Is Partial Slip Under Transverse Oscillatory Loading a Transient Event? Frederick Meyer, 1 Daniel Forchheimer, 2 Arne Langhoff, 1 Diethelm Johannsmann* 1 1 Institute of Physical Chemistry, TU Clausthal, Germany 2 Intermodulation Products AB, Sweden - High-frequency nonlinear contact mechanics - Intermodulation products - Sudden impacts - Crystallization • High-frequency micromechanics: Sylvia Hanke, Rebekka König, Judith Petri, Jana Vlachová, Frederick Meyer • Intermodulation: Daniel Forchheimer • QCM work in general : Arne Langhoff 1
The 2 nd -Generation Quartz Crystal Microbalance I ~ U ~ quartz plate electrodes f = f res : large amplitude of motion large current 8 conductance G [mS] unloaded • Shifts in frequency and bandwidth : D f , DG 6 loaded G • many overtones 4 D f ( n ), DG ( n ) Df • dependence on amplitude 2 D f ( n, u 0 ), DG ( n , u 0 ) G 0 • Higher harmonics 11.999 12.000 12.001 frequency [MHz] • Intermodulation products 2
Small Load Approximation G Complex Resonance Frequency f f i r r conductance G(f) -1 f r current I(t) Fourier p r exp 2 if t Transform f r G (2 pG ) -1 frequency time ˆ Z " load impedanc e" Small-load approximation L ˆ v D ˆ ˆ stres f i i s Z ˆ p p ˆ L v velo city f Z Z v 0 q q ˆ ^: c o mp lex amplitud e ( (t) = exp(i t) ) QCM: The Quartz Crystal periodic stress, Micro Stress -Balance in-phase, out-of-phase Mason, W.P., Piezoelectric Crystals and their Applications to Ultrasonics 1948 Pechhold, W Acustica 1959, 9, 48 Johannsmann, D., The Quartz Crystal Microbalance in Soft Matter Research, Springer 2014 Analogous equations exist in atomic force microscopy, valid if the perturbations are small 3
Small Load Approximation Sauerbrey D D ˆ f i inertial stress i i v m p p ˆ f Z velocity Z v 0 q q D D D m :Mass per unit area of film f m m Z / 2f :Mass per unit area of crystal f m q q 0 q Stress might go back to viscous drag elastic forces , ... acoustic multilayers, interfackal high-frequency rheology ˆ D p f i area The stress can be averaged over area : ˆ f Z v 0 q N ˆ ˆ ˆ F with F the periodic forc e Discrete objects: ar ea A time 2 F t exp i t D p f i N The force can be averaged over time : ˆ f Z A v 0 q QCM covers nonlinear force-displacement relations 4
Stiffness of Sphere-Plate Contacts air or water added clamp deformation only glass spheres weight close to contact diameter Coating 1 or 2 mm soft link, heavy sphere P D f N p f A Z 0 q Elastic Load SiO spheres r= 2.2 mm 300 200 D f * air Hertz-Mindlin: 2 G a 100 water a : contact radius JKR Fits 1 2 1 2 1 0 1 2 : effective modu l us 0 1 2 3 * 4 G 4 G G 1 2 added weight [g] Vlachová, J.; König, R.; Johannsmann, D. Beilstein J. Nanotechnol. 2015, 6, 845. 5
Contact Stiffness Contact Strength Coulomb friction Quasi-static, force control: tangential stick-slip transition is an instability force strength: µ S F N µ D F N tangential displacement stiffness Oscillatory motion, strain control: Partial slip not an instability tangential partial slip force gross slip tangential displacement stiffness 6
Small Load Approximation Sauerbrey D D ˆ f i inertial stress i i v m p p ˆ f Z velocity Z v 0 q q D D D m :Mass per unit area of film f m m Z / 2f :Mass per unit area of crystal f m q q 0 q Stress might go back to viscous drag elastic forces , ... acoustic multilayers, QTM high-frequency polymer rheology ˆ D p f i area The stress can be averaged over area : ˆ f Z v 0 q N ˆ ˆ ˆ F with F the periodic forc e Discrete objects: ar ea A time D p 2 F t exp i t f i N time The force can be averaged over : ˆ f Z A v 0 q QCM covers nonlinear force-displacement relations 7
The QCM and Nonlinear Response D 2 t exp i t f i p time The stress can be averaged over time : ˆ f Z v 0 q Loads are small strain contr ol F(t) 1 1 u u cos t dt du 0 2 1 u u / 0 Stress and force can be averaged over displacement, u D f ~ F t cos i t time time u u / u 0 0 partial F u u , , F u u , , 0 0 force 2 slip 1 u u / F 0 u DG ~ F t sin i t visco- time F elastic F u u , , F u u , , 0 0 u displacement u D D G f , are weighted averages of friction l o op shape of friction loop uncertain Hanke, S .; Petri, J.; Johannsmann, D., PRE 2013, 032408 Johannsmann, D., Springer 2014 8
