ISSP International Workshops on Soft Matter Physics –structural rheology- (2010.8.10, ISSP, Chiba, JAPAN)

Criteria for the validity of Amontons Coulomb’s law;
Study of friction
using dynamics of driven vortices of superconductor
Department of BasicSciences,
University of Tokyo

RIKEN
S. Savelev
F. NoriAtsutaka MAEDA
Daisuke Nakamura
Ryo Tanaka
Y. Imai
CRIEPI
Ecole Polytechnique
M. Konczykowski
I. Tsukada
JAERI
S. Okayasu

C. J. van der Beek

Outline1) Background：
Problems in physics of friction
Collective dynamics of quantum condensate
Dynamics of driven vortices in superconductors
2) Physics of friction by using dynamics of driven vortices
ModelKinetic friction as a function of velocity
Broadened dynamical phase transition
A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Static friction as a function of waiting time

Criteria for the validity of Amontons-Coulomb friction
D. Nakamura et al.: arXiv 0906.3086.

3) Response to ac (µ-wave) driving force
4) Conclusion, perspective

Microscopic friction vs Macroscopic friction
Fs : maximum static friction force
Fk : kinetic friction force

Amontons-Coulomb’s law (A-C law)

microscopic
friction

D. M. Tanner et al., MEMS Reliability :Infrastructure, Test Structures, Experiments, and Failure Modes
2000-2091 (Sandia National Laboratories, 2000).

Fs is constant
Fk does not depend on v

macroscopic
friction

In some cases, eventhe macroscopic friction
does not obey the A-C law

What is the limit of the validity of Amontons-Coulomb’s law?

Amontons-Coulomb friction vs real friction
Amontons-Coulomb’s law

Real friction system

To understand friction from microscopic point of view

Need model system with which systematic experiments arepossible in a repeated manner (free from wear)

Microscopic formulation of friction

H. Matsukawa and H. Fukuyama:
PRB 49, 17286 (1994)←a displacement of upper atom i ui , massma←b displacement of lower atom i vi , mass mb
eq. motion
for an upper atom i
eq. motion
for a lower atom i

dissipation from a representative DF to others
steady statesumming up for all atoms
time averaged

friction: sum of interatomic (pinning) forces

1 D model for clean surfaces
numerical solution for the above equation
・clean surface
finite Fk even for zero Fs
・disordered surface
less velocity dependent
similar to Amontons-Coulomb’s law

Fk as a function of velocity

clean surface(normal) dirty surface

Model systems for friction study in quantum condensate in solids
Charge-density wave (CDW)

1D

ui : displacement of i-th electron in the CDW

m: mass of the i-th electron
Fp: pinning force for i-th electron
Vortex lattice of superconductor

2DB

ui : displacement of i-th votex in the lattice

m: mass of the i-th vortex in the lattice
Fp: pinning force for i-th votexVortex dynamics - the ideal model system of friction
viscous force η<ｖ>pinning force FPIN

J, E

creep motion

driving force J ×Φ0flux flow

moving direction

flux flowelastic pinningcreep motion
Friction force on solid-solid interface

H. Matsukawa and H. Fukuyama PRB 49, 17286 (1994).

Common feature

・ multi-internal-degrees of freedom
・ non-equilibrium・ non-linearity

A. Maeda et al., PRL 94, 077001 (2005).

Advantage・ scan H, T and F continuously
・ intrinsically reproducible--- no wear, no debrisPurpose of study
Driven vortices as a model system of solid-solid friction(1) Measure kinetic friction as a function of velocity(2) Static friction as a function of waiting time
Change temperature, magnetic field, pinning, system size(3) Find the criteria for the validity of Amontons-Coulomb friction(4) Ac dynamics (µ-wave to THz)
Microscopic understanding of elementary excitation
at the interface

High-Tc cuprate superconductor：variety of H-T phase diagram

Expressing solid-solid friction in terms of vortex motion
necessary to make correspondence with theory

viscous force η<ｖ>Φ0: flux quantum
J: current density
ρ: resistivity

kinetic friction FFRIC（pinning force）

Driving force
J ×Φ0

direction of vortex motion

Flux flow resistivity

I -V measurement and viscosity ,η , measurement can deduce kinetic friction

Sample preparation
La2-xSrxCuO4 （LSCO）(x = 0.12, 0.15)
Crystal structure

thickness 〜 3000 Å
single crystal, thin film

Fabrication of samples
Rectangular 〜 0.5mm〜 1.5mm

PLD method

Bridgephotolithography ＋ chemical etching
Bridge wih columnar defects
Pb ionS. Komiyama (Univ. of Tokyo）

5.8 GeV Pb ion for the columnar defects@ Grand Accelerateur National d’Ions Lourds(GANIL)
Ecole Polytechnique, M. Konczykowski and C. J. van der BeekFk (v)

