1. F act, S ir, Not F iction: A F raction of F rictions O bey S trictures of F racture - Philip (Flip) Kromer ∙ 7 Feb 2002 - On A New Theory of Friction By M. Marder and E. Gerde, And on My Numerical Work Using Code By D. Holland. - Center for Nonlinear Dynamics - Background Talk on Numerical Simulation of A New Theory of Friction Based on Traveling, Self-Healing Cracks You may view the slides for this talk at http://www.mrflip.com/research/talks

2. Macroscopic: Friction is amazingly simple Microscopic: Friction is amazingly complicated B ig i dea  F= µN  Where µ: Depends only on material Is independent of contact area Is near 1 Is independent of sliding velocity How could Something so Simple Come from Something so Complicated?

3. Macroscopic: Friction is amazingly simple Microscopic: Friction is amazingly complicated  Roughness at every length scale  Plastic and Elastic deformations  Adhesion and Interlocking of Asperities  Oxide Layers, Adsorbed Layers, Hydrocarbon Chains  Phononic and Electronic Drag  Many-body Quantum Mechanical Problem  Aspects of the problem are too big for QM… A Polished surface: bumps up to ≈ 200nm: too large  … Yet too small for Statistical Mechanics. How could Something so Simple Come from Something so Complicated?

4. A B rief H istory of F riction Early History  da Vinci1496  Amonton’s Laws of Friction1699  Coulomb’s Theory of Friction 1781 … Traditional Theory  Bowden and Tabor Model1940 Bowden and Tabor, Friction & Lubrication of Solids Persson, Sliding Friction … A New Theory of Friction  Marder and Gerde2001  Numerical Investigations2002-200? Truth is stranger than Friction

5. da Vinci is Da Man Da Vinci made careful, quantitative studies of friction ca. 1490-1496 “Friction produces double the amount of effort if the weight be doubled.” [Codex Forster III 72 r] “The friction made by the same weight will be of equal resistance at the beginning of its movement although the contact may be of different breadth or lengths.” [Codex Forster II, 133r & 133v]

6. da Vinci is Da Man Also: Lubricants; Roller Bearings; Inclined Plane with Friction; Abrasion; Roughness

7. Friction is Simple: Amontons’ Laws F= µN µ depends on materials only Independent of contact area µ k is largely independent of sliding velocity Coefficient of Friction is rarely outside.02-3  Very small range for a material property

8. Coulomb is Cool, Mon Friction due to Interlocking Asperities  Lifting over asperities  Bending asperities  Breaking asperities (not whole truth) Interlocking: important for metals; not primary Adhesion is primary friction mechanism  Adhesion should increase with area of contact Asperity just means “Bump”

9. Profilometer Scans of Steel Surfaces (note scale) STM Scans of Silver T he T raditional T heory of F riction Bowden and Tabor, 1940: Solid surfaces are highly irregular True area of contact far less than apparent area

10. True Area Increases with Load Ex: steel cube 10 cm on a side, on steel table  True area δA ≈ 0.1 mm 2, 10 -5 of apparent area  Junctions have diameter ≈ 10 µm about 1000 junctions [Persson p47]

11. Interlocking is not important

12. We can now produce F = µ N This yields the Coulomb equation The true area is proportional to load Friction is from shearing cold-welded junctions Since τ and σ are usually similar in magnitude, this explains why typically µ ≈ 1

13. A N on- T raditional T heory of F riction Marder & Gerde, Nature 413, 285-288 (2001) Science Friction

14. Analytical Continuum Model

15. Problem with Continuum Model?  Infinite Oscillations Approaching Crack Tip Displacement |u y |: Linear Scale Distance x: Log Scale Crack Tip Open End

16. Analytical Lattice Model

17. Envelope of Solutions Gives µ! Build a Catalog of Matched Solutions  Lattice: Crack Tips  Continuum: Self- Healing Only matching solutions accepted Envelope is a linear threshold of -σ vs. τ:  Static Coefficient of Friction!

