Multi-scale Modeling of NanocrystallineMaterials

N Chandra and S Namilae

Department of Mechanical EngineeringFAMU-FSU College of Engineering

Florida State UniversityTallahassee, FL 32312 USA

Presented atICSAM2003, Oxford, UK, July 28, 03

Nano-crystalline materials and Nanotechnology ?

Richard Feynman in 1959 predicted that “There is a lot of room below…” Ijima in 1991 discovered carbon nanotubes that

conduct heat more than Copper conduct electricity more than diamond has stiffness much more than steel has strength more than Titanium is lighter than feather can be a insulator or conductor just based on geometry

Nano refers to m (about a few atoms in 1-D) It is not a miniaturization issue but finding new science, “nano-science”-new phenomenaAt this scale, mechanical, thermal, electrical, magnetic, optical and electronic effects interact and manifest differently The role of grain boundaries increases significantly in nano-crystalline materials.

910

Mechanics at atomic scale

Physical Problem

Molecular Dynamics-Fundamental quantities (F,u,v)

Born Oppenheimer

Approximation

Compute Continuum quantities-Kinetics (,P,P’ )-Kinematics (,F)-EnergeticsUse Continuum Knowledge- Failure criterion, damage etc

Stress at atomic scale

Definition of stress at a point in continuum mechanics assumes that homogeneous state of stress exists in infinitesimal volume surrounding the point

In atomic simulation we need to identify a volume inside which all atoms have same stress

In this context different stresses- e.g. virial stress, atomic stress, Lutsko stress,Yip stress

Virial Stress

1 1 1

2 2

N Ni j

ij i j

r r Vm v v

r r

Stress defined for whole system

For Brenner potential:

1 1 1

2 2

N N

ij i j i jm v v f r

Total Volume

if Includes bonded and non-bonded interactions

(foces due to stretching,bond angle, torsion effects)

BDT (Atomic) Stresses

Based on the assumption that the definition of bulk stress would be valid for a small volume around atom

1 1 1

2 2

N

ij i j j imv v r f

Atomic Volume

- Used for inhomogeneous systems

Lutsko Stress

1 1

1 1 1

2 2

N Nlutskoij i j j iLutsko

mv v r f

r

- fraction of the length of - bond lying inside the averaging volume

Averaging Volume

-Based on concept of local stress in statistical mechanics-used for inhomogeneous systems-Linear momentum conserved

l

Strain calculation

Displacements of atoms known

Lattice with defects such as GBs meshed as tetrahedrons

Strain calculated using displacements and derivatives of shape functions

Borrowing from FEM Strain at an atom

evaluated as weighted average of strains in all tetrahedrons in its vicinity

Updated lagarangian scheme used for MD

Mesh of tetrahedrons

GB

GB as atomic scale defect …

Grain boundaries play a important role in the strengthening and deformation of metallic materials.

Some problems involving grain boundaries : Grain Boundary Structure Grain boundary Energy Grain Boundary Sliding Effect of Impurity atoms

We need to model GB for its thermo-mechanical (elastic and inelastic) properties possibly using molecular dynamics and statics.

Equilibrium Grain Boundary Structures

[110]3 and [110]11 are low energy boundaries, [001]5 and [110]9 are high energy boundaries

[110]3 (1,1,1) [001]5(2,1,0)

[110]9(2,21) [110]11(1,1,3)

GB

GB

GB

GB

Experimental Results1

1 Proceeding Symposium on grain boundary structure and related phenomenon, 1986 p789

Grain Boundary Energy Computation

Calculation

GBE = (Eatoms in GB configuration) – N Eeq(of single atom)

0

1

2

3

0 20 40 60 80 120 140 160 180

100

(b)

(

111)

(113

)

(

112)

Egb

,eV

/A2

Egb

,eV

/A2

S5

(55)

S

(44) S27

(552)

S9

()

S27

(5)

S(

)

S

(8)

S(

2)

S

(225)S7

(4)

S4

(5)

S4

(556)

S9(2

2)

S

(2)

S4

(44)

S

(2)

S(

0)

S(0

0)

Elastic Deformation-Strain profiles

Position (A)

Str

ain

-40 -20 0 20 40

0.005

0.01

0.015

0.02

Position (A)

Str

ain

-40 -20 0 20 40

0.005

0.00505

0.0051

0.00515

0.0052

Strain intensification observedAt the grain boundary

9(2 2 1) Grain boundary Subject to in plane deformation

Stress profile

Position (A)

Str

ess

(eV

/A)

-40 -20 0 20 40

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

3

Stress Calculated in various regions calculated using lutsko stress

Stress Concentration observed at the grain boundary

Stress concentration present at 0 % strain indicating residual stress due to formation of grain boundary

Stress-Strain response of GB Stress Strain response of bicrystal bulk and at grain boundary Grain boundary exhibits lower modulus than bulk

Strain

No

rma

lize

dS

tre

ss(e

V/A

)

0 0.005 0.01 0.0150

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

3

9 2 2 STGB

Bulk

GB

Grain Boundary Sliding Simulation

4 5 o

Y ’

X ’

Y

X

Z [1 1 0 ]

GB

Generation of crystal for simulation of sliding. Free boundary conditions in X and Y directions, periodic boundary condition in Z direction.

