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Contact Mechanics ME6260/EECE7244 1 SEM Image of Early Northeastern University MEMS Microswitch Asperity ME6260/EECE7244 2 SEM of Current NU Microswitch Asperities ME6260/EECE7244 3 Two Scales of the Contact • Contact Bump (larger, micro-scale) • Asperities (smaller, nano-scale) Nominal Surface ME6260/EECE7244 4 Basics of Hertz Contact The pressure distribution: p ( r ) p0 1 ( r / a ) 2 , r a p(r) p produces a parabolic depression on the surface of an elastic body. (1 2 ) Depth at center p0 a 0 r 2E Curvature in contact region Pressure Profile a 2 1 (1 ) p0 R 2 Ea Resultant Force P a 0 2 2 p(r )2 rdr a p0 3 ME6260/EECE7244 5 Basics of Hertz Contact Elasticity problem of a very “large” initially flat body indented by a rigid sphere. P R rigid r δ R a r z R R2 r 2 We have an elastic half-space with a spherical depression. But: (r R) w(r ) ( R R 2 r 2 ) R(1 1 r 2 / R 2 ) r 2 / 2R ME6260/EECE7244 6 Basics of Hertz Contact So the pressure distribution given by: p ( r ) p0 1 ( r / a ) 2 , r a 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 2 2 1 (1 ) p0 R 2 Ea (1 ) p0 a 2E But wait – that’s not all ! Same pressure on a small circular region of a locally spherical body will produce same change in curvature. ME6260/EECE7244 7 Basics of Hertz Contact P ME6260/EECE7244 8 Hertz Contact Hertz Contact (1882) 3P * 1/ 2 4E R P 2/3 Interference 1/ 3 3PR a * 4 E 1 1 12 1 22 * E E1 E2 1 1 1 R R1 R2 Contact Radius Effective Young’s modulus Effective Radius of Curvature ME6260/EECE7244 E2,2 R2 2a R1 E1,1 9 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 ????? ME6260/EECE7244 10 Onset of Yielding Yielding initiates below the surface when VM = Y. Fully Plastic (uncontained plastic flow) 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 2.8Y). ME6260/EECE7244 11 Round Bump Fabrication Shipley 1818 Photo Resist Before Reflow Shipley 1818 The shape of the photo resist is transferred to the silicon by using SF6/O2/Ar ICP silicon etching process. Photo Resist After Reflow Silicon Bump Silicon Bump • Critical issues for profile transfer: – Process Pressure – Biased Power – Gas Ratio O2:SF6:Ar=20:10:25 ME6260/EECE7244 O2:SF6:Ar=15:10:25 12 Evolution of Contacts After 10 cycles After 102 cycles After 103 cycles ME6260/EECE7244 After 104 cycles 13 Elasto-Plastic Contacts (L. Kogut and I Etsion, Journal of Applied Mechanics, 2002, pp. 657-662) 4E aC KH C * R, K 0.454 0.41 , H 2.8 Y , aC C R , PC 3R 2E 2 * 3 c, aC, PC 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 P 1.03 PC C P 1.40 PC C 1.425 , 1.263 , A 0.93 AC C A 0.94 AC C 1.136 , 1 6 C , 6 110 C 1.146 Note when /c=110,ME6260/EECE7244 then P/A=2.8Y 14 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 = a2/2R ME6260/EECE7244 15 Nanoindenters Hysitron Ubi® Hysitron Triboindenter® ME6260/EECE7244 16 Nanoindentation Test Indent Force vs. displacement ME6260/EECE7244 17 Depth-Dependent Hardness Depth Dependence of Hardness of Cu 12 10 H h* 1 H0 h 0 (H/H )2 8 6 H0=0.58 GPa 4 h*=1.60m 2 0 0 1 2 3 4 1/h (1/ m) 5 6 7 Data from Nix & Gao, JMPS, Vol. 46, pp. 411-425, 1998. ME6260/EECE7244 18 Surface Topography Mean of Asperity Summits Mean of Surface 1 L 2 Standard Deviation of Surface Roughness 0 ( z m) dx L 2 Standard Deviation of Asperity Summits 1 N S ( z Si z S ) 2 N i 1 2 Scaling Issues – 2D, Multiscale, Fractals ME6260/EECE7244 19 Contact of Surfaces Flat and Rigid Surface d Reference Plane Mean of Asperity Summits Typical Contact ME6260/EECE7244 20 