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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