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

A chemist’s perspective on cracking, fatigue failure, and surface reactions. Robin L. Hayes New York University.

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

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  1. A chemist’s perspective on cracking, fatigue failure, and surface reactions Robin L. Hayes New York University 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 horse the rider was lost,For the want of a rider the battle was lost,For the want of a battle the kingdom was lost--And all for the want of a horseshoe nail. -Benjamin Franklin

  2. NDSEG Fellowship • DOD-MURI via AFOSR • DOE-ASCI • NSF CHE-0121375 • NSF CHE-0310107 Acknowledgements NIST NYU Princeton Caltech Emily A. Carter Mark Tuckerman Michael Ortiz Emily Jarvis

  3. Density Functional Theory (DFT) Total Energy Single Particle Kinetic Energy Hartree Electron - Electron Energy Exchange-Correlation Functional Electon-Ion Coulombic Interaction Ion-Ion Energy Self-Consistent Equations: Exc small, but not known exactly pseudopotential Veff[r]

  4. Which flavor of DFT Pseudopotentials: Replace chemically unimportant core electrons with numerically tractable potential Kohn Sham: Expand density in orbitals PRB, 136, 864 (1964); PRA, 140, 1133 (1965).

  5. t* d* Macroscopic Crack Models Oxidation induced cracking Metal Stress corrosion cracking in an Al aerospace part Oxide Crack tip Cohesive elements Attraction between surfaces Mode 1 Cracking

  6. Fittingparameters Universal Binding Energy Relationship (UBER) UBER describes cohesion, adhesion of unrelaxed surfaces, chemisorption, diatomic molecules Rose, Smith, Ferrante, Phys Rev. B28, 1835 (1983). Expressed as initial crack, dur: Bulk Crystal Unrelaxed Crack UBER for Unrelaxed Crack equilibrium interlayer spacing d dur twice the unrelaxed surface energy = 2 * (E-Ebulk)/(2 Asurf)

  7. Energies of (0001) a-Al2O3 DFT Unrelaxed UBER Fit Energy [J/m2] DFT Relaxed dur (Å) sc Wad dc Traditional Cohesive Law Inadequate for Continuum Model • UBER fails for QM energies of relaxed surfaces • DE∞, reduction in surface energy due to relaxation, is HUGE for Al2O3 Failure criterion differ by several orders of magnitude [1] This work and Evans, Hutchinson, Wei, Acta Mater. 47, 4093 (1999).

  8. Uniformly Expanded Crack Equilibrium Set up the crack problem non-local Total displacement local non-local behavior only important near crack For specific material behavior, need first-principles calculation Hayes, R.L., Ortiz, M. and Carter, E.A. PRB,69 (2004) 172104.

  9. Assumptions • Periodic unit cell • Uniaxial tensile stress only (mode 1 cracking) • No dislocations – brittle fracture • Convex on interval 0≤ d ≤ d0 • Inflection point atd0 • Concave ford > d0 • f0dominates bulk crystal behavior,f1locally perturbs near crack surfaces Typical interatomic potential Nguyen and Ortiz solved for f0 in the limit of large N J. Mech. & Phys. Solids50, 1727 (2002).

  10. Exaggerated view Exaggerated view d d Solution for Local Part Unrelaxed Surface Energy Uniaxial Moduli Cracked regime Healed regime Purely Harmonic Elastic Deformation Rigidly Separate Surfaces

  11. Replace g0withgr Matched Asymptotic Expansion ~ ~ Solution for Total Energy Relaxed Surface Energy Traction used in engineering simulations of cracking to account for surface-surface interactions Macroscopic failure criteria Theory in line with experiment

  12. Further Generalizations to Universal Curve • If … • unit cell remains periodic • steady state process on time-frame of crack formation • Then hi, extra degrees of freedom, can be eliminated Do constrained minimization to reduce outhi Minimize f(d1,...,dN)as before Examples of hi: • Bravais sublattices (i.e. Al2O3) • impurity concentration • tangential displacement,D, if constrained to

  13. Three Representative Materials Metals Semiconductors Ceramics • (0001) surface of a-Al2O3 • surface severely relaxes inward by ~33% (~0.7 Å) • strong bulk cohesion • brittle – dislocations do not form easily • (111) surface of fcc Al • surface remains at bulk termination (~1% outward expansion) • weak bulk cohesion • ductile - dislocations form easily • (100)-2x1 surface of cubic diamond Si • surface relaxes inward by ~2% & reconstructs into rows of buckled dimers • brittle – dislocations do not form easily sapphire ruby

  14. Insert dat the crack, fix the unit cell, and allow ions to relax • Either start at ideal bulk termination or the relaxed structure from a largerd • use fat largestdas 2gr • plot d*vsf* • Vary dby uniformly stretching the material • fit tof0to get C • plotd* vsf* DFT Comparison Calculations Uniform Expansion Introduce Crack and Relax

  15. Al QM Al2O3 QM Si QM f* f* = 1 f* = d*2 Deviations from universal behavior around d* = 1 (due to small Nin QM calculations…) filled  healed open  cracked d* Universality of Asymptotic Binding Energy Relationship for Relaxed Surfaces Metals, semiconductors, and ceramic fall on universal curve! ONLY the uniaxial elastic constant (C), relaxed surface energy (gr) and number of layers (N), needed to describe cracks with relaxed surfaces (slow cracking) Nguyen, O. and Ortiz, M. J. Mech. & Phys. Solids50, 1727 (2002) Hayes, R.L., Ortiz, M. and Carter, E.A. PRB,69 172104 (2004).

