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SLACS-INFM/CNR Sardinian Laboratory for Computational Materials Science www.slacs.it SLACS Atomically informed modeling of the microstructure evolution of nanocrystalline materials A. Mattoni [email protected]
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Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Page 1: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR Sardinian Laboratory for Computational Materials Science

www.slacs.itSLACS

Atomically informed modeling of the microstructure evolution of nanocrystalline materials

A. Mattoni

[email protected]

Page 2: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

SLACSCNR-INFM

CRSLNLR Regional Laboratories

Atomistic investigation: large scale molecular dynamics simulations Large scale electronic structure calculationsContinuum modeling: models for growth, interface mobilities

http://www.slacs.it

•Division: Material Physics (Microstructure evolution of nanostructured materials)(6 members,

www.dsf.unica.it/colombo)

Page 3: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

OUTLINE

The microstructure of interest for nanocrystalline materialsBoundaries between order/disordered phase

The theoretical frameworkMolecular dynamics atomistic simulations

Modeling the growth of nanocrstals embedded into an amorphousmatrix

Page 4: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Molecular dynamics

The material of interest is The material of interest is described as an assembly of described as an assembly of molecular constituentsmolecular constituents

Page 5: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Molecular dynamics

An interatomic depending on atomic positionsAn interatomic depending on atomic positions

rF i = mi

r ˙ ̇ r i

rF i = −

r ∇ iU(

r r 1,...,

r r N )

U(r r 1,...,

r r N )

The interatomic forces are calculated accordinglyThe interatomic forces are calculated accordingly

Newton’s equations of motion are integrated Newton’s equations of motion are integrated numerically (“Verlet velocity”) numerically (“Verlet velocity”)

Choose dt “judiciously” (~1fs) and iterate in time (“ad nauseam”)Choose dt “judiciously” (~1fs) and iterate in time (“ad nauseam”)€

r(t + dt) = r(t) + v(t)dt + 0.5a(t)(dt)2

v(t + dt) = v(t) + 0.5[a(t) + a(t + dt)]dt

Page 6: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Interatomic potentials

U(r r 1,...,

r r N ) = V2(rij )

i< j

V2(rij ) = 4εσ

r

⎝ ⎜

⎠ ⎟

12

−σ

r

⎝ ⎜

⎠ ⎟6 ⎡

⎣ ⎢

⎦ ⎥

””6-12” Lennard-Jones potential: repulsive core 6-12” Lennard-Jones potential: repulsive core 1/r1/r1212 ; VdW attraction 1/r ; VdW attraction 1/r66 r>r r>reqeq

””6-12” Lennard-Jones potential: prototypical 6-12” Lennard-Jones potential: prototypical interatomic force model for close-packed metalsinteratomic force model for close-packed metals

QuickTime™ e undecompressore TIFF (Non compresso)

sono necessari per visualizzare quest'immagine.

Professor Sir John Lennard-Jones (FRS), one of the founding fathers of molecular orbital theory

Page 7: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Interatomic potentials

U(r r 1,...,

r r N ) = V2(rij )

i< j

∑ + V3(rij ,rik,cosϑ ijk )i< j<k

Stillinger-Weber potential for anysotropic Stillinger-Weber potential for anysotropic covalent bonding (1985)covalent bonding (1985)

QuickTime™ e undecompressore TIFF (Non compresso)

sono necessari per visualizzare quest'immagine.

F. StillingerDepartment of Chemistry Princeton University Princeton, NJ 08540

T.A. Weber

U(r r 1,...,

r r N ) = V2(rij ,Zi)

i< j

∑ + V3(rij ,rik,cosϑ ijk,Zi)i< j<k

(EDIP) Environment dependent interatomic (EDIP) Environment dependent interatomic potential (1998)potential (1998)

Z i= u(rli)l≠ i

Page 8: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

MD comes of age…K. Kadau et al. Int. Journal of Modern Physics C 17 1755 (2006)

B. J. Alder and T. E. Wainwright,

J. Chem. Phys.27,1208(1957)

Stillinger-WeberStillinger-WeberLennard-JonesLennard-JonesTersoffTersoff

EDIPEDIP

320 BILLION ATOM SIMULATION ON BlueGene/LLos Alamos National Laboratory

Page 9: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

MD comes of age… more or less

Compromise between Compromise between accuracy accuracy and and computational workloadcomputational workload

The bottleneck of standard molecular dynamics: The bottleneck of standard molecular dynamics: time and length scalestime and length scales

In order to properly reproduce fracture related properties of covalent In order to properly reproduce fracture related properties of covalent materials of group IV materials (Si, Ge, C) it is necessary to take into materials of group IV materials (Si, Ge, C) it is necessary to take into account interactions as long as the second nearest neighbors distanceaccount interactions as long as the second nearest neighbors distance

