1. ME6260/EECE7244 1 Contact Mechanics

2. ME6260/EECE7244 2 SEM Image of Early Northeastern University MEMS Microswitch Asperity

3. ME6260/EECE7244 3 SEM of Current NU Microswitch Asperities

4. ME6260/EECE7244 4 Two Scales of the Contact Nominal Surface Contact Bump (larger, micro-scale) Asperities (smaller, nano-scale)

5. ME6260/EECE7244 5 Basics of Hertz Contact The pressure distribution: produces a parabolic depression on the surface of an elastic body. Resultant Force Pressure Profile p(r) r a p0p0 Depth at center Curvature in contact region

6. ME6260/EECE7244 6 Basics of Hertz Contact Elasticity problem of a very “large” initially flat body indented by a rigid sphere. rigid We have an elastic half-space with a spherical depression. But: R r

7. ME6260/EECE7244 7 Basics of Hertz Contact  So the pressure distribution given by: gives a spherical depression and hence is the pressure for Hertz contact, i.e. for the indentation of a flat elastic body by a rigid sphere with  But wait – that’s not all !  Same pressure on a small circular region of a locally spherical body will produce same change in curvature.

8. ME6260/EECE7244 8 Basics of Hertz Contact P

9. ME6260/EECE7244 9 P Hertz Contact Hertz Contact (1882) 2a R1R1 R2R2  E 1, 1 E 2, 2 Interference Contact Radius Effective Radius of Curvature Effective Young’s modulus

10. ME6260/EECE7244 10 Assumptions of Hertz  Contacting bodies are locally spherical  Contact radius << dimensions of the body  Linear elastic and isotropic material properties  Neglect friction  Neglect adhesion  Hertz developed this theory as a graduate student during his 1881 Christmas vacation  What will you do during your Christmas vacation ?????

11. ME6260/EECE7244 11 Onset of Yielding  Yielding initiates below the surface when  VM =  Y. Elasto-Plastic (contained plastic flow)  With continued loading the plastic zone grows and reaches the surface  Eventually the pressure distribution is uniform, i.e. p=P/A=H (hardness) and the contact is called fully plastic (H  2.8  Y ). Fully Plastic (uncontained plastic flow)

12. ME6260/EECE7244 12 Round Bump Fabrication Critical issues for profile transfer: –Process Pressure –Biased Power –Gas Ratio Photo Resist Before ReflowPhoto Resist After Reflow The shape of the photo resist is transferred to the silicon by using SF 6 /O 2 /Ar ICP silicon etching process. Shipley 1818 O 2 :SF 6 :Ar=20:10:25O 2 :SF 6 :Ar=15:10:25 Silicon Bump Shipley 1818

13. ME6260/EECE7244 13 Evolution of Contacts After 10 cyclesAfter 10 2 cyclesAfter 10 3 cyclesAfter 10 4 cycles

14. ME6260/EECE7244 14  c, a C, P C are the critical interference, critical contact radius, and critical force respectively. i.e. the values of , a, P for the initiation of plastic yielding Curve-Fits for Elastic-Plastic Region Note when  /  c =110, then P/A=2.8  Y Elasto-Plastic Contacts (L. Kogut and I Etsion, Journal of Applied Mechanics, 2002, pp. 657-662)

15. ME6260/EECE7244 15 Fully Plastic Single Asperity Contacts (Hardness Indentation)  Contact pressure is uniform and equal to the hardness (H)  Area varies linearly with force A=P/H  Area is linear in the interference  = a 2 /2R

16. ME6260/EECE7244 16 Nanoindenters Hysitron Triboindenter® Hysitron Ubi®

17. ME6260/EECE7244 17 Nanoindentation Test Force vs. displacement Indent

18. ME6260/EECE7244 18 Depth-Dependent Hardness H 0 =0.58 GPa h*=1.60  m Data from Nix & Gao, JMPS, Vol. 46, pp. 411-425, 1998.

19. ME6260/EECE7244 19 Surface Topography Standard Deviation of Surface Roughness Standard Deviation of Asperity Summits Scaling Issues – 2D, Multiscale, Fractals Mean of Surface Mean of Asperity Summits

20. ME6260/EECE7244 20 Contact of Surfaces d Reference Plane Mean of Asperity Summits Typical Contact Flat and Rigid Surface

21. ME6260/EECE7244 21 Typical Contact Original shape 2a  P R Contact area

22. ME6260/EECE7244 22 Multi-Asperity Models (Greenwood and Williamson, 1966, Proceedings of the Royal Society of London, A295, pp. 300-319.) Assumptions  All asperities are spherical and have the same summit curvature.  The asperities have a statistical distribution of heights (Gaussian).  (z) z

23. ME6260/EECE7244 23 Multi-Asperity Models (Greenwood and Williamson, 1966, Proceedings of the Royal Society of London, A295, pp. 300-319.) Assumptions (cont’d)  Deformation is linear elastic and isotropic.  Asperities are uncoupled from each other.  Ignore bulk deformation.  (z) z

24. ME6260/EECE7244 24 Greenwood and Williamson

25. ME6260/EECE7244 25 Greenwood & Williamson Model  For a Gaussian distribution of asperity heights the contact area is almost linear in the normal force.  Elastic deformation is consistent with Coulomb friction i.e. A  P, F  A, hence F  P, i.e. F =  N  Many modifications have been made to the GW theory to include more effects  for many effects not important.  Especially important is plastic deformation and adhesion.

