Is Partial Slip Under Transverse Oscillatory Loading a Transient Event?

Transcription

1 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

2 conductance G [ms] The 2 nd -Generation Quartz Crystal Microbalance U ~ I~ quartz plate electrodes f = f res: large amplitude of motion large current loaded G unloaded G frequency [MHz] Shifts in frequency and bandwidth:, DG many overtones (n), DG(n) dependence on amplitude (n, u ), DG (n, u ) Higher harmonics Intermodulation products 2

3 conductance G(f) current I(t) Complex Resonance Frequency f f ig G f r frequency Small-load approximation i i ZL f pz pz q q ˆ ˆv Fourier Transform Small Load Approximation r r f r -1 (2pG) -1 exp 2 if t p r time ˆ Z " load impedance" L vˆ ˆ stress ˆv velocity ^: complex amplitude ( (t) = ˆ exp(it) ) QCM: The Quartz Crystal Micro Stress-Balance periodic stress, 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 214 Analogous equations exist in atomic force microscopy, valid if the perturbations are small 3

4 Sauerbrey i inertial stress i iv ˆ Dm f pz velocity pz vˆ q q Small Load Approximation Dm D m : Mass per unit area of film f m m q q Z q / 2f : Mass per unit area of crystal Stress might go back to viscous drag, elastic forces... acoustic multilayers, interfackal high-frequency rheology The stress can be averaged over area : i ˆ f p Z vˆ q area Discrete objects: ˆ area N F ˆ with F ˆ the periodic force A The force can be averaged over time: q time i N 2 F t exp i t f p Z A vˆ QCM covers nonlinear force-displacement relations 4

5 air or water Coating added weight Stiffness of Sphere-Plate Contacts glass spheres diameter 1 or 2 mm deformation only close to contact clamp soft link, heavy sphere f P N ApZ q Elastic Load 3 SiO spheres r= 2.2 mm Hertz-Mindlin: 2 a: contact radius 1 G * G 4G * Ga : effective modulus 2 air 1 water JKR Fits added weight [g] Vlachová, J.; König, R.; Johannsmann, D. Beilstein J. Nanotechnol. 215, 6,

6 Coulomb friction tangential force Contact Stiffness Contact Strength Quasi-static, force control: stick-slip transition is an instability strength: µ S F N stiffness µ D F N tangential displacement Oscillatory motion, strain control: Partial slip not an instability tangential force partial slip stiffness gross slip tangential displacement 6

7 Sauerbrey i inertial stress i iv ˆ Dm f pz velocity pz vˆ q q Small Load Approximation Dm D m : Mass per unit area of film f m m q q Z q / 2f : Mass per unit area of crystal Stress might go back to viscous drag, elastic forces... acoustic multilayers, QTM high-frequency polymer rheology The stress can be averaged over area : i ˆ f p Z vˆ q area Discrete objects: ˆ area N F ˆ with F ˆ the periodic force A The force can be averaged over time: q time i N 2 F t exp i t f p Z A vˆ QCM covers nonlinear force-displacement relations 7

8 The stress can be averaged over time: Loads are small strain control 1 1 u u cost dt du 2 1 u / u t i 2 p Z Stress and force can be averaged over displacement, u ~ F t cos i time The QCM and Nonlinear Response f q t expit ˆv time F(t) time,,,, t DG ~ F t sin i F u u F u u,,,, time F u u F u u u / u 1 u / u, DG are weighted averages of friction loop shape of friction loop uncertain u 2 u force F partial slip F viscoelastic u displacement u Hanke, S.; Petri, J.; Johannsmann, D., PRE 213, 3248 Johannsmann, D., Springer 214 8

9 Data fit to Mindlin model They might also be explained by a temperature-induced softening of the contact This question can be answered with 3 rd harmonic generation Shape of Friction Loop? force force force force displacement displacement displacement displacement Ghosh, S. K. et al. ; Biosensors & Bioelectronics 211, 29, 145 Berg, S.; DJ, Surface Science 23, 541, 225 9

10 partial / total slip tangential force slip stick Partial Slip displacement Cattaneo, C., Rendiconti dell' Academia Nationale dei Lincei 1938 Mindlin, R.D.; Deresiewicz H.: J. Appl. Mech Johnson, K. L., Contact Mechanics 1985 Savkoor, A. R. Tech. University Delft, 1987 Varenberg, M.; Etsion, I.; Halperin, G., Tribology Letters 25 When transient: Transition state between stick and slip, mixed lubrication, When slow: Contact aging, compaction, soil mechanics, granular media, When oscillatory: Fretting wear 1