A. Mattoni, M. Ippolito and L. Colombo, B 76, 224103 (2007)

ReliabilityReliability of the model potentials of the model potentials

Page 10: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Computational Effort

CMPToolCMPTool: : a set of highly efficient parallel a set of highly efficient parallel numerical libraries for computational materials numerical libraries for computational materials science developed in collaboration with science developed in collaboration with CaspurCaspur, , RomeRomeGroup of materials scienceGroup of materials science (M. Rosati, S. Meloni, L. Ferraro, M. Ippolito)(M. Rosati, S. Meloni, L. Ferraro, M. Ippolito)

Typical simulation parametersTypical simulation parametersnumber of atomsnumber of atoms > > 10 1055 Runs as long asRuns as long as 6 10 6 1066 iterations (6 ns) iterations (6 ns)

1ns annealing of 100000 atoms1ns annealing of 100000 atoms requires of the order of requires of the order of 1000 CPU1000 CPU hours on state-of-the-art AMD - Opteron Dual core Linux clusterhours on state-of-the-art AMD - Opteron Dual core Linux cluster

A. Mattoni et al. Comp. Mat. Sci. 30 143 (2004)S. Meloni et al. Comp. Phys. Comm. 169 462 (2005)

Page 11: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Nanocrystalline materials

Crystalline materials

0-D

Points:I,V, clusters, dots

Lines:Dislocations

1-D

Interfaces:Grain boundaries

2-D 3-D

Amorphous materials

In the amorphous phase (isotropic) the concept of dislocation is lost The microstructure evolution is controlled by: Recrystallization, normal grain growth

Plastically deformed materials

Ion implantation

Page 12: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Mixed phase nanocrystalline systems

Nanocrystalline materials (nc-Si) Nanocrystalline materials (nc-Si) may be prepared through the may be prepared through the crystallization of amorphous crystallization of amorphous (disordered) (disordered) nc grains are embedded into a second nc grains are embedded into a second phase matrixphase matrix

Experimentally it is found that the smallest grain size is obtained when the amorphous samples are annealed at a crystallization temperature that is close to half the bulk melting temperatureQ. Jiang, J. Phys.: Condens. Matter 13 (2001) 5503–5506

nc

Embedding amorphous matrix

Page 13: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Nc-Si for photovoltaics

Nano-crystalline silicon (nc-Si) Nano-crystalline silicon (nc-Si) consists in a distribution of grains consists in a distribution of grains embedded into an amorphous embedded into an amorphous matrixmatrix

Observation of domains separated by amorphous boundaries and (in some cases texturing)

Bright field TEM micrograph

S. Pizzini et al.Mat. Sci. Eng. B 134 p. 118 (2006)

Page 14: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Modeling the a-/nc- evolution

During annealing of amorphous bulk it is difficult to deconvolve nucleation from growth (impurities, control the temperatures, grains impingement) C. Spinella et al. J. Appl. Phys. 84 5383 (1998)

Atomistic simulation as a tool to perform numerical experiment under perfectly controlled conditions of temperature and purity

What is the equation of motion of an isolated a-c boundary (planar or curved)?

Silicon as a prototype of a covalently bonded material

Mattoni and Colombo, Phys. Rev. Lett. 99, 205501 (2007)

Page 15: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Why does a grain grow?

a-Si/c-Si is a metastable system

M. G. Grimaldi M. G. Grimaldi et alet al. Phys. Rev. B 44 1546 (1991). Phys. Rev. B 44 1546 (1991)

ga−c = ga − gc ~ 0.1 eV/atom1 kJ/mole=1.03 10-2 eV/atom

Page 16: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Driving force pa-c

Driving force Driving force : specific free-energy : specific free-energy differencedifference

dG = −ga−cdVc + γ a−cdS < 0

pa−c = −dG

dVc

> 0

pa−c = ga−c − γ a−c

dS

dVc

dVc

Page 17: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Equation of motion

Transition state theory

κ ~ Mpa−c

dR ~ κdt

Interface limited growth

Equation of motion of the a-c displacementEquation of motion of the a-c displacement

dR

dt= M(ga−c − γ a−c

dS

dV)

Eb

a-Sia-Si c-Sic-Si

pa−cVatom

Page 18: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Transition State Theory

Eb

a-Sia-Si c-Sic-Si

κ =κ+ −κ−

κ =ωe−

Eb

kT −ωe−

Eb +Vatom pa−c

kbT

pa−cVatom

κ ~ ωVatom

e−

Eb

kbT

kbT

⎜ ⎜ ⎜

⎟ ⎟ ⎟pa−c = Mpa−c

Page 19: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Curved a-c boundary

r =R

R*

dR

dt= υ SPE (1−

R*

R)

