Simulation: Powerful tool for Microscale Materials Design and Investigation
Dr. Xining Zang ME 138/ME238
Prof. Liwei Lin Professor, Dept. of Mechanical Engineering Co-Director,
Berkeley Sensor and Actuator Center The University of California, Berkeley, CA94720
e-mail: [email protected] http://www.me.berkeley.edu/~lwlin
Time Scale and Length Scale
Modeling and Simulation Modeling: describes the classical scientific method of formulating a simplified imitation of a real situation with preservation of its essential features In other words, a model describes a part of a real system by using a similar but simpler structure
Simulation: putting numbers into the model and deriving the numerical end-results of letting the model run on a computer. A simulation can never be better than the model on which it relies.
Computational Materials Science
Micro scale Meso scale Atomic scale Electron scale
Experiments
Materials Science Question
Model System
Simulation
Result Analysis
Theory
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Dr. Kristin Persson (LBNL) Prof. Gerbrand Ceder
Molecular Dynamics Key: Newton second law & Hamiltonian
Sir William Rowan Hamilton (1805-1865 Ireland).
Sir Isaac Newton (1643-1727 England)
Molecular Dynamics• Using the information gained from r, v and application
of the forces:
• Initial positions are advanced towards a lower energy state.
• Through some time step delta t
• New positions are obtained.
• New velocities are obtained
• REPEAT.
• This is repeated as many times as is needed to obtained equilibrium.
MD for mechanical properties
Young’s ModulusYoung’s Modulus
Yield Strength Yield Strength
Interface structure Interface structure
Dislocation-interface interactionDislocation-interface interaction
An example: ultralow elastic modulus
10μm
GPanm
10~20nm thickness, one order decrease)
Mechanical propertyNanocrystalline softening
Has been limitedly studied in typical FCC metal Cu/Ti etcGrain boundary dominated sliding > dislocation strengthening 2D stress field
Reverse Hall-PeigeNeed to be tested out
Nature 391, 561-563 (5 February 1998) International Journal of Solids and StructuresVolume 49, Issue 26, 15 December 2012, Pages 3942-3952
Why atomistic simulation?
A.Misra, M.J,Demkowicz,et,al JOM,2008 April
from • Hall-petch
to• Single dislocation• Nanocrystalline softening?
thickness
Dislocation
outline
Simulation process outline
Calculation details
Interface structure
Dislocation-interface interaction
Methodology
Results and discussion
Difficulties and questions
Simulation process outline
interface construction
relaxation [quenching MD]
calculation certain properties
Interface structure
interface
Coherent ——transparent
Semi-coherent——half-transparent
Incoherent interface ——opaque
examples
Cu-Ni (Fcc-Fcc) {Medyanik 2009}{Rao 2000}
Cu-Ag(Fcc-Fcc) {Hoagland 2002}Cu-Nb(Fcc-Bcc)
{Wang, 2008} {Wang, 2008}
KS
KS1
KS2
KSmin
plane strain adjoining atom layer
Incoherent Cu-Nb
|| , ||Cu Nb Cu Nb (111) (110) 110> 111>
Kurdjumov–Sachs (KS)
Cu and Nb slabs translate
minimize the energy
Positions of Cu and Nb not relaxed
perfect Cu and Nb
M.J.Demkowicz, Jour.Necluar.Mater.372(2008)
KS1
a strained monolayer
3 Cu, 2 Nb 1 Nb
homogeneous in-plane deformation
KS2
Requires– Direction of the monolayer rows A||1– The rows C || Nb– d(A) is chosen such that s(C)=s(Nb)– d(C) is chosen such that s(A)=s(Nb)
M.J.Demkowicz, Jour.Necluar.Mater.372(2008)
Results– 1 atoms less per 188 atoms– Energy per atom to create the artificial monolayer
0.03eV/atom– The A row lies between the valley of NbAnd the C rows lies between the Nb
• Two-body Morse
• EAM ,CuNb in CsCl, first-principle
using VASP.
Potential for the bi-layer atoms ( )( ) (1 )M Mr RCuNb M Mr D e D
' ( ) ( ) ( )
1cut
CuNb CuNb CuNb cut
m
cut CuNb
r rcut
r r r
r drm r dr
M.J.Demkowicz, Jour.Necluar.Mater.372(2008)
Cu-Nb J.wang, et,al, Acta 56(2008)3111
Cu-Nb J.wang, et.al, Acta 56(2008)5685
Cu-Nb Hoagland et.al, Philos.Mag, 2006
Cu-Ni / Cu-Ag Hoagland et.al, Philos.Mag, 2002
Cu-Nb S. Shao, et.al, Model.Sim.MSE 2010
Details about the structure construction in papers
J.wang, et.al,Acta,2008
region 1 ——region 2
PBC:
in the x and z directions.
