Application of Asymptotic Expansion Homogenization to Atomic Scale

N Chandra and S Namilae

Department of Mechanical EngineeringFAMU-FSU College of Engineering

Florida State University

Why link atoms and continuum ?

“Nanotechnology”

Atomic details (structural and Material) have profound influence on properties

-Thermomechanical-Physical, electrical, magnetic

However computational problems-100 nm cube of Si ~ billion atoms

Macroscopic phenomenon effected by atomic scale details

Fracture Crack tip

Plasticity Dislocation

Grain boundaries

“Materials by design”

Creep /SP

Problems in Atomic scale domain

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

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(22)

S

(2)

S4

(44)

S(2)

S(0)

S(00)

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

74

922

555

4556

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

Mg Segregation in Al Grain Boundaries

Y distance from GB (A)

E10

(eV

/A)

0 5 10 15 20

2.86

2.88

2.9

2.92

2.94

2.96

2.98

3

GB

X

-22

E for Pure Al = 2.82 10 (eV/A )GB

x-2 2

1 2 S9 (2 2 )STGB

Mg Atom

Y

Position 1

Position 2

Y distance from GB (A)0 2 4 6 8 10 12 14

0.75

0.755

0.76

0.765

0.77

0.775

0.78

E10

(eV

/A)

GB

x

22

EGB For Pure Al =0.65 x 10-2 (eV/A2)

Mg Atom

Y

S ( ) STGB

Segregation of Mg atoms to particular locations in grain boundary is based on size effect and hydrostatic pressure

Variation of grain boundary energy in presence of Mg atom

Hydrostatic Stress and Segregation Energy

Grain boundary energy and segregation are influenced by changes in coordination of atoms at grain boundary

Simulation results also indicate that there is an increase In grain boundary sliding when Mg atoms are present

Effect of Mg on sliding

(r/r )

Num

ber

ofA

tom

s

0 1 2 30

2

4

6

8

10

12

e (r/r )

Num

ber

ofA

tom

s

0 1 20

2

4

6

8

10

e(r/r )

Num

ber

ofA

tom

s

0 1 2 30

2

4

6

8

10

e

Distribution of atoms around impurity atom in 9 STGB

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

Sinclair (1975) Hoagland et.al (1976)

Mullins (1982)

Gumbusch et.al. (1991)

Tadmor et.al. (1996), Shenoy et.al. (1999)

Flexible Border Technique

Finite element Atomistic method

FE-At method

Quasicontinuum method

Continuum-Atomics linking

Rafii Tabor (1998)

Broughton et.al. (2000)

Lidorkis et. al. (2001)

Friesecke and James

Three scale model

Coarse grained molecular dynamics

Handshaking methods -CLS

Multiscale scheme

Continuum-Atomics linking

Other efforts: CZM based, description of continuum in atomic Regions, lipid membranes etc

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

AEH idea

+uy=

y

ue ux

e x

= +

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

AEH literature Functional analysis

Bensoussan et.al. (1978), Sanchez Palencia (1980) Elasticity well established

Kikuchi et. al. (1990) Adaptive mesh refinement Hollister et. al. (1991) Biomechanics Application Ghosh et. al. (1996),(2001) AEH combined with

VCFEM Buannic et. al. (2000) Beam theory with AEH

Inelastic Problems Fish et.al. (2000) Plasticity Chung et.al. (2001) Viscoplasticty

Transport Problems in Porous media

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

Contd

xy

0 1 22, , , ..u u x y u x y u x y

The functions u(i) (x,y) are Y periodic in variable y. and are independent of the scaling parameter .

The basic concept in AEH is to expand the primary variables as an asymptotic series. Using the expansion for displacement u

From the definition of the scaling parameter, for any g(x,y)

, , ,1g x y g x y g x y

x yx

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

Microscale EquationUsing the following transformation

0

1 uu

x

Micro equation can be solved as

0

. 0u

Cy y x

Y Y

v CC dY v dYy y y

In Variational form

corrector term in macro scale due to microscale perturbations. series of vectors

Microscale Equation• The Y scale here is composed of atoms interacting through an interatomic potential.

• If we consider a finite element mesh refined to atomic scale in the Y region then, would denote the atomic level stiffness matrix • W is the total strain energy density of the Y scale and q dente the displacements of individual atoms. • Micro equation can be solved as

2

q q

W

2

T locB Cq q

W

CY

at atomic level (6xN)BTCloc

q Atomic displacements

Cloc Local elastic constants determined from MD

Macroscale equationGiven by

apply the mean operator on this equation, by virtue of Y-periodicity of u(2) equation reduces to

1 2 0 1

0u u u u

C C fy x y x x y

0

0H uC f

x x

C H is the homogenized elasticity matrix for the overall region given by

1.H

YC dy

Y y

(A)

Equation (A) solved by FEM with appropriate BC gives solution corrected for atomic scale effects

Local Elastic Constants

Based on Kluge et al J. of App. Phy. (1990)

Knowing local strain and local stress in a small region V of MDSimulation local elastic constants

0i

loc

F

dVC

e

system of N interacting atoms in a parallelepiped whose edges are described by vectors a, b and c with H=(a,b,c)

Constant strain application H=Ho to H=Ho+ Ho

(Parinello –Rahman Variable cell MD)

Local Elastic constants

Local stress in small area defined as

, ,1 i j

ij ijij ij

g r r sUr r

r r

, , ( )

( )

i

i j ij i j

i

s r

g r r s s R s r s r

s r

volume , rij distance between ith and jth atoms,

U interatomic potential functionunit step function Dirac delta function Rij center of mass of particles i and j

This Method has been applied to grain boundaries using EAM and pair potentials

Computational Procedure Create an atomic 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

Summary Incorporating atomic-scale effects in

determining the material behavior is important in a number of engineering applications.

Grain boundaries structure and deformation characteristics can be studied at atomic scale. 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.

Work is underway to implement the computational methodology.