11 DG Partial Slip and Gross Slip humidity polymer film (T g ~18 or 37 C) added weight diameter µm amplitude of oscillation 2 nm Small spheres (d = 5 µm) soft substrate (T g ~ 37 C) 1 Hz 2 Hz Medium size spheres (14 µm) soft substrate (T g ~ 37 C) Partial Slip 1 Hz 2 Hz Large spheres (275 µm) hard substrate (T g ~ 18 C) 1 Hz Gross Slip 2 Hz amplitude [nm] amplitude [nm] 11

12 Partial Slip Nonlinear Stress-Strain Relations Quantitative models for the force-displacement relation exist (but: quasi-static) Stress in sliding zone follows Coulomb ( S = µp) Cattaneo, C., Rendiconti dell' Academia Nationale dei Lincei 1938 Mindlin, R.D.; Deresiewicz H.: J. Appl. Mech Stress in sliding zone constant ( S = const.) Savkoor, A. R. Tech. University Delft, 1987 p x stick partial slip Cattaneo-Mindlin S = µp force [a.u.] Cattaneo-Mindlin Savkoor umin F+ F- partial slip Savkoor S = const = Fmin -1 1 u/umax 12

13 Partial Slip Nonlinear Stress-Strain Relations p x stick partial slip Cattaneo-Mindlin S = µp force [a.u.] Cattaneo-Mindlin Savkoor F+ F- Cattaneo-Mindlin partial slip Savkoor N u 1 DG Fmin S = const = A2np Z 3µ F q N u Savkoor N A2np 2 Z q 4 9 p µf 2 N 5 u f u 1 A2 2 2 np Z q 8 4a D DG u 2 N u N 8 u A2np Z 6p q 2 u 2 a 2 DG u/u Cattaneo-Mindlin Cattaneo- Mindlin Savkoor Savkoor amplitude, u DG 2 Hz 5 1 amplitude [nm] Also: Leopoldes, J.; Jia, X., PRL 21 1 Hz 13

14 Intermodulation Products AB, Sweden Multifrequency Lockin Analysis A) Frequency combs A fast (milliseconds) way to probe resonances 42 freqs, fit Lorentzians, DG (no calibration required) B) Intermodulation products Excite with 2 frequencies beating signal (fast amplitude ramps) Signal [a.u.] frequency [Hz] linear real imag Nonlinearities signals at f = 2f 1 f 2 and 2f 2 f 1 Fast Signal resonantly enhanced, response function well understood C. Hutter et al. Phys. Rev. Lett. 21, 14, 581 Calibration issues nonlinear 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

15 air or water Coating added weight glass spheres diameter 1 or 2 mm [Hz] Partial Slip 2 reference tripod tripod + 2.7g Mindlin model works here (does not always work) There is a 3 rd harmonic signal There is a weak 2 nd harmonic signal (normal forces involved) Here: 5 MHz (fundamental mode) results are different on overtones Strong intermodulation products DG [Hz] rd Intermodulation Product x1 3 3 rd Harmonic x Apparent Amplitude [nm] 15

16 air or water Coating added weight glass spheres diameter 1 or 2 mm Partial Slip [Hz] DG [Hz] reference tripod tripod + 2.7g rd Intermodulation Product x rd Harmonic x Mindlin model works here (does not always work) Apparent Amplitude [nm] Red: Mindin-Theory Calibration issues Partial slip is at least partially transient normalized 3 rd harmonic amplitude rd Harmonic Generation Cattaneo Mindlin slope 2 dry silica sphers.1.1 normalized amplitude gross slip 16

17 air or water Coating added weight glass spheres diameter 1 or 2 mm Rich phenomenology at small amplitudes No 3 rd harmonic generation Intermodulation products seen Partial Slip in Liquids [Hz] DG [Hz] H 2 O reference tripod tripod + 2.7g rd Intermodulation Product x rd Harmonic x Apparent Amplitude [nm] 17

18 Sudden Impacts 1 Hz DG Transient friction events accessible Nonlinearities only at impact rd intermodulation product, real part Different experiment rd intermodulation product, abs 1E-3 1E-4 1E Time [s] 18

19 Crystallization [Hz] -3-6 [Hz] DG [Hz] Coupled Resonances 1 2 Time [hs] 5 Mhz CaCO 3 (aq), T = 23 C thiolated gold surface, quiescent solution initial supersaturation S = 1.5 Conventional QCM: Coupled Resonances DG [Hz] rd IMP x1 3 Bare surface 1.5 h 25.5 h Apparent Amplitude [nm] Multifrequency Lockin Amplifier (separate experiment) (u), DG (u) do not fit Mindlin model no 3rd harmonic generation Intermodulation products seen 19

20 Conclusions Fast QCM measurements (Dt = 4 ms) 3 rd harmonic generation: Partial slip is a transient event. Intermodulation products probe nonlinear mechanics. Impact of particles on QCM surface can be monitored. Nonlinearities are most pronounced during first impact. CaCO 3 nanoparticles (crystals) formed as individual objects Undergo rocking motion with nonlinear response. 2