The capillarity is expected to The capillarity is expected to be sizeable up to R~Rbe sizeable up to R~R** and and there give rise to anthere give rise to anAccelerated -> uniform growthAccelerated -> uniform growth

R* =γ a−c

ga−c

In silicon RIn silicon R**< 1 nm< 1 nm

Page 20: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Planar a-c boundary

Uniform motionUniform motion: the a-c velocity is constant: the a-c velocity is constant

A. Mattoni et al. EPL 62 862 (2003)

vSPE (T) =M0

kbTga−ce

−Eb

kTExponential dependence on TExponential dependence on T with E with Ebb=2.6eV=2.6eVEXP G. L. Olson Mater Sci. Rep. 3, (1988) EXP G. L. Olson Mater Sci. Rep. 3, (1988)

AS N. Bernstein et al. PRB 61 6696 (2000AS N. Bernstein et al. PRB 61 6696 (2000))

QuickTime™ and aCinepak decompressor

are needed to see this picture.

dR

dt= vSPE (T)

Page 21: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Curved a-c boundary

c-Si/a-Si: c-Si/a-Si: Isolated Crystalline fiber Isolated Crystalline fiber embedded into the amorphous phaseembedded into the amorphous phase

nc-Si/a-Sinc-Si/a-Si: Crystalline fiber embedded : Crystalline fiber embedded into an amorphous phaseinto an amorphous phase[1 0 0] case[1 0 0] case

Page 22: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Characterization of the a-nc system

Page 23: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Structure Factor

T/Tm

1.00.5 1.50.0

amorphous

Page 24: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Analysis

Θ

Page 25: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Crystallinity

Crystallinity of a mixed a-Si/nc-SiCrystallinity of a mixed a-Si/nc-Si: relative number of crystalline atoms: relative number of crystalline atoms

Θ=χCΘC + (1− χ C )Θα

χC =Θ − Θα

ΘC − Θα α

α

Θ−ΘΘ−Θ

≈)(

),(

T

tT

C

χC (T, t)∝ A = πR2

R(t,T) =χ C (t,T)

πL2

Page 26: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Fiber recrystallization

Page 27: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

Power law model

υ(R) = Mλ

R

⎝ ⎜

⎠ ⎟

1

q−1

ga−c (1−R*

R)

υ ~ R1−

1

q

Power law model Power law model the model describes both decreasing and increasing the model describes both decreasing and increasing nonuniform growthnonuniform growth

υ(R) = M ga−c (1−R*

R)

υ ~ 1

R ~ t

R ~ t q

Page 28: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Fiber recrystallization

R ~ t q

Page 29: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Fiber recrystallization

ThereThere is a is a dependence of dependence of the the growth growth exponents on exponents on temperaturetemperature and there is a and there is a clear transition clear transition close to the close to the amorphous amorphous meltingmelting

Page 30: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Fiber recrystallization

υ =∂R

∂t(t,T) =

∂R

∂t(t(R,T),T) = υ (R,T)

Page 31: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Fiber recrystallization

Page 32: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

SLACS-INFM/CNR

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MATHEMATICAL MODELS FOR DISLOCATIONS

Characterization of defects

Page 33: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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Rome, December 13, 2007

MATHEMATICAL MODELS FOR DISLOCATIONS

A simple explanation

dR ~ a

dR

dt~ υ τ

n f

a

R

n f ~ Rα +1

dR

dt~ Rα

dR

dt~

1

R

n f ~ n0

n f ~ R

dR

dt~ υ 0

dt ~L f

υ τ

dR

dt~

a

L f

υ τ

Page 34: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Conclusions

• Molecular dynamics simulation are emerging as a powerfool tool to Molecular dynamics simulation are emerging as a powerfool tool to help the characterization of the microstructure evolution of help the characterization of the microstructure evolution of nanostructured materialsnanostructured materials

• An atomically informed continuum model is found to describe An atomically informed continuum model is found to describe recrystallization in both the cases of isolated grain and distribution of recrystallization in both the cases of isolated grain and distribution of grainsgrains

Contact: [email protected]

EU-STREP “NANOPHOTO”CASPUR-ROME and CINECA-BOLOGNA computational support

A. Mattoni and L. Colombo, Phys. Rev. Lett. 99, 205501 (2007)

M. Fanfoni and M. Tomellini, Phys. Rev. B 54, 9828 (1996)

www.dsf.unica.it/colombo)

C. Spinella et al. J. Appl. Phys. 84 5383 (1998)

Page 35: Atomically informed modeling of the microstructure evolution of nanocrystalline materials

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MATHEMATICAL MODELS FOR DISLOCATIONS

Recrystallization