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a Transient Event? Frederick Meyer, 1 Daniel Forchheimer, 2 Arne - - PowerPoint PPT Presentation

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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,

slide-1
SLIDE 1

1

  • High-frequency nonlinear contact mechanics
  • Intermodulation products
  • Sudden impacts
  • Crystallization

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 micromechanics:

Sylvia Hanke, Rebekka König, Judith Petri, Jana Vlachová, Frederick Meyer

  • Intermodulation: Daniel Forchheimer
  • QCM work in general: Arne Langhoff
slide-2
SLIDE 2

2

The 2nd-Generation Quartz Crystal Microbalance

electrodes quartz plate U~

I~

f = f res: large amplitude of motion  large current

Df G loaded

11.999 12.000 12.001 2 4 6 8

G unloaded conductance G [mS] frequency [MHz]

  • Shifts in frequency and

bandwidth: Df, DG

  • many overtones

Df(n), DG(n)

  • dependence on amplitude

Df(n, u0), DG (n , u0)

  • Higher harmonics
  • Intermodulation products
slide-3
SLIDE 3

3

Small Load Approximation

conductance G(f) frequency

G

fr

time

current I(t)

(2pG)-1 fr

  • 1

Fourier Transform

r r

Complex Resonance Frequency f f i   G

 

p r exp 2 if t

L

ˆ ^: c ˆ

  • Z

" mp load lex impedanc amplitud v ˆ v velo e ( (t) = exp(i t) e" city ˆ stres ) s

ˆ

   

 

  p p  D

q L q

Z i i f Z Z ˆ f ˆ v

Small-load approximation

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

QCM: The Quartz Crystal Micro Stress-Balance

periodic stress, in-phase, out-of-phase

slide-4
SLIDE 4

4

Small Load Approximation

 

q q q q q

Sauerbrey ˆ f i inertial stress i i v m ˆ f Z velocity Z v m :Mass per unit area of film f m m Z / 2f :Mass per unit area of crystal f m D  D   p p D  D D         

    time

q

2 F t exp i t f i N The force can be averaged over : ˆ f Z A v QCM covers force-displacement relations  D  p time nonlinear

area ar q ea

Discrete objects:

f i The stress can be averaged over ˆ f Z v N ˆ ˆ F with F the periodic forc A ˆ ˆ e D  p    area : Stress might go back to , ... acoustic multilayers, interfackal high-frequency rheology viscous drag elastic forces

slide-5
SLIDE 5

5

Stiffness of Sphere-Plate Contacts

1 2 1 2 * *

Hertz-Mindlin: : contact radius 1 2 1 2 1 : effective modu 2 l 4 4 us         G a G G a G

deformation only close to contact

  clamp

q P

soft link, heavy sphere f N f A Z    D p  Elastic Load    

added weight Coating glass spheres diameter 1 or 2 mm air or water

Vlachová, J.; König, R.; Johannsmann, D. Beilstein J. Nanotechnol. 2015, 6, 845.

1 2 3 100 200 300

added weight [g]

air water JKR Fits

Df

SiO spheres r= 2.2 mm

slide-6
SLIDE 6

6

Contact Stiffness  Contact Strength

tangential displacement µD FN tangential force Coulomb friction stiffness strength: µ

SFN

tangential displacement tangential force stiffness partial slip gross slip

Oscillatory motion, strain control:  Partial slip not an instability Quasi-static, force control: stick-slip transition is an instability

slide-7
SLIDE 7

7

Small Load Approximation

 

q q q q q

Sauerbrey ˆ f i inertial stress i i v m ˆ f Z velocity Z v m :Mass per unit area of film f m m Z / 2f :Mass per unit area of crystal f m D  D   p p D  D D         

    time

q

2 F t exp i t f i N The force can be averaged over : ˆ f Z A v QCM covers force-displacement relations  D  p time nonlinear

area ar q ea

Discrete objects:

f i The stress can be averaged over ˆ f Z v N ˆ ˆ F with F the periodic forc A ˆ ˆ e D  p    area : Stress might go back to , ... acoustic multilayers, QTM high-frequency polymer rheology viscous drag elastic forces

slide-8
SLIDE 8

8

           

