A chemist’s perspective on cracking, fatigue failure, and

surface reactionsRobin L. Hayes

New York University

For the want of a nail the shoe was lost,For the want of a nail the shoe was lost,For the want of a shoe the horse was lost,For the want of a shoe the horse was lost,For the want of a horse the rider was lost,For the want of a horse the rider was lost,For the want of a rider the battle was lost,For the want of a rider the battle was lost,For the want of a battle the kingdom was lost--For the want of a battle the kingdom was lost--And all for the want of a horseshoe nail. And all for the want of a horseshoe nail.

-Benjamin Franklin-Benjamin Franklin

Acknowledgements

CaltechPrinceton

Emily A. Carter Michael Ortiz

NYU

Mark Tuckerman

• NDSEG Fellowship• DOD-MURI via AFOSR• DOE-ASCI

• NSF CHE-0121375• NSF CHE-0310107

NIST

Emily Jarvis

Density Functional Theory (DFT)

]r[E)]r([E)]r([E)]r([J)]r([T)]r([E iiextxcs Total

EnergySingle

Particle Kinetic Energy

Hartree Electron - Electron Energy

Exchange-Correlation Functional

Electon-Ion Coulombic Interaction

Ion-Ion Energy

Self-Consistent Equations:

][)(

][

)(

][

)(

][

)(

][rV

r

E

r

J

r

T

r

Eext

xcs

Veff[]Exc small,

but not known exactlypseudopotential

Which flavor of DFT

Kohn Sham: Expand density in orbitals

occ

i*i )r()r()r(

2/N

ii

2is

e

21

2)]r([T

Pseudopotentials:

Replace chemically unimportant core electrons with numerically tractable potential

PRB, 136, 864 (1964); PRA, 140, 1133 (1965).

Macroscopic Crack Models

Oxidation induced cracking

Metal

Mode 1 Cracking

Stress corrosion cracking in an Al aerospace part

Attraction between surfaces

t*

*

Crack tipOxide Cohesive elements

Universal Binding Energy Relationship (UBER)

UBER describes cohesion, adhesion of unrelaxed surfaces, chemisorption, diatomic molecules

ao eaEE )0.1(

Fitting parameters

Bulk Crystal

Unrelaxed Crack

dur

equilibrium interlayer spacing

Rose, Smith, Ferrante, Phys Rev. B 28, 1835 (1983).

uraur

urourur eaEE )0.1(

Expressed as initial crack, ur:

twice the unrelaxed surface energy = 2 * (E-Ebulk)/(2 Asurf)

UBER for Unrelaxed Crack

Traditional Cohesive Law Inadequate for Continuum Model

• UBER fails for QM UBER fails for QM energies of relaxed energies of relaxed

surfacessurfaces

• EE∞∞, reduction in , reduction in surface energy due to surface energy due to relaxation, is HUGE relaxation, is HUGE

for Alfor Al22OO33

DFT1 Continuum

c (Å) 0 – 3 104

c (GPa) 10 0.04 – 0.8

Wad (J/m2) 1 1

[1] This work and Evans, Hutchinson, Wei, Acta Mater. 47, 4093 (1999).

Failure criterion differ Failure criterion differ by several orders of by several orders of

magnitudemagnitude

Energies of (0001) -Al2O3

ur (Å)

Ene

rgy

[J/m

2]

DFT Unrelaxed

DFT Relaxed

UBER FitE

c

c

Wad

Set up the crack problem

NN

iiN

totE ,,,, 111

01

local non-local

N

ii

1

,,1

0 N

Uniformly Expanded

Nd

Equilibrium

d

N ,,11

non-local behavior only

important near crack

Total displacement

For specific material behavior, For specific material behavior, need first-principles calculationneed first-principles calculation

Crack

id

1 id

Hayes, R.L., Ortiz, M. and Carter, E.A. PRB, 69 (2004) 172104.

Assumptions

Typical interatomic potential

• Periodic unit cell• Uniaxial tensile stress only (mode 1 cracking)• No dislocations – brittle fracture• Convex on interval 0 ≤ ≤ 0

• Inflection point at 0

• Concave for > 0

• 0 dominates bulk crystal behavior,

1 locally perturbs near crack surfaces

Nguyen and Ortiz solved for 0 in the limit of large N

J. Mech. & Phys. Solids 50, 1727 (2002).

Solution for Local PartUniaxial Moduli

Exaggerated viewd

N

otherwise

ifNC

N

C

,2

,22,

2min

0

02

02

0

CN00 2 Unrelaxed Surface Energy

Purely Harmonic Elastic Deformation Rigidly Separate Surfaces

Healed regime Cracked regime

Exaggerated viewd

0

Solution for Total Energy

Matched Asymptotic Expansion Replace 0 with r

otherwise

ifNC

N

C

r

rr

,2

,22,

2min

22

CNrr 2

otherwise

ifNCt rr

,0

, NCNC rrr 2

Relaxed Surface Energy

Traction used in engineering simulations of cracking to account for surface-surface interactions

r N~

r N1~

Macroscopic failure criteria

Theory in line with experiment

Further Generalizations to Universal Curve

NNNN

,,;,,min,, 11,,

11

01

N

ii

If …• unit cell remains periodic• steady state process on time-frame of crack formationThen i, extra degrees of freedom, can be eliminated