18. N umerical S imulations Or, How to take Simple Physics and make it Difficult, Expensive, and Time-Consuming

19. Molecular Dynamics Simulations Boring Details:  Single Verlet – 4 th order in dt.  Temperature by kicking/scaling  To prevent O(N 2 ) problem, build Neighbor Lists  Neighbor Lists by cell method with shell  No long-range forces included See Ph.D. thesis of D. Holland for whole story

20. Computers are Too Small Space requirements Consider a sample 0.1 mm x 0.01 mm x 1 µm … Hardly macroscopic, but still 10 15 atoms! For 100 bytes/atom, need 100,000,000 GB (3D)! Time requirements Characteristic timescale of chemical interactions: 1 fs Therefore 1 µs of simulation takes 10 9 timesteps CPU Speeds Units are GFlops, Giga Floating point Operations Per Second Measured 20 min / G atom · timestep · GFlop 1200 op / atom · timestep Therefore, 1 µs of simulation takes 20 min/atom·GFlop We need about a GFlop/atom! Preposterously too slow! In all, we need 10 24 atom·timesteps

21. My Brand-New Computer: Too Small Tick.ph, an extremely fast workstation Needs\ 2·10 8 x more RAM 1.5· 10 10 years Dual 1500 MHz, 512MB RAM

22. UT’s Supercomputer: Too Small Golden, Here in Texas: #340 in world † † Rankings from the “World’s Fastest Supercomputer” list, Nov 2001: http://www.top500.org/ Needs\ 3·10 6 x more RAM 3.3· 10 8 years 272 Node Cray T3-E, 128 MB/node

23. Even This is Too Small ASCI White, Lawrence Livermore: #1 in World † † Rankings from the “World’s Fastest Supercomputer” list, Nov 2001: http://www.top500.org/ Needs\ 1·10 5 x more RAM 5.3· 10 6 years 8192 Node 1024 MB/node

24. Computers are Too Small † Golden data from http://www.tacc.utexas.edu/resources/systems/ ‡ ASCI White data from http://www.llnl.gov/asci/platforms/white/

25. Impatience demands Inexactitude Approximations  Effective Potential  Effective Potential with Cutoff  Simple Effective Potential with Cutoff Snapping Hooke Springs  Two Dimensions  Scaling Argument

26. Impatience demands Cleverness Scaling Argument  Use Nonlinear Dynamics arguments  Matched Asymptotics Multi-scale Modeling  MD Simulation of particles for atomistic regime  Adaptive Finite Element Mesh for continuum regime  Tight-Binding region for quantum regime (Not for us, though)

27. R esults Oooh, Pretty Pictures

28. Spontaneous Pure Shear Crack

30. Seeded, Ignited Shear Crack

31. Does this Model Qualitatively Explain Friction? Coefficient of Friction  Do we get a threshold -σ vs. τ for traveling cracks? That is, do we get a Coefficient of Friction µ? Is robustness or nucleation imposing the threshold?  Is µ Independent of Contact Area?  Is µ a Sensible Physical Value?  Is µ k Independent of speed? What are the size effects?

32. F uture W ork Three different issues:  Does it capture qualitative physics of friction?  Does it make quantitatively correct predictions?  If it does succeed in the general sense, why? That is, how can something so simple replace something so complicated? OK, Then is any of it True?

33. How might something so Simple model something so Complicated? Carefully prepared atomically flat surfaces  Might be boring: Laboratory Curiosity  Special case: Nanomachines Earthquakes  Original inspiration for the new theory Model of the process for individual asperities Surface behaves as simple aggregate flat interface Right Picture, for Simple Reasons

34. How might something so Simple model something so Complicated? This could be a picture of asperities, not atoms  Traveling crack hops from asperity to asperity Right Picture, for Complicated Reasons

35. How might something so Simple model something so Complicated? Clever Mathematical Mapping  Surface is fractal – contact at every scale  (Polished surfaces: half wavelength ≈ 300nm: large)  Conjecture: A Renormalization argument might imply that the fractal surface ends up being a boring flat surface Right Picture, for Complicated Reasons

36. S ummary Laziness  Discard “Correct” Model for “Simple” Flat Rigid Surfaces Impatience  Numerical Work requires either way, way too much time, or just the right amount of cleverness Hubris  Chutzpah to assert this model is correct  Work will give insight to where & how it is correct. For those who fell asleep after the first slide You may view the slides for this talk at http://www.mrflip.com/research/talks Our research is guided by the Three Virtues of Programming: Laziness, Impatience, Hubris [Wall 91]