Materials with voids T.A. Abinandanan & R. Mukherjee Department of Materials Engineering Indian Institute of Science Bangalore, India.

Outline Voids, cavities, cracks Void growth and shrinkage Key feature: Vacancies are both conserved and non-conserved. Void evolution under stress Void growth under stress Sintering of nanoparticle clusters.

Voids Late Stages of high temperature deformation

Voids

Nucleation Growth Coalescence

Overall Goal A phase-field model of polycrystals with voids Applications: Failure under during temperature deformation Sintering powder compacts

Features Multiple grains: Grain boundaries Voids: Free surface Externally applied stress Enhanced diffusivity at grain boundaries and surfaces Most important: vacancy source term.

Atomistic Picture Crystal – Void system: Lattice gas model Polycrystal with grain boundaries: Potts model

Grain 1, η 1, Grain 2 η 2 Void Atomistic Picture

Approach : Phase Field Model ρ: Vacancy Concentration  Material & Cavity η 1,η 2 : Order Parameter  Grain Orientation Continuum Analogue Lattice Gas Model -> Cahn-Hilliard Model with Atoms and Vacancies Potts Model - > Fan-Chen Model

Total Free Energy F : Total Free Energy F ch : Chemical Contribution To Free Energy F el : Elastic Contribution To Free Energy

Chemical Contribution To Free Energy f: Bulk Free Energy Density ρ : Vacancy Concentration η1, η2 : Order Parameters Κρ : Gradient Energy Coefficient for Gradient in ρ Κ η1, Κ η2 : Gradient Energy Coefficient for Gradient in η1, η2

Approach : Phase Field Model ρ=0, η1=1, η2=0 ρ=1, η1=0, η2=0 ρ=0, η1=0, η2=1

Free energy plots near equilibrium phases Minima are located at (η1,η2)=(1,0) And (0,1), for ρ=0.0 Matrix Minima are located at (η1,η2)=(0,0), for ρ=1.0 Void

Bulk Free Energy Density Grain I : ρ=0, η1=1, η2=0 Cavity: ρ=1, η1=0, η2=0 Grain I I: ρ=0, η1=0, η2=1

Approach : Phase Field Model Along ABAlong CD

Formulation: Kinetics Cahn-Hilliard Equation (Vacancy Concentration ) Allen-Cahn Equation (For Grain Orientation) J. W. Cahn, Acta Metallurgica, 1961 S. M. Allen and J. W. Cahn, Acta Metallurgica, 1979

Vacancies Conserved during diffusion. They can also be created and annihilated at GBs. Existing vacancies – compressive eigenstrain Created vacancies – dilatational eigenstrain.

Algorithm At each time-step: Creation / Annihilation: Compute  v and create in proportion to  v. Re-scaling: Compute homogeneous strain and re-scale the system dimensions. Diffusion: Compute diffusion potential, allow vacancy diffusion.

Variable Mobility M : Mobility M : Mobility ρ : Vacancy Concentration ρ : Vacancy Concentration η1, η2 : Order Parameters η1, η2 : Order Parameters P,Q,R,S: Constants P,Q,R,S: Constants Vacancy Diffusion Enhanced Mobility at the grain boundary and the surface Enhanced Mobility at the grain boundary and the surface Cavity Surface Grain Boundary Matrix

Dihedral Angle (Simulation)I II III Example: Dihedral Angle

Single Grain With Cavity Grain Boundary Cavity With Uniaxial Tensile Stress Void Evolution under stress

Note: No vacancy source / sink. Only diffusion.

Analysis of Schmidt and Gross: Elongation direction of second phase under a applied stress in elastically inhomogeneous system Very soft inhomogeneity elongates normal to the applied stress I. Schmidt and D Gross, Proceedings of Royal Society (London) A, 1999 Bicrystal with Cavity

Cavity shape change during grain growth (No vacancy source / sink; only diffusion)

Void Growth under Tension

Void Shrinkage under Compression

A final example Sintering of Nanoparticle Clusters The small size of the cluster allows us to study sintering without worrying about vacancy source/sink terms. The small size of the cluster also allows 3D simulations!

Experimental Results E.A. Anumol and N. Ravishankar, 2010

Initial Configuration ~400 spherical particles Closely packed

Fully densified compact

Hollow Polycrystalline Aggregate

Multiple Holes

High Surface Diffusivity

High GB diffusivity

Nanoparticle Sintering Full densification is always the end result. Hollow structures of various forms (one compact hole, one interconnected hole, multiple holes) are intermediate configurations. Hollow: High surface diffusivity

Sintering Map

Conclusions A comprehensive model for a polycrystalline material with voids is being developed. It incorporates enhanced diffusivity at surfaces and grain boundaries. Vacancies are conserved and non-conserved. It is being used for studying a wide variety of phenomena –high temperature deformation, void growth, sintering, hot pressing, …