Cu:42 185.9530 A ;19 48.5677A;
Nb: 23 185.9614 A ; 17 48.5958A;[112]Nba
[110]2Cua
[111]2Cua
[112]2Cua
KS1
KS2
u / x 0.04785 u / 0.09209z / x 0.02763w / 0.05317w z displacemnt gradient
Cu inserting strained monolayer
additional PBC displacemnt gradient
u / x 0.4144 4e u / 0.4867 3z e
/ x 0.1441 4w e / 0.8811 4w z e
Cu-Nb J.wang, et,al, Acta 56(2008)3111
X,z=18.6 nm y=4.86
Cu-Nb J.wang, et,al, Acta 56(2008) 5685
Cu-Nb Hoagland et.al, Philos.Mag, 2006
Ksmin
Relaxation:Translate in 3 directions, but rotation is not allowed Excess energy:(relative to their respective cohesive energy. )
Another way to construct the interfaceFixing bottom
wide enough
Cu-Nb S. Shao, et.al, Model.Sim.MSE 2010
Two sets of Slip plane.10.5
dislocations through interface
Dislocation transmission
Interface-dislocation interaction
Modulus interactionModulus interaction
Size interactionSize interaction
Chemical interactionChemical interaction
“ ”ineraction“ ”ineraction
Modulus interaction
modulus mismatch (Koehler barrier) introduce an image force
Koehler stress to move a screw dislocation from a softer layer to a harder layer against its elastic image
(a/2) screw dislocations are considered to be split into Shockley partials
1 2( ) /kf E E h
* 1 1 2
2 1
( )4 ( )k
bh
Size interactionvan der Merwe misfit dislocations /coherency stresscoherency stress in multilayers:
lattice parameter misfit between the lamellae, elastic constants of the individual lamellae, volume fraction of the lamellae and wavelength of the multilayer system
Misfit dislocations ——forest-type obstacles
* ,
0,
d c
c
a ba
Chemical interactiongamma surface mismatch introduce a localized force on the gliding dislocations
the negative elastic interaction energy between the Shockley partials
the total energy of the stacking fault bounding the partials.
The stress on the leading dislocation of the a pile-up
the stacking fault energy
el chE E E
elE
chE
* 2 , , , ( )ich i i Cu Ni Cuwhereb b
i
Calculation details
relaxation
defect visualizatin
thermodynamic stress
elastic constant
strain related
interface constructioninterface construction
Relaxation[quenching MD]
Relaxation[quenching MD]
calculation certain properties
calculation certain properties
{Wang, et.al, Acta (2008)5685}
Interface dislocation displacement field: Barnett-Lothe solution.
field is applied to both the movable and fixed region
Dislocation line is parallel to Z axis
Relaxation Quenching MD until the force on any atom
Lattice glide dislocation enters the interface
disregistry analysis
Pairs of atoms straddle the desired shear plane
relaxed system
atoms in the sheared systems
The disregistry
r r
ij i jr r r
d d d
ij i jr r r
d
ij ijr r
{Shao, Modelling Simul. Mater.Sci.Eng.18(2010)} conjugate gradient procedure
energy minimization
parallel atomistic simulation code LAMMPS
visualization of the defects: atomic excess energy, coordinate number;
centro-symmetry parameter
• centro-symmetry parameter:
• Atomeye function to compute P
• pairs of the opposite nearest neighbors
2
1,
| |i ii
P R R
2
1,
2
1,2
| |
2 | |
i ii
jj
R Rc
R
i iR andR
Indentation lattice glide dislocation
Indentation on Cu-Nb bi-layer
[adsorption]
Indentation on Nb-Cu bi-layer
[transmission]
• rigid spherical indenter with radius R = 40Å• quadratic repulsive force• center of the top surface
DB 1.45nm from the interface in Cu Array relaxed by energy minimization Shockley partials B D
•The lead Shockley partials enter the interface leaving trailing Shockley in the Cu near original site of DB•If the Burgers changed into BD, the enters the interface instead•The observation that one of the partials enters the interface, irrespective of sign, denies the possibility that this response is caused by small residual stresses or coherency stresses local to the interface.
B D
Results and Discussion
Slip systems in Cu/Nb
Core spreading of lattice glide dislocation within the interfaces
Sheared interface attracts lattice glide dislocation Interface
Interface shear strength
Elastic interaction between a dislocation and interface