Typical Contact P Original shape Contact area 2a R ME6260/EECE7244 21 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 ME6260/EECE7244 (z) 22 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 ME6260/EECE7244 (z) 23 Greenwood and Williamson ME6260/EECE7244 24 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. ME6260/EECE7244 25 Contacts With Adhesion (van der Waals Forces) Surface forces important in MEMS due to scaling Surface forces ~L2 or L; weight as L3 Surface Forces/Weight ~ 1/L or 1/L2 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 ME6260/EECE7244 26 Forces of Adhesion Important in MEMS Due to Scaling Characterized by the Surface Energy () and the Work of Adhesion () 1 2 12 For identical materials 2 Also characterized by an inter-atomic potential ME6260/EECE7244 27 Adhesion Theories (A simple point-of-view) 1.5 Some inter-atomic potential, e.g. Lennard-Jones 1 Z0 / TH 0.5 0 Z -0.5 -1 0 1 2 3 Z/Z 0 For ultra-clean metals, the potential is more sharply peaked. ME6260/EECE7244 28 Two Rigid Spheres: Bradley Model P R2 1 1 1 R R1 R2 PPullOff 2 R Bradley, R.S., 1932, Philosophical Magazine, 13, pp. 853-862. R1 P ME6260/EECE7244 29 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 P1 a a ME6260/EECE7244 30 Derivation of JKR Model Stored Elastic Energy Mechanical Potential Energy in the Applied Load Surface Energy Total Energy ET Equilibrium when 3 a K P 3R 6RP (3R) 2 , R a2 8a R 3K dET 0 da K 4 * E 3 PPullOff 1.5R ME6260/EECE7244 31 JKR Model Deformed Profile of Contact Bodies Pressure Profile P p(r) • Hertz model Hertz Only compressive stresses can exist in the contact area. a a 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. r P p(r) JKR a ME6260/EECE7244 r a 32 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. 6R a 0 K 2 1 3 0 4 R a0 2 3R 3K 2 2 2 1 3 3. F has a minimum value to meet the equilibrium equation Pmin 3 R 2 1/ 3 min 3 R 2 2 2K i.e. the pull-off force. ME6260/EECE7244 33 DMT Model 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. DMT model p(r) Tensile stresses exist outside the contact area. Stress profile remains Hertzian inside the contact area. a r Applied Force, Contact Radius & Vertical Approach a3 K P 2R, R a2 R PPullOff 2 R ME6260/EECE7244 34 JKR-DMT Transition Tabor Parameter: 1/ 3 R *2 3 E Z0 2 1 DMT theory applies (stiff solids, small radius of curvature, weak energy of adhesion) 1 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. ME6260/EECE7244 35 Maugis Approximation 1.5 TH , Z Z 0 h0 0, Z Z 0 h0 Maugis approximation 1 0.5 / TH where h0 TH 0 h0 -0.5 h0 Z 0 -1 0 1 2 3 Z/Z 0 ME6260/EECE7244 36 Elastic Contact With Adhesion ME6260/EECE7244 37 Elastic Contact With Adhesion w= ME6260/EECE7244 38 Elastic Contact With Adhesion ME6260/EECE7244 39 Adhesion of Spheres 1.5 JKR Tabor Parameter Maugis / TH 0.5 1/ 3 R *2 3 E Z0 1 Lennard-Jones 0 2 JKR valid for large DMT valid for small DMT -0.5 -1 0 1 2 Z/Z0 3 and TH are most important E. Barthel, 1998, J. Colloid Interface Sci., 200, pp. 7-18 ME6260/EECE7244 40 Adhesion Map K.L. Johnson and J.A. Greenwood, J. of Colloid Interface Sci., 192, pp. 326-333, 1997 ME6260/EECE7244 41 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. ME6260/EECE7244 42 Surface Tension ME6260/EECE7244 43 ME6260/EECE7244 http://www.unitconversion.org/unit_converter/surface-tension-ex.html 44 = 0.072 N/m for water at room temperature ME6260/EECE7244 45 ME6260/EECE7244 46 p ME6260/EECE7244 47 ME6260/EECE7244 48 ME6260/EECE7244 49 ME6260/EECE7244 50 ME6260/EECE7244 51