La2-xSrxCuO4

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

（up to 〜1 km/s）
1) Fk changes with B and T
in a reproducible manner
good as a model system
2) very much different fromthe Amontons-Coulomb behavior
similar to “clean surface”
3) Fk saturates and decreases
existence of a peak in Fk(v)4) smaller Fk in irradiated samplesinconsistent with the behavior
at low velocities ?
BΦ=3T Columnar defects
200 MeV Iodine
Data points with crosses denotepulsed measurements

pristine

3T irradiatedBi2Sr2CaCu2Oy (bulk single crystals)

Universal to many SCs

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Minimal model to explain the data : overdamped equation of motion
S. Savel’ev and F. Nori

: position of vortices: viscosity of vortices: substrate pinning potential: inter-vortex interaction: driving force: thermal random force: temperature

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Numerical simulation for 1D vortex array at finite temperatures
S. Savel’ev and F. Noria peakA peak in the kinetic friction Fk(v)velocity at the peak

S. Savel’ev and F. Noriin good agreement with experiment
Potential energy plays an important role for Fk(v).

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

“Inversion” of kinetic friction at intermediate velocities！

frictionsample with strong pinninghigher static friction
lower kinetic friction
more gradual dependence on v

pristine
3ＴirradiatedvelocityS. Savel’ev and F. Nori

Q/l〜ηv~200m/s

A. Maeda et al.: Phys. Rev. Lett. 94 (2005) 077001.

Physical origin of the peak

Fkkineticstaticchanging parameterschange transition between
static and kinetic regimev

broaden the transition

increasing magnetic field
increasing temperature
decreasing system size (macro to micro)

N strongly coupled systemcollective coordinate
new stochastic variable
effective temperature(L : system size)

Suggesting the importance of thermal fluctuation in microscopic friction

Time domain response for various tw
D. Nakamura et al.: arXiv 0906.3086.
Input sawtooth signal, changing t W , measure IcµMeasurement of Fs(tw)Rectangular-type thin film
Driving currentxTc (K)# rectangular A 0.12

35.20

# rectangular B 0.15

35.06

Waveforms become different with tW
Ic depends on T → normalize by Ic(max tW)High T
characteristic
timescale

D. Nakamura et al.: arXiv 0906.3086.

Low T

log tw〜10-1 sTg

Logarithmic

Characteristic timescale

The origin of logarithmic dependence near TgThermally Assisted Flux Flow

P. W. Anderson and Y. B. Kim, Rev. Mod. Phys. 36, 39(1964).
P. H. Kes et al., Supercond. Sci. Technol. 1, 242(1989).

At I < Ic,
Critical state

In TAFF region, spatial magnetic profile
obeys diffusion equation of motion
Blatter et al. RMP (94).

This timescale can be estimated aslog tw at high T

D. Nakamura et al.: arXiv 0906.3086.

Rapid relaxation at low T with characteristic timescale
Low temp.

Slow tW* cannot be explanined by the motion of a single vortex

Bundle relaxationh : thickness, η: viscous coefficient
ξ: coherence length# rect. A〜10-1 s

energy loss by theviscous motion

depinning energy

Length scale of coherent relaxation(cf. vortex-vortex pacing : ~ 50nm)
Dynamic coherence length : 30µm
(from noise study in Bi-2212)

Bundle relaxation picture confirmed

A. Maeda et al., Phys. Rev. B 65, 054506 (2002).

D. Nakamura et al.: arXiv 0906.3086.

D. Nakamura et al.: arXiv 0906.3086.

Size effect of the relaxationµcrossover line of the relaxationw
no relaxation by the edge pinning effectsmaller relaxation region for the narrower bridge sample

Interplay between the size of coherent bundle and the system size
crucially affects the validity of Amontons-Coulomb’s law

D. Nakamura et al.: arXiv 0906.3086.

Bridge sample with columnar defects# Columnarx

Tc (K)

0.15

35.11

2.0

Dramatic changing for the relaxation
Sample with columnar defects →no relaxation at all

Strong pinning force also leads tothe Amontons-Coulomb’s type friction

D. Nakamura et al.: arXiv 0906.3086.

Summary of Fs(tw) experiments

The size of vortex bundle

Rectangular film
Bridge film
Bridge filmwith columnar defects

high T

low T

log tWcharacteristic
timescalelog tWno relaxation

no relaxation

no relaxation

size effectcolumnar pineffect

The origin of the relaxation phenomena
Thermally Assisted Flux Flow of vortex bundle
Bundle size determines the characteristic timescale
no relaxation
very strong pinning center(Amontons-Coulomb like)D. Nakamura et al.: arXiv 0906.3086.

