Multi-scale Modeling of Nanocrystalline Materials N Chandra and S Namilae Department of Mechanical Engineering FAMU-FSU College of Engineering Florida State University Tallahassee, FL 32312 USA Presented at ICSAM2003, 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 phenomena At 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.

Mechanics at atomic scale Molecular Dynamics -Fundamental quantities (F,u,v) Compute Continuum quantities -Kinetics (,P,P’ ) -Kinematics (,F) -Energetics Use 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 Stress defined for whole system For Brenner potential: 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  - Used for inhomogeneous systems

Lutsko Stress - fraction of the length of - bond lying inside the averaging volume Based on concept of local stress in statistical mechanics used for inhomogeneous systems Linear momentum conserved

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 GB Mesh of tetrahedrons

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 GB GB [110]3 (1,1,1) [001]5(2,1,0) GB GB [110]9(2,21) [110]11(1,1,3)

Grain Boundary Energy Computation GBE = (Eatoms in GB configuration) – N  Eeq(of single atom) Calculation Experimental Results1 1 Proceeding Symposium on grain boundary structure and related phenomenon, 1986 p789

Elastic Deformation-Strain profiles 9(2 2 1) Grain boundary Subject to in plane deformation Strain intensification observed At the grain boundary

Stress profile 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 GB

Grain Boundary Sliding Simulation Generation of crystal for simulation of sliding. Free boundary conditions in X and Y directions, periodic boundary condition in Z direction. Y’ X’ Simulation cell contains about 14000 to 15000 atoms BC for sliding are shown. Experimental studies are performed on bicrystals oriented at 45 deg. Similar BC were used and shear state of stress was applied. Temp of 450K . The crystals contained about 15000 atoms and 5000 iterations were done. A state of shear stress is applied T = 450K 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 Grain boundary sliding is more in the boundary, which has higher grain boundary energy Fig.6 Extent of sliding and Grain boundary energy Vs misorientation angle Monzen et al1 observed a similar variation of energy and tendency to slide by measuring nanometer scale sliding in copper Reversing the direction of sliding changes the magnitude of sliding 1 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)

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) 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 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 Overall problem decoupled into Micro Y scale problem and Macro 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

Hierarchical Equations Strain can be expanded in an asymptotic expansion Substituting in equilibrium equation , constitutive equation and separating the coefficients of the powers of  three hierarchical equations are obtained as shown below. 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.