Materials with voids

T.A. Abinandanan & R. Mukherjee

Department of Materials EngineeringIndian 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

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, η1,

Grain 2 η2

Void

Atomistic Picture

Approach : Phase Field Model

ρ: Vacancy ConcentrationMaterial & Cavity

η1 ,η2:Order ParameterGrain Orientation

Continuum Analogue

Lattice Gas Model -> Cahn-Hilliard Model with Atoms and VacanciesPotts Model - > Fan-Chen Model

Total Free Energyelch FFF

dVF elij

elijV

el 21=

F : Total Free Energy Fch : Chemical Contribution To Free Energy Fel : Elastic Contribution To Free Energy

dVfFV

ch ])()()(),,([= 222

211

221

Chemical Contribution To Free Energy

f: Bulk Free Energy Densityρ : Vacancy Concentrationη1, η2 : Order Parameters Κρ : Gradient Energy Coefficient for

Gradient in ρΚ η1, Κ η2 : Gradient Energy Coefficient

for Gradient in η1, η2

,])()()(),,([= 222

211

221 dVfF

V

ch

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

22222 )()(1)(1=),( iii ZBAf

0.25]2[]24

[=)( 2224

jiii

i

Approach : Phase Field Model

Along AB Along CD

Formulation: Kinetics

Cahn-Hilliard Equation(Vacancy Concentration)

Allen-Cahn Equation(For Grain Orientation)

,.= DMt

)/(= VNF

D

)/(

= VNFLt

J. W. Cahn, Acta Metallurgica, 1961S. 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 ρ : Vacancy Concentration η1, η2 : Order Parameters P,Q,R,S: Constants

2/122

22

21

21

2222 )]1()1([)1()1( SRQPM

Vacancy Diffusion

Enhanced Mobility at the grain boundary and the surface

Cavity

SurfaceGrain Boundary

Matrix

Dihedral Angle

(Simulation)

I 0.7362 0.7154 61.94 60.00

II 0.5970 0.4125 69.79 69.50

III 0.5387 0.2405 77.10 77.00

s gb

s

gb

2

cos 1

Example: Dihedral Angle

Single Grain With Cavity

Grain Boundary Cavity With Uniaxial Tensile Stress

Void Evolution under stress

cAA c

AA

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 particlesClosely 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, …