Interaction(pinning)

Our achievements on this model approach
Amontons-Coulomb’s
type friction

Fs(tW)

Fk(v)Bundle size

Degrees of freedom

Key parameters of the validity of Amontons-Coulomb’s law
・ pinning strength and thermal fluctuation
・ size of the coherently moving object and system sizea universal parameter :a criteria of the validity of A-C law :Friction vs ac dynamics
Friction at the interface

Dissipation by elementary excitation
at the interface

Ac dynamics at the interface
vs microscopic understanding of friction
1) Importance of lattice dynamics
Phonons
Phonons in disordered systems
Corresponding excitations in CDWs and vortices in SC
2) data at µ-waves for vorticesNormal modes by periodic array of atoms: Phonon
Dispersion relation

Angular frequency
Wave number (well defined)(C. Kittel : “Introduction to Solid State Physics”)

Optical response by a bound mode: Lorentz oscillator(F. Wooten : “Optical Properties of Solids”)

for small ω

For pinned CDW H. Fukuyama and P. A. Lee : Phys. Rev. B17 (1978) 535.

P. A. Lee, T. M. Rice and P. W. Anderson: SSC 14 (1974) 703.

Experiments in CDWs
(D. Reagor et al.: Phys. Rev. B34 (1986) 2212.)(R. J. Cava et al.: Phys. Rev. B30 (1984) 3228.)(W. Wu et al.: Phys. Rev. Lett. 52 (1984) 2382.)(R. J. Cava et al.: Phys. Rev. B31 (1985) 8325.)(also J. P. Stokes et al.: Phys. Rev. B32 (1985) 6939.)

Mean-field model of vortex motion
P. Martinoli et al. Physica B 165&166, 1163 (1990),
M. W. Coffey and J. R. Clem: PRL 67 (1991) 386, PRB 45 (1992) 9872.

massless
overdamped
viscous force η<ｖ>pinning force FPIN

J, E
driving force J ×Φ0
moving direction

remarks(a) Magnus force, vortex-vortex interaction are not considered.
(b) Vortex mass is considered to be zero.
(c) Condensate wave function is considered to be s-wave.
(d) From microscopic point of view, the two-fluid picture is not a good approximation.
(Eschrig-Rainer-Sauls)Ac resistivity by the vortex motion
dissipative(flux flow)

reactive(pinned)

Our expectation
Non AC like
Weaker pinning

Digergent ε(ω)

Smaller bundle size

Remarkable nonlinearity

Stronger pinning

Lorentz-oscillator like ε(ω)
no nonlinearity

Larger bundle size
AC likeExperiments in vortices in superconductors (THz)
B. Parks et al. Phys. Rev. Lett. 74 (1995) 3265.

YBCO

overdamped

Experiments in vortices in superconductors (µ-wave)

overdamped

Conventional superconductor
Pb-In, Nb-Ta(J. C. Gittleman and B. Rosenblum:
Phys. Rev. Lett. 16 (1968) 734.)

High-Tc superconductor(Y. Tsuchiya et al.: Phys. Rev. B63 (2001) 184517-1.)

YBCO(Y. Matsuda et al.: PRB66(2002)014527.)

TBCO

Well describedby the mean-field modelwith field independent ωcoSeems to exhibit no apparent nonlinearity at present
Need to be investigated systematically in more detailAc impedance of SC: vortex motion plus superfluid
M. W. Coffey and J. R. Clem, PRL 67, 386 (1991).

δ nf : normal-fluid skin depth
λ : penetration depthParametersη : viscous drag coeff.
τp–1 : pinning frequency
ε : flux-creep factor
λ : penetration depth
Measurement of S11(ω)
byVector Network
Analyzer

normal-fluid
response
flux flowfrom ρ1from ρ2

modified 2.4mm J-J adapter
coax.
cable
Sample with Corbino
disk geometry

LSCO: viscous drag ηDissipation: similar to othercuprates (YBCO, BSCCO etc.)Y. Tsuchiya et al.: PRB 63 (2001) 184517.
A. Maeda et al.: JPSJ 76 (2007) 094708.
A. Maeda et al.: Physica C 460-562 (2007) 1201.

η depends on B.

Conclusion（１）Discuss dynamics of driven vortices and physics of friction utilizing driven
vortices of superconductor（２）Kinetic friction as a function of velocity:
understood in terms of simple overdamped oscillator model
Broadened transition of dynamic phase transition (Amontons-Coulomb behavior)suggesting the importance of thermal fluctuation for microscopic friction（３）Static friction as a function of waiting time: relaxation of vortex bundle
stronger pinning, large bundle size ⇔ Amontons-Coulomb like behavior（４）Criteria for the validity of Amontons-Coulomb friction
Presence of a universal parameter :（５）Ac dynamics (µ-wave to THz) of vortices
no remarkable anomaly nor nonlinearity such as CDW system at present

Future perspective（１）Change pinning, system size more systematicallyintroduce stick-slip motion in driven vortices
change velocity dependence of kinetic friction largely
Feedback to physics of friction‐Basic step for understanding friction, control of friction‐（２）Need to investigate
correspondence between ac dynamics and other friction properties‐comparative studies of Fk(v), Fs(tw) and σ(ω)‐