Diffuse Interface Field Approach (DIFA) to Modeling and Simulation of Particle-based Materials Processes

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1 Diffuse Interface Field Approach (DIFA) to Modeling and Simulation of Particle-based Materials Processes Yu U. Wang Department Michigan Technological University

2 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)

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

4 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

5 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

6 Simulation Particle sedimentation and stacking

7 Simulation Particle sedimentation and stacking

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

9 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

10 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

11 Simulation Particles of same charge: repulsion

12 Simulation Particles of opposite charges: attractive self-assembly dipolar stable mutually induced dipoles

13 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

14 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

15 Simulation Particles of opposite charges: non-spherical shapes

16 Simulation Stacking of charged particles under external fields g

17 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

18 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

19 Simulation Dipolar particles: agglomeration

41 Simulation Dipolar particles: field-directed self-assembly

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63 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

64 Simulation Phase field model of dielectric/magnetic composites

Particle-Filled Polymer-Matrix Composites Alignment of

Dielectric: PZT fillers Electro-Optic: PbTiO 3

equiaxed irregular needle-shaped irregular Duenas et

random aligned Shanmugham et al, J. Mater. Res.

65 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, Or et al, J. Appl. Phys., 93, 8510, random aligned Shanmugham et al, J. Mater. Res., 19, 795, 2004.Or et al, J. Magn. Magn. Mater., 262, L181, 2003.

66 Model Formulation Particles in multi-phase liquid: capillary forces ({ } { }) 1 2 F = f cα, ηβ + κα cα dv α 2 Landau polynomial 2 2 ({ } { }) f c, η = A ( 3c 4c ) + ( 3η 4η ) + 6 χcc 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

67 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 β 2 β

68 Simulation Irregular-shaped particle at curved fluid interface

69 Simulation Particle self-assembly directed by fluid interface: encapsulation negative pressure zero pressure positive pressure

70 Simulation Bijel: bicontinuous interfacially jammed emulsion gels

71 Simulation Bijel: bicontinuous interfacially jammed emulsion gels

72 Simulation Capillary bridges for in-situ firming of colloidal crystals 100 nm, 10,000 Pa

73 Simulation Capillary bridges for in-situ firming of colloidal crystals

74 Acknowledgement NSF DMR ; 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, , 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, , 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, 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, , 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, , Y.U. Wang, Phase Field Model of Dielectric and Magnetic Composites, Appl. Phys. Lett., 96, , Y.U. Wang, Computer Modeling and Simulation of Solid-State Sintering: A Phase Field Approach, Acta Mater., 54, , 2006.

75 Diffuse Interface Field Approach (DIFA) to Modeling and Simulation of Particle-based Materials Processes Yu U. Wang Department Michigan Technological University

Contents. Preface XI Symbols and Abbreviations XIII. 1 Introduction 1

Contents. Preface XI Symbols and Abbreviations XIII. 1 Introduction 1 V Contents Preface XI Symbols and Abbreviations XIII 1 Introduction 1 2 Van der Waals Forces 5 2.1 Van der Waals Forces Between Molecules 5 2.1.1 Coulomb Interaction 5 2.1.2 Monopole Dipole Interaction