) Hr H nmk ne dr ( ) = ( ) ( ) ( ) dr+ ( ) ( ) ( ) ikr { } ( ) T α µ M α Hr η r ; α µ r r α M α Hr η r ; α dr mag 3 c 3 0 V 0 V
Simulation Dipolar particles: agglomeration
Simulation Dipolar particles: field-directed self-assembly
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Processing-Microstructure Relationship Mechanisms of Filler Particle Self-Assembly 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.) attraction repulsion
Simulation Phase field model of dielectric/magnetic composites
Particle-Filled Polymer-Matrix Composites Alignment of irregular-shaped functional filler particles Dielectric: PZT fillers Electro-Optic: PbTiO 3 nanoparticles Magnetostrictive: Terfenol-D particles equiaxed irregular needle-shaped irregular Duenas et al, J. Appl. Phys., 90, 2433, 2001. Or et al, J. Appl. Phys., 93, 8510, 2003. random aligned Shanmugham et al, J. Mater. Res., 19, 795, 2004.Or et al, J. Magn. Magn. Mater., 262, L181, 2003.
Model Formulation Particles in multi-phase liquid: capillary forces ({ } { }) 1 2 F = f cα, ηβ + κα cα dv α 2 Landau polynomial 2 2 ({ } { }) 4 3 4 3 2 2 2 2 f c, η = A ( 3c 4c ) + ( 3η 4η ) + 6 χcc 1 2 + λ cη α= 1 β β α= 1 α β α α β β α α β c α δ F = Mα t δ cα Cahn-Hilliard dp = c dµ + c dµ p p = c + c A A B B 0 1µ 1 2µ 2 µ = f c α α Gibbs-Duhem p = γ R Young-Laplace
Model Formulation Particles in multi-phase liquid: capillary forces Laplace pressure LP df (, r β) = κp η(, r β) p() r dv interfacial tension IT df ( β) = κt[ c ( c ηβ )] dv = κ [( c η ) c c η ] dv ( ) c= cc c c T 1 2 1 2 β 2 β
Simulation Irregular-shaped particle at curved fluid interface
Simulation Particle self-assembly directed by fluid interface: encapsulation negative pressure zero pressure positive pressure
Simulation Bijel: bicontinuous interfacially jammed emulsion gels
Simulation Bijel: bicontinuous interfacially jammed emulsion gels
Simulation Capillary bridges for in-situ firming of colloidal crystals 100 nm, 10,000 Pa
Simulation Capillary bridges for in-situ firming of colloidal crystals
Acknowledgement NSF DMR-0968792; TeraGrid supercomputers. 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 Simulation of Self-Assembly of Charged Colloidal Particles of Various Shapes and Sizes, Acta Mater., 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 for Firming Compact Colloidal Microstructures in Phase Separating Liquids: A Computational Study, Langmuir, 28, 2696-2703, 2012. T.L. Cheng, Y.U. Wang, Shape-Anisotropic Particles at Curved Fluid Interfaces and Role of Laplace Pressure: A Computational Study, J. Colloid Interface Sci., 402, 267-278, 2013. 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 Phase Field Approach, Acta Mater., 54, 953-961, 2006.
Diffuse Interface Field Approach (DIFA) to Modeling and Simulation of Particle-based Materials Processes Yu U. Wang Department Michigan Technological University