Do constrained minimization to reduce out i

Minimize ,...,N as before

Examples of i:

• Bravais sublattices (i.e. Al2O3)• impurity concentration• tangential displacement, , if constrained to

Three Representative Materials

• (111) surface of fcc Al

• surface remains at bulk termination (~1%

outward expansion)

• weak bulk cohesion

• ductile - dislocations form easily

sapphire ruby

• (100)-2x1 surface of cubic diamond Si

• surface relaxes inward by ~2% & reconstructs into rows of buckled dimers

• brittle – dislocations do not form easily

• (0001) surface of -Al2O3

• surface severely relaxes inward by ~33% (~0.7 Å)

• strong bulk cohesion

• brittle – dislocations do not form easily

Metals Semiconductors Ceramics

• Vary by uniformly stretching the material • fit to 0 to get C • plot vs

DFT Comparison Calculations

Uniform Expansion Introduce Crack and Relax

• Insert at the crack, fix the unit

cell, and allow ions to relax• Either start at ideal bulk termination or the relaxed structure from a larger • use at largest as 2r • plot vs

Universality of Asymptotic Binding Energy Relationship for Relaxed Surfaces

r* 2

ONLY the uniaxial elastic constant (C), relaxed surface energy (ONLY the uniaxial elastic constant (C), relaxed surface energy (rr) and number ) and number

of layers (N), needed to describe cracks with relaxed surfaces (slow cracking)of layers (N), needed to describe cracks with relaxed surfaces (slow cracking)

Metals, semiconductors, and Metals, semiconductors, and ceramic fall on universal curve!ceramic fall on universal curve!

Al QM

Al2O3 QM

* = 1* = *2

Si QM

filled healedopen cracked

*

* Deviations from universal Deviations from universal behavior around behavior around ** = 1 = 1(due to small (due to small NN in QM in QM

calculations…)calculations…)

2/1

r

*

N

C

2

1

Nguyen, O. and Ortiz, M. J. Mech. & Phys. Solids 50, 1727 (2002)Hayes, R.L., Ortiz, M. and Carter, E.A. PRB, 69 172104 (2004).

Cracked DFT points at * < 1

Source of DeviationCracked DFT points at * < 1

Crack cannot heal until eCrack cannot heal until e-- density density bridges crack bridges crack crack surfaces crack surfaces

“see” each other and heal“see” each other and heal

ChemPhysChem 2, 55 (2001).

““healing” distance nearly healing” distance nearly independent of independent of NN

shift “crack” curve to smaller * as N increases

Al2O3 Al

• Arises from surface – surface Arises from surface – surface interactions immediately before interactions immediately before crack heals crack heals• Stronger bulk cohesion (i.e. AlStronger bulk cohesion (i.e. Al22OO33) )

surfaces approach closer surfaces approach closer larger variation in energy

Crack Forms Crack Heals*

*

cracks

*

*

heals

Universal curve valid for all cracking Universal curve valid for all cracking and healing ifand healing if ** > 1> 1

Si

N1* ~

Universal form for relaxed surface attractive forces in the limit of large N

Work of Adhesion (Wad) is independent of the number of layers!Experiment and theory should match

t*

tcrack* = 0

telast* = 2*

Wad = area under curve

12*1*21 adW

Energy units = 2r

*

0.1

1

10

100

1000

10000

100000

[Å]

• Uniaxial moduli have the correct orderingUniaxial moduli have the correct ordering• Surface energies are the correct order of magnitudeSurface energies are the correct order of magnitude

• Renormalization brings the failure criteria in line with experimental valuesRenormalization brings the failure criteria in line with experimental values

Do Theory and Expt. Agree?

r r

0.1

1

10

100

1000

10000

100000

[MPa]

0

0.5

1

1.5

2

2.5

[Å]

Lattice Constant

0

50

100

150

200

250

[GPa/

Å]

Uniaxial moduli

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

[J/m̂

2]

Relaxed surface energy

Al (this work)Al (expt)

Al2O3 (this work)Al2O3 (expt)

Si (this work)Si (expt)

Intelligently Informing Macroscopic Crack Models

Universal Binding Energy Relationship for Relaxed

Surfaces

Interlayer Spacing, d

Uniaxial Elastic

Constant, C

Relaxed Surface

Energy, r

Expt. Expt. Expt.