X’

Y’

A state of shear stress is applied

L

NMMM

O

QPPP

0 0

0 0

0 0 0 T = 450K

Simulation cell contains about 14000 to 15000 atoms

Grain boundaries studied: 3(1 1 1), 9(2 2 1), 11 ( 1 1 3 ), 17 (3 3 4 ), 43 (5 5 6 ) and 51 (5 5 1)

Sliding Results

0 20 40 60 80 100 120 140

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

E10

(eV

/A)

Slid

ing

Dis

tanc

e(A

)

GB

x2

2

7

4

92

2

5

55

4

556

Fig.6 Extent of sliding and Grain boundary energy Vs misorientation angle

Sliding Distance

Grain Boundary energy

EG

BX

10-2

(eV

/A2)

Grain boundary sliding is more in the boundary, which has higher grain boundary energy

Monzen et al1 observed a similar variation of energy and tendency to slide by measuring nanometer scale sliding in copper

Monzen, R; Futakuchi, M; Suzuki, T Scr. Met. Mater., 32, No. 8, pp. 1277, (1995)Monzen, R; Sumi, Y Phil. Mag. A, 70, No. 5, 805, (1994)Monzen, R; Sumi, Y; Kitagawa, K; Mori, T Acta Met. Mater. 38, No. 12, 2553 (1990)

1

Reversing the direction of sliding changes the magnitude of sliding

Problems in macroscopic domain influenced by atomic scale

MD provides useful insights into phenomenon like grain boundary sliding

Problems in real materials have thousands of grains in different orientations

Multiscale continuum atomic methods required

A possible approach is to use Asymptotic Expansion Homogenization theory with strong math basis, as a tool to link the atomic scale to predict the macroscopic behavior

Homogenization methods for Heterogeneous Materials

Heterogeneous Materials e.g. composites, porous materials

Two natural scales, scale of second phase (micro) and scale of overall structure (macro)

Computationally expensive to model the whole structure including fibers etc

Asymptotic Expansion Homogenization (AEH)

Overall Structure

Microstructure

Schematic of macro and micro scales

Three Scale models to link disparate scales

Conventional AEH approach fails when strong stress or strain localizations occur (as in crack problem)

molecular dynamics in the region of localization

Conventional non-linear/linear FEM for macroscale

Displacements, energies and forces are discontinuous across the interface connecting two descriptions.

Handshaking method handshaking methods to join the two regions

Y ScaleAEH Region

AtomisticComputation

HandshakingRegion

A three scale modeling approach using non-linear FEM with or without AEH to model

macroscale and MD to model nano scale and a handshaking method to model the transition

between macro to nano scale.

AEH idea

+uy=

y

ue ux

e x

= +

Overall problem decoupled into Micro Y scale problem andMacro X scale problem

Formulation Let the material consist of two scales, (1) a micro Y

scale described by atoms interacting through a potential and (2)a macro X scale described by continuum constitutive relations.

Periodic Y scale can consist of inhomogeneities like dislocations impurity atoms etc

Y scale is Scales related through Field equations for overall material given by

X

_ _

0 on (Equilibrium)

on (Constitutive Eqn)

on

Boundary Conditions

on and on u t

fx

C e

ue

x

u u n t

xy

Hierarchical Equations Strain can be expanded in an asymptotic expansion

0 0 1 1 21

...u u u u u

e uy x y x y

Substituting in equilibrium equation , constitutive equation and separating the coefficients of the powers of three hierarchical equations are obtained as shown below.

0

0 1 0

1 2 0 1

0

0

0

uC

y y

u u uC C

y x y x y

u u u uC C f

y x y x x y

Micro equation

Macro equation

Computational Procedure Create an atomically

informed model of microscopic Y scale

Use molecular dynamics to obtain the material properties at various defects such as GB,

dislocations etc. Form the matrix and homogenized material properties

Make an FEM model of the overall (X scale) macroscopic structure and solve for it using the homogenized equations and atomic scale properties

•Y scale as polycrystal with 7 grains as shown above (50A)•Grain boundary 2A thick•Elastic constants informed from MD• E for GB =63GPA •Homogenized E=71 GPA

Summary Nanoscience based nanotechnology offers a great

challenge and opportunity. Combining superplastic deformation with other physical

phenomena in the design/manufacture/use of nanoscale devices (not necessarily large structures) should be explored.

MD/MS based simulation can be used to understand the mechanics (static and flow) of interfaces, surfaces and defects including GBs.

Using Molecular Dynamics it has been shown that extent of grain boundary sliding is related to grain boundary energy

The formulation for AEH to link atomic to macro scales has been proposed with detailed derivation and implementation schemes.