Shape of Friction Loop? Data fit to Mindlin model force They might also be explained by a temperature-induced softening of the contact displace- ment This question can be answered with 3 rd harmonic generation force displace- ment Ghosh, S. K. et al. ; Biosensors & Bioelectronics 2011 , 29, 145 Berg, S.; DJ, Surface Science 2003 , 541, 225 9
Partial Slip partial / total slip tangential slip stick force Cattaneo, C., Rendiconti dell' Academia Nationale dei Lincei 1938 Mindlin, R.D.; Deresiewicz H.: J. Appl. Mech. 1953 Johnson, K. L., Contact Mechanics 1985 displacement Savkoor, A. R. Tech. University Delft, 1987 Varenberg, M.; Etsion, I.; Halperin, G., Tribology Letters 2005 When transient: Transition state between stick and slip, mixed lubrication, … When slow: Contact aging, compaction, soil mechanics, granular media, … When oscillatory: Fretting wear http://www.mr2oc.com/208-aef-engine-powertrain/455990-calling-m-e-s- fretting-failure-mode-clutch-hub-female-spline-e153-mr2-turbo-trans.html 10
Partial Slip and Gross Slip added humidity weight diameter polymer film 50 275 µm (T g ~108 or 37°C ) amplitude of oscillation 0 20 nm Small spheres (d = 50 µm) 100 Hz 20 Hz soft substrate (T g ~ 37°C) Medium size spheres (140 µm) DG Partial Slip D f 100 Hz 20 Hz soft substrate (T g ~ 37°C) Gross Slip Large spheres (275 µm) 100 Hz 20 Hz hard substrate (T g ~ 108°C) 0 5 10 0 5 10 amplitude [nm] amplitude [nm] 11
Partial Slip Nonlinear Stress-Strain Relations Quantitative models for the force-displacement relation exist (but: quasi-static) stick Stress in sliding zone follows Coulomb ( S = µp) p Cattaneo, C., Rendiconti dell' Academia Nationale dei Lincei 1938 Mindlin, R.D.; Deresiewicz H.: J. Appl. Mech. 1953 partial slip Stress in sliding zone constant ( S = const.) Cattaneo-Mindlin S = µp Savkoor, A. R. Tech. University Delft, 1987 x partial slip Cattaneo-Mindlin Savkoor force [a.u.] Savkoor S = const = 0 F + u min F - F min -1 0 1 u/u max 12
Partial Slip Nonlinear Stress-Strain Relations stick Cattaneo-Mindlin force [a.u.] Savkoor p F + partial slip F - Cattaneo-Mindlin S = µp x partial slip F min Savkoor S = const = 0 -1 0 1 u/u 0 Cattaneo-Mindlin D f N D f u 1 u Savkoor 100 Hz 0 p 0 D f 2 3µ F A2n Z N q N 4 DG u u p p 0 0 2 Cattaneo-Mindlin A2n Z 9 µF DG q N 20 Hz Savkoor 2 0 5 10 N 5 u DG D 0 f u 1 amplitude [nm] p 0 2 2 Cattaneo- A2 n Z 8 4 a q 0 Mindlin 2 u Also: Leopoldes, J.; Jia, X., N 8 Savkoor DG 0 u PRL 2010 p p 0 2 2 A2n Z 6 2 a q 0 amplitude, u 0 13
Multifrequency Lockin Analysis Signal [a.u.] Intermodulation Products AB, Sweden real imag A) Frequency combs A fast (milliseconds) way to probe resonances 42 freqs, fit Lorentzians D f, DG (no calibration required) 4991700 4992000 4992300 4992600 frequency [Hz] B) Intermodulation products linear Excite with 2 frequencies beating signal (fast amplitude ramps) Nonlinearities signals at f = 2f 1 – f 2 and 2f 2 – f 1 nonlinear Fast Signal resonantly enhanced, response function well understood C. Hutter et al. Phys. Rev. Lett. 2010 , 104, 050801 Calibration issues C) 2 nd and 3 rd Harmonic Generation Excite at f, probe at 2f, 3f, etc… (also: determine background) Signal not resonantly enhanced Probes MHz dynamics 14
Partial Slip air or water added glass spheres weight diameter Coating 1 or 2 mm D f [Hz] reference 200 tripod tripod + 2.7g 0 0 DG [Hz] Mindlin model works here -20 (does not always work) There is a 3 rd harmonic signal -40 There is a weak 2 nd harmonic signal rd Intermodulation 9 3 (normal forces involved) 3 Here: 5 MHz (fundamental mode) Product x10 6 3 results are different on overtones Strong intermodulation products 0 rd Harmonic x10 3 1.5 3 1.0 0.5 0.0 0 5 10 15 20 25 Apparent Amplitude [nm] 15
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