Bulk geometry optimizati

on

Uniaxial Expansion

Infinitely separated surfaces

Macroscopic models of cracking

Example: Hydrogen Embrittlement of Steel

hydrogen

Initial crack

Crack speed depends on hydrogen concentration

Final crack

Serebrinsky, Carter, Ortiz J. Mech. Phys. Sol. 52 (2004) 2403.

Fatigue failure in Si?

• Stress-assisted surface oxide dissolution Oxide forms on crack surface, preferentially dissolves, grooves nucleate cracks [Shrotriya, Allamech, Brown, Zuo, and Soboyejo Exp. Mech. 50 (2003) 289.]

• Reaction layer fatigue Oxide preferentially forms in high stress regions which then develop microcracks [Muhlstein, Stach, and Ritchie Acta Mater. 50 (2002) 3579.]

• Mechanically induced subcritical cracking Subcritical crack growth in Si → accumulation of damage at crack tip [Kahn, Ballarini, Bellante, and Heuer, Science 298 (2002) 1215.]

Fatigue cracking usually in ductile materials → surprise in Silicon

Monotonic loading Cyclic loading

time

load *

time

load *

Connally and Brown Science 256 (1992) 1537.

Si (100) Surface ReconstructionTop View Side View

Tilted Dimer

Phys. Rev. B 55 4731 (1997).

Expt. LEED at 120 K – 190 K

Phys. Rev. Lett 89 286104 (2002).

STM at 4 K

p-type substrate

(2x1)

c(4x2)

(2x1)

n-type substrate

(2x1)

p(2x2)

(2x1)Some form of tilted dimer is likely the ground state

of Si near 0 K

Model Assumptions(where new work is needed)

• Crack can be represented by parallel slabs• (100)2x1 reconstruction captures enough

of the physics [cracks actually form in (110) or (111) planes]

• Series of static calculations can capture millions of fatigue cycles

• Absence of Oxygen and H2O does not alter the conclusions.

Hysteresis in Silicon

For 12 layers, healed and cracked (100) Si

have the same energy

Follow uniform expansion curve when

crack first forms

Reconstructed surface prevents ideal bulk

crystal from reforming

Energy barrier prevents lower energy reconstructed surface

from forming

DFT Renormalized Energy – Displacement Curve

Reconstructed surface causes hysteresis during load cycling

3% strain1.13 J/m2

Surface reconstruction prevents perfect healing

Improper interfacial healing suggests that mechanically induced subcritical crack formation may be the primary mechanism of fatigue failure in Si.

Hayes and Carter JCP (2005) in press.

Organic-Semiconductor Interfaces

Nano-lithographyExample: - Passivate Si(100) surface with benezene - Create 2 nm wide patterns with STM tip - React with vinyl ferroceneKruse and Wolkow Appl. Phys. Lett. 81 (2002) 4422.

Self-assembled nanowires and other nanostructuresExample: - styrene forms lines on H-Si(100) - precursor to molecular electronicsDiLabio, Piva, Kruse, and Wolkow JACS 126 (2004) 16048.

MonolayersExample: - monolayer of 1,5-cyclooctadiene absorbed on Si(100) - -bond on surface available for further rxns - precursor to molecular sensorDiLabio, Piva, Kruse, and Wolkow JACS 126 (2004) 16048.

Proposed Reaction Mechanism of 1,3-cyclohexadiene addition to the (100)Si-2x1 surface

Minary, Tuckerman JACS 127 (2005) 1110.

“+” charge on down Si

Reaction proceeds through a stepwise zwitterionic mechanism

Resonance → 2 locations for carbocation

ji

ijjiijI

Ii

ii RERML,

2 ],[2

1

Car-Parrinello Molecular Dynamics

electrons orthogonality constraint

The results are trustworthy if …

DFTatoms

• Basis set converged• Pseudopotential• Exchange-correlation function• Boundary conditions1 • Thermostat• Fictitious electron mass• Time step

- with plane waves, ↑ the kinetic energy cutoff

- gives physical results (geometry, elastic constants, vibrations)

- ex. energy conservation, small cp temperature, small cp forces

[1] Minary, Tuckerman, Pihakari, Martyna JCP 116 (2002) 5351.

Reaction kinetically controlled

A 11±3 15

B 16±7 15

C 31±6 30

D 10±6 30

E 12±9

Pre

dic

ted d

ecr

easi

ng p

op.

Thermo-dynamic

Expt. STM (%)

STM data from Teague and Boland, TSF 464 (2004) 1. Theory from Minary and Tuckerman, JACS 127 (2005) 1110.

Product Distribution of 1,3 cyclohexadiene on

Si(100)Theory, CPMD 1,3-butadiene (%)

A1 site

B1 site

C2 sites

D1 site

E2 sites

Preliminary results

Reaction proceeds through an asymmetric transition state

Conclusions

• First principles traction vs. separation relationship for FEM simulations

• Atomic scale reconstructions may cause fatigue failure in Si

• Reaction mechanisms for organic-semiconductor interfaces