2

exp i i The stress can be averaged over : ˆ v Loads are small 1 1 cos 1 / Stress and force can be averaged over ~ cos i 2 time displacement, u   D  p         D  

time time q

t f f Z u u t dt du u u f t strain contr F t

  • l

t

   

 

         

 

2

, , , , , , , are weighted averages of friction / 1 / ~ sin i l ,

  • p

,

  

  DG  D D  G       shape of friction loop uncertain

u time u

u u u u F u u F u u F t F u u F u u f t

The QCM and Nonlinear Response

Hanke, S.; Petri, J.; Johannsmann, D., PRE 2013, 032408 Johannsmann, D., Springer 2014

F(t) u0 displacement u F F

visco- elastic partial slip

force time

slide-9
SLIDE 9

9

force displace- ment force displace- ment

Shape of Friction Loop?

Data fit to Mindlin model They might also be explained by a temperature-induced softening of the contact This question can be answered with 3rd harmonic generation

Ghosh, S. K. et al. ; Biosensors & Bioelectronics 2011, 29, 145 Berg, S.; DJ, Surface Science 2003, 541, 225

slide-10
SLIDE 10

10

stick slip

Partial Slip

displacement tangential force partial / total slip Cattaneo, C., Rendiconti dell' Academia Nationale dei Lincei 1938 Mindlin, R.D.; Deresiewicz H.: J. Appl. Mech. 1953 Johnson, K. L., Contact Mechanics 1985 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

slide-11
SLIDE 11

11

Partial Slip and Gross Slip

added weight polymer film (Tg~108 or 37°C) diameter 50  275 µm humidity amplitude of oscillation 0  20 nm

5 10 5 10 100 Hz 20 Hz

DG Df

100 Hz 20 Hz 100 Hz amplitude [nm] amplitude [nm] 20 Hz

Small spheres (d = 50 µm) soft substrate (Tg ~ 37°C) Medium size spheres (140 µm) soft substrate (Tg ~ 37°C) Large spheres (275 µm) hard substrate (Tg ~ 108°C)

Partial Slip Gross Slip

slide-12
SLIDE 12

12

Partial Slip  Nonlinear Stress-Strain Relations

  • 1

1

umin Fmin F-

Cattaneo-Mindlin Savkoor

force [a.u.] u/umax F+

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. 1953

Stress in sliding zone constant (S = const.)

Savkoor, A. R. Tech. University Delft, 1987

 stick

p

   partial slip Cattaneo-Mindlin

S = µp

x

partial slip Savkoor

S = const = 0

slide-13
SLIDE 13

13

 stick

p

   partial slip Cattaneo-Mindlin

S = µp

x

partial slip Savkoor

S = const = 0

  • 1

1

Fmin F-

Cattaneo-Mindlin Savkoor

force [a.u.] u/u0 F+

   

2 q N 2 q N

Cattaneo-Mindlin u 3µ N F 4 u 9 µF f u 1 A2n Z N u A2n Z   D      p   DG   p  p 

   

2 2 q 2 2 q 2 2

Savkoor u 4 a u 2 N 5 f u 1 8 A2 a n Z N 8 u 6 A2n Z                D      p   DG  p p      

Partial Slip  Nonlinear Stress-Strain Relations

amplitude, u0 Df  Cattaneo-Mindlin Savkoor DG Cattaneo- Mindlin Savkoor

5 10

DG Df

100 Hz

amplitude [nm]

20 Hz

Also: Leopoldes, J.; Jia, X., PRL 2010

slide-14
SLIDE 14

14

Multifrequency Lockin Analysis

Intermodulation Products AB, Sweden A) Frequency combs A fast (milliseconds) way to probe resonances 42 freqs, fit Lorentzians  Df, DG (no calibration required)

4991700 4992000 4992300 4992600

real imag

Signal [a.u.] frequency [Hz]