26. ME6260/EECE7244 26 Contacts With Adhesion (van der Waals Forces)  Surface forces important in MEMS due to scaling  Surface forces ~L 2 or L; weight as L 3  Surface Forces/Weight ~ 1/L or 1/L 2  Consider going from cm to  m  MEMS Switches can stick shut  Friction can cause “moving” parts to stick, i.e. “stiction”  Dry adhesion only at this point; meniscus forces later

27. ME6260/EECE7244 27 Forces of Adhesion  Important in MEMS Due to Scaling  Characterized by the Surface Energy (  ) and the Work of Adhesion (  )  For identical materials  Also characterized by an inter-atomic potential

28. ME6260/EECE7244 28 Adhesion Theories Z 0123 -0.5 0 0.5 1 1.5 Z/Z 0  /  TH Some inter-atomic potential, e.g. Lennard-Jones Z0Z0 (A simple point-of-view) For ultra-clean metals, the potential is more sharply peaked.

29. ME6260/EECE7244 29 Two Rigid Spheres: Bradley Model P P R2R2 R1R1 Bradley, R.S., 1932, Philosophical Magazine, 13, pp. 853-862.

30. ME6260/EECE7244 30 JKR Model Johnson, K.L., Kendall, K., and Roberts, A.D., 1971, “Surface Energy and the Contact of Elastic Solids,” Proceedings of the Royal Society of London, A324, pp. 301-313. Includes the effect of elastic deformation. Treats the effect of adhesion as surface energy only. Tensile (adhesive) stresses only in the contact area. Neglects adhesive stresses in the separation zone. P a a P1P1

31. ME6260/EECE7244 31 Derivation of JKR Model Total Energy E T Stored Elastic Energy Mechanical Potential Energy in the Applied Load Surface Energy Equilibrium when

32. ME6260/EECE7244 32 JKR Model Hertz modelHertz model Only compressive stresses can exist in the contact area. Pressure Profile Hertz ar p(r) Deformed Profile of Contact Bodies JKR model JKR model Stresses only remain compressive in the center. Stresses are tensile at the edge of the contact area. Stresses tend to infinity around the contact area.JKR p(r) a r P a a P

33. ME6260/EECE7244 33 JKR Model 1.When  = 0, JKR equations revert to the Hertz equations. 2.Even under zero load (P = 0), there still exists a contact radius. 3.F has a minimum value to meet the equilibrium equation i.e. the pull-off force.

34. ME6260/EECE7244 34 DMT Model DMT model Tensile stresses exist outside the contact area. Stress profile remains Hertzian inside the contact area. p(r) ar Derjaguin, B.V., Muller, V.M., Toporov, Y.P., 1975, J. Coll. Interf. Sci., 53, pp. 314-326. Muller, V.M., Derjaguin, B.V., Toporov, Y.P., 1983, Coll. and Surf., 7, pp. 251-259. Applied Force, Contact Radius & Vertical Approach

35. ME6260/EECE7244 35 Tabor Parameter: JKR-DMT Transition DMT theory applies (stiff solids, small radius of curvature, weak energy of adhesion) JKR theory applies (compliant solids, large radius of curvature, large adhesion energy) Recent papers suggest another model for DMT & large loads. J. A. Greenwood 2007, Tribol. Lett., 26 pp. 203–211 W. Jiunn-Jong, J. Phys. D: Appl. Phys. 41 (2008), 185301.

36. ME6260/EECE7244 36 Maugis Approximation where 0123 -0.5 0 0.5 1 1.5 Z/Z 0  /  TH Maugis approximation h0h0

37. ME6260/EECE7244 37 Elastic Contact With Adhesion

38. ME6260/EECE7244 38 Elastic Contact With Adhesion w= 

39. ME6260/EECE7244 39 Elastic Contact With Adhesion

40. ME6260/EECE7244 40 Adhesion of Spheres JKR valid for large  DMT valid for small  Tabor Parameter 0123 -0.5 0 0.5 1 1.5 Z/Z 0  /  TH Maugis JKR DMT Lennard-Jones  and  TH are most important E. Barthel, 1998, J. Colloid Interface Sci., 200, pp. 7-18

41. ME6260/EECE7244 41 Adhesion Map K.L. Johnson and J.A. Greenwood, J. of Colloid Interface Sci., 192, pp. 326-333, 1997

42. ME6260/EECE7244 42 Multi-Asperity Models With Adhesion Replace Hertz Contacts of GW Model with JKR Adhesive Contacts: Fuller, K.N.G., and Tabor, D., 1975, Proc. Royal Society of London, A345, pp. 327-342. Replace Hertz Contacts of GW Model with DMT Adhesive Contacts: Maugis, D., 1996, J. Adhesion Science and Technology, 10, pp. 161-175. Replace Hertz Contacts of GW Model with Maugis Adhesive Contacts: Morrow, C., Lovell, M., and Ning, X., 2003, J. of Physics D: Applied Physics, 36, pp. 534-540.

43. ME6260/EECE7244 43 Surface Tension

44. ME6260/EECE7244 44 http://www.unitconversion.org/unit_converter/surface-tension-ex.html

45. ME6260/EECE7244 45  = 0.072 N/m for water at room temperature

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47. ME6260/EECE7244 47 pp

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