  16. Cracked DFT points ats*< 1 Cracked DFT points atd*< 1 cracks heals f* f* d* d* d* ~ Universal curve valid for all cracking and healing ifd*> 1 Source of Deviation Crack cannot heal until e- density bridges crack  crack surfaces “see” each other and heal • Arises from surface – surface interactions immediately before crack heals • Stronger bulk cohesion (i.e. Al2O3)  surfaces approach closer  larger variation in energy Crack Heals Crack Forms Si Al2O3 Al “healing” distance nearly independent of N  shift “crack” curve to smallerd*as N increases ChemPhysChem2, 55 (2001).

  17. Energy units = 2gr telast* = 2d* t* Wad = area under curve tcrack*= 0 d* 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

  18. Lattice Constant Uniaxial moduli Relaxed surface energy Do Theory and Expt. Agree? Al (this work) Al (expt) Al2O3 (this work) Al2O3 (expt) Si (this work) Si (expt) • Uniaxial moduli have the correct ordering • Surface energies are the correct order of magnitude • Renormalization brings the failure criteria in line with experimental values

  19. Intelligently Informing Macroscopic Crack Models Uniaxial Elastic Constant, C Interlayer Spacing, d Relaxed Surface Energy, gr Uniaxial Expansion Bulk geometry optimization Infinitely separated surfaces Expt. Expt. Expt. Universal Binding Energy Relationship for Relaxed Surfaces Macroscopic models of cracking

  20. Example: Hydrogen Embrittlement of Steel Initial crack hydrogen Final crack Crack speed depends on hydrogen concentration Serebrinsky, Carter, Ortiz J. Mech. Phys. Sol. 52 (2004) 2403.

  21. * * load load time time Fatigue failure in Si? Monotonic loading Cyclic loading Fatigue cracking usually in ductile materials → surprise in Silicon Connally and Brown Science256 (1992) 1537. • 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.]

  22. Si (100) Surface Reconstruction Top View Side View Tilted Dimer Expt. LEED at 120 K – 190 K Phys. Rev. B55 4731 (1997). STM at 4 K p-type substrate n-type substrate (2x1) (2x1) c(4x2) p(2x2) Some form of tilted dimer is likely the ground state of Si near 0 K (2x1) (2x1) Phys. Rev. Lett89 286104 (2002).

  23. 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.

  24. Hysteresis in Silicon DFT Renormalized Energy – Displacement Curve Energy barrier prevents lower energy reconstructed surface from forming For 12 layers, healed and cracked (100) Si have the same energy Reconstructed surface prevents ideal bulk crystal from reforming 3% strain 1.13 J/m2 Follow uniform expansion curve when crack first forms Reconstructed surface causes hysteresis during load cycling

  25. 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.

  26. Organic-Semiconductor Interfaces Nano-lithography Example: - Passivate Si(100) surface with benezene - Create 2 nm wide patterns with STM tip - React with vinyl ferrocene Kruse and Wolkow Appl. Phys. Lett. 81 (2002) 4422. Self-assembled nanowires and other nanostructures Example: - styrene forms lines on H-Si(100) - precursor to molecular electronics DiLabio, Piva, Kruse, and Wolkow JACS 126 (2004) 16048. Monolayers Example: - monolayer of 1,5-cyclooctadiene absorbed on Si(100) - p-bond on surface available for further rxns - precursor to molecular sensor DiLabio, Piva, Kruse, and Wolkow JACS 126 (2004) 16048.

  27. Proposed Reaction Mechanism of 1,3-cyclohexadiene addition to the (100)Si-2x1 surface Resonance → 2 locations for carbocation “+” charge on down Si Reaction proceeds through a stepwise zwitterionic mechanism Minary, Tuckerman JACS 127 (2005) 1110.

  28. Car-Parrinello Molecular Dynamics orthogonality constraint atoms electrons DFT The results are trustworthy if … • 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.

  29. Reaction kinetically controlled Product Distribution of 1,3 cyclohexadiene on Si(100) A 1 site Expt. STM (%) Theory, CPMD 1,3-butadiene (%) Thermo- dynamic B 1 site A 11±3 15 B 16±7 15 C 31±6 30 D 10±6 30 E 12±9 C 2 sites Predicted decreasing pop. D 1 site E 2 sites STM data from Teague and Boland, TSF 464 (2004) 1. Theory from Minary and Tuckerman, JACS 127 (2005) 1110.

  30. Preliminary results Reaction proceeds through an asymmetric transition state

  31. 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

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