Lecture: P1_Wk2_L5 Contact Mechanics. Ron Reifenberger Birck Nanotechnology Center Purdue University 2012

Transcription

1 Lecture: P_Wk_L5 Contct Mechnics Predict the stresses nd deformtions which rise when the surfces of two solid bodies re brought into contct, subject to surfce constrints. Ron Reifenberger Birck Nnotechnology Center Purdue University 0

2 Action of point force (Boussinesq, 885) www. tem-nnotec.de r (r) E, ν P_Wk_L5 (r) ν = π E r Singulr t r=0!

3 Action of punch with circulr cross-section r www. tem-nnotec.de P_Wk_L5 = re * ν ν + = E E E * 3

4 Action of cone-shped punch θ P_Wk_L5 E* = π tn θ ν ν + = E E E * 4

5 At microscopic scle, for smll indenttions.... AM Problem Contct Mechnics Literture R d ts Contct rdius, 0 (x)=indenttion Use contct mechnics to mke predictions for two quntities s function of the pplied force P_Wk_L5 5

6 R The bsic problem Quntities of interest: Contct rdius () Adhesive orce ( d ) Contct rdius t zero lod (=0) Contct rdius t seprtion Pull-off force d Three Clssic Theories Hertz Johnson-Kendll-Roberts erjguin-muller-toporov (JKR) (MT) P_Wk_L5 6

7 Need to evelop Tip-smple Interction Model () (b) Equilibrium (c) Pull-off R tip smple 3 Hertz JKR pull-off Rigid tip-rigid smple vs. eformble tip nd rigid smple* rom the erjguin pproximtion for rigid tip intercting with rigid smple () we hve tip smple( o ) = dhesion = πrtipu ( o ) πrtip W3 = πrtip ( γ3 + γ3 γ ) Rel tips nd smples re not rigid (see b,c bove). Severl theories hve been developed to better ccount for this fct * Theories lso pply to deformble smples; rigid smple is shown only to demonstrte key quntities clerly. or exmple is the combined tip-smple deformtion in (b) P_Wk_L5 7

8 elstic Clssic Tip-Smple Interction Models (flt plne = R ) elstic, with dhesion in contct region vdw with rigid spheres elstic, dhesive nd vdw R rigid plne R Hertz (88) JKR (97) Brdley (93) MT 975 Neglects surfce forces nd dhesion Assumes linerly elstic sphere indenting n elstic surfce Neglects long-rnge interctions outside contct re Applicble to soft smples with high dhesion Considers two rigid spheres intercting vi Lennrd-Jones 6- potentil Elstic sphere ginst rigid plne surfce Includes vn der Wls forces outside the contct region. Applicble to stiff smples with low dhesion. P_Wk_L5 K.L. Johnson, Contct Mechnics (Cmbridge University Press, 985) 8

9 Surfce forces give rise to surfce energies Work of dhesion nd cohesion: work done to seprte unit res of two mterils nd from contct to infinity in vcuum. If nd re different then W is the work of dhesion; if nd re the sme then W is the work of cohesion. Surfce energy: This is the free energy chnge when the surfce re of mteril is incresed by unit re: W = γ = γ J m (see lecture P_Wk_L) When seprting dissimilr mterils, the free energy chnge in destroying the existing interfcil re per unit re is known s the interfcil energy γ Work of dhesion in third medium: ( ) in vcuum, W = γ + γ γ W3 = γ3 + γ3 γ 3 P_Wk_L5 Before 3 After 9

10 Equilibrium Stndrd results E, ν E, ν Hertz JKR 3 ν ν 3 = + = 4 E E 4 E " springs in series " E tot W 3 = γ = γ 3 + γ 3 γ * pull-off = d () Contct Rdius (b,c) eformtion(c) Hertz 0 R tip Hertz = E tot 3 Hertz = R Hertz tip MT πr =W MT d tip 3 R tip + MT = Et ot MT d 3 MT = R MT tip JKR JKR 3π d = R W tip 3 Rtip JKR JKR = d + + E tot JKR d 3 JKR JKR 4 d JKR = - R 3 R E JKR tip tip tot P_Wk_L5 Notes: () No surfce forces W 3 =0 (b) Hrd to define ccurtely when contcts re smll (c) Adhesive correction 0

11 Adhesion rises fundmentlly from the short rnge interction between molecules. In the JKR model, pull-off occurs when contct rdius JKR =0 JKR Adhesion - consequences JKR eformble smple Wrning - mjor limittion of clssicl results: Surfce roughness P_Wk_L5

12 Exmple Hertz contct: R tip = 30 nm; pp = nn E tip =E sub =00 GP; Poisson rtio = ν tip =v sub =0.3=v ν ν 3 sub tip 3 ν = + = E tot 4 Esub Etip E = = Etot = 46.5GP (00 GP) 00GP Contct rdius: R tip =60 nm pp Hertz Hertz R 3 tip = = 0.59 nm Etot eformtion: Hertz P_Wk_L5 /3 - limits resolution in AM = = = pm Pull-off orce=0 Rtip RtipEtot Contct Pressure?: P = 0.9 GP 9000 tmos. π Hertz

13 Which contct model to choose? Rtip Hertz = Etot 3 o JKR define new prmeter width of elsticlly deformed neck tomic spcing P_Wk_L5 R tipw 3 3 Rtip W 3 3π = πetot = Etot πetot µ 4.7 /3 /3 3 RtipW3 9R tipw 9 λ = = 0 6πEtot 0 6 /3 /3 0 0 [ no dimensions] Rtip W 3 /3 = Etot πetot.65 πetot 0 /3 [ no dimensions] 3

14 Vlidity of different models pplied force dhesion = π W3Rtip 3 ν ν = + Etot 4 Es Et s t λ dhesion elsticity 0 = ( typicl tomic spcing) P_Wk_L5 4

15 Trnsition from MT to JKR: Mugis-ugdle Theory Non-contct Contct R tipw3 λ o πetot /3 3 ν ν s t = + Etot 4 Es Et = intertomic distnce o dhesion elsticity Contct Rdius Non-contct Contct Loding orce pproch Repulsive Attrctive Penetrtion λ 0: MT (stiff mterils) λ : JKR (soft mterils). Mugis, J. Colloid Interfce Sci. 50, 43 (99). P_Wk_L5 Penetrtion 5

16 Up Next: Combining contct mechnics with intermoleculr interctions 6

17 Appendix: A few comments on these theories In the limit of smll dhesion JKR -> MT Most equtions of JKR nd Hertz nd MT hve been tested experimentlly on moleculrly smooth surfces nd found to pply extremely well The limittion for ppliction to AM is tht no tip is perfectly smooth sphere, smll sperities mke big difference. Hertz, MT describe conservtive interction forces, but in JKR, the interction itself is non-conservtive (why?) for force to be considered conservtive it must be written s grdient of potentil energy. P_Wk_L5 7