C) 2nd and 3rd Harmonic Generation  Excite at f, probe at 2f, 3f, etc… (also: determine background)  Signal not resonantly enhanced  Probes MHz dynamics B) Intermodulation products Excite with 2 frequencies  beating signal (fast amplitude ramps) Nonlinearities  signals at f = 2f1 –f2 and 2f2 –f1  Fast  Signal resonantly enhanced, response function well understood

  • C. Hutter et al. Phys. Rev. Lett. 2010, 104, 050801

 Calibration issues linear nonlinear

slide-15
SLIDE 15

15 added weight Coating glass spheres diameter 1 or 2 mm air or water

Partial Slip

  • 40
  • 20

5 10 15 20 25 0.0 0.5 1.0 1.5 3 6 9 200 DG [Hz]

Apparent Amplitude [nm]

3

rd Harmonic x10 3

3

rd Intermodulation

Product x10

3

Df [Hz] reference tripod tripod + 2.7g

 Mindlin model works here (does not always work)  There is a 3rd harmonic signal  There is a weak 2nd harmonic signal (normal forces involved)  Here: 5 MHz (fundamental mode) results are different on overtones  Strong intermodulation products

slide-16
SLIDE 16

16 added weight Coating glass spheres diameter 1 or 2 mm air or water

Partial Slip

  • 40
  • 20

5 10 15 20 25 0.0 0.5 1.0 1.5 3 6 9 200 DG [Hz] Apparent Amplitude [nm] 3

rd Harmonic x10 3

3

rd Intermodulation

Product x10

3

Df [Hz] reference tripod tripod + 2.7g

 Mindlin model works here (does not always work)  Red: Mindin-Theory  Calibration issues  Partial slip is at least partially transient

0.01 0.1 10

  • 5

10

  • 4

10

  • 3

10

  • 2

dry silica sphers normalized 3

rd harmonic amplitude

normalized amplitude

slope 2

gross slip

Cattaneo Mindlin 3

rd Harmonic Generation

slide-17
SLIDE 17

17 added weight Coating glass spheres diameter 1 or 2 mm air or water

Partial Slip in Liquids

50 100 1 2 3 4 5

  • 1

1 2 0.0 0.1 0.2

  • 150
  • 100
  • 50

DG [Hz]

H2O

Apparent Amplitude [nm]

3

rd Harmonic x10 3

3

rd Intermodulation

Product x10

3

Df [Hz] reference tripod tripod + 2.7g

 Rich phenomenology at small amplitudes  No 3rd harmonic generation  Intermodulation products seen

slide-18
SLIDE 18

18

2 4 6 8 10 2.2 2.4 2.6

  • 0.10
  • 0.05

0.00 0.05 0.10 2 4 6 8 10 1E-5 1E-4 1E-3 0.01 0.1

DG 100 Hz Df 3rd intermodulation product, real part 3rd intermodulation product, abs Time [s]

Sudden Impacts

 Transient friction events accessible  Nonlinearities only at impact Different experiment

slide-19
SLIDE 19

19

Crystallization

 CaCO3(aq), T = 23°C thiolated gold surface, quiescent solution initial supersaturation S = 1.5  Conventional QCM: Coupled Resonances

  • 6000
  • 4000
  • 2000

10 20 2000 4000 Time [hs] Df [Hz]

5 Mhz 15 25 35 45 55 Coupled Resonances

DG [Hz]

  • 6000
  • 3000

500 1000 0.0 0.5 1.0 1.5

  • 0.05

0.00 0.05 0.10 0.15

Df [Hz] DG [Hz]

3

rd IMP x10 3

Apparent Amplitude [nm]

Bare surface 1.5 h 25.5 h

Multifrequency Lockin Amplifier (separate experiment)  Df(u), DG (u) do not fit Mindlin model  no 3rd harmonic generation  Intermodulation products seen

slide-20
SLIDE 20

20

Conclusions

  • Fast QCM measurements (Dt = 4 ms)
  • 3rd 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.

  • CaCO3 nanoparticles (crystals) formed as individual objects

Undergo rocking motion with nonlinear response.