Upload
meredith-fields
View
218
Download
0
Tags:
Embed Size (px)
Citation preview
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.