Transition from Stick-slip to Smooth Sliding: An Earthquake-like Model

An earthquake-like model presented below exhibits a “transition” from stick-slip motion to smooth sliding at a velocity of the order of those observed in experiments. The results show that experimentally observed smooth sliding at the macroscopic scale must correspond to microscopic-scale stick-slip motion (see also ).

The problem goto top

                                    vc  is the critical velocity of the transition from stick-slip to smooth sliding.

Simulation (soft lubricant,  Vll= 1/9,  Nl = 2,  kspring= 3·10–4):  vc~ 10–2c,  or  vc~ 1 to 10 m/s

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Experiment (friction force versus time at increasing sliding velocity for grafted chain molecules; Ruina 1983, Israelashvili 1993; from Persson’s survey 1999):  vc ~ 0.1 to 1 μm/s

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Earthquake-like model goto top

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Minimal model goto top

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The main idea is to introduce in some way a macroscopic-scale characteristic time  τ  (e.g., of order seconds) which will determine the transition between two qualitatively different regimes, the regime of smooth sliding and the regime of stick-slip motion. The most reasonable way is to assume that the static frictional force depends on the time of stationary contact, for example, let us assume that if the sliding-to-locked transition has occur at  t=0,  then the static frictional force will evolve as

fs(t) = fs1 + (fs2 – fs1)(1 – e–t/τ) .

(Fs)

One of possible mechanisms of a slow  fs  change with the time of stationary contact (after the sliding-to-locked transition) is a plastic deformation of a softer material, e.g., the lubricant. Possible mechanisms may be
-   the increase of the area of real contact with time
-   squeezing out for simple (spherical) molecules [recall our simulation result:  fs(Nl) ~ exp(–...Nl)]
-   inter-diffusion when the surfaces are covered by layers of long-chain polymers (e.g. fatty acid monolayers)

 

Variants of the model:

(a)   model: 1D ↔ 2D

(b)   lattice: regular ↔ random

(c)   interaction: short-ranged ↔ long-ranged

(d) “stimulated” model (when a junction relaxes, it emits a wave burst which stimulates other junctions to relax, too)

 

A minimal model: to reproduce experimental dependencies,

(a)   the model must be 2D

(b)   the spatial distribution of junctions  must be random

(c)   it must exist an interaction between the junctions (which may be short-ranged)

(d)   the model must incorporate a dependence of the static frictional force  fs(t)  on the time of stationary contact, e.g., according to Eq.(Fs)

 

Simulation goto top

A typical dependence goto top

 

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Details of the transition goto top

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Here  Ncount /N   is a “size” of the avalanche. Note that the transition is smooth.

 

Avalanche sizes goto top

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Figure: the distribution of avalanche sizes  P(s/N)  at  (a) v = 0.1 (solid curve) and  v = 0.2 (dotted curve), and  (b) v = 3 (solid curve) and  v = 5 (dotted curve).  The inset in (b) is a log-linear plot showing the exponential dependence.

Estimation goto top

Avalanche size at “smooth” sliding goto top

Let us at time t a given (i-th) junction, pinned for an "age" τi, relaxes. Then the force on a neighboring (j-th) junction abruptly increases by the amount  Δfj = [fsi(τi) – fb]b,  where b = kij /(k + ∑j≠ikij′).  For the triangular lattice with the parameters  kij~ k  we have  ‹b›≈1/7. The nearest neighboring junction,  j, will relax too (and, thus, the avalanche will start) if  fj(t+0) = fj(t–0) + Δfj  ≥ fsi(τj).  In the high-velocity regime we can put  fsi≈ fs,  because  ‹τi› << τ.  The distribution of forces  P(fi)  in this case has a simple form (Persson 1995): it is constant for forces within the interval  fb< fi< fs  and zero outside it. Thus, the probability  p  that the j-th junction will relax, i.e. that  fj(t–0) ≥ (1–b) fs,  is equal to  p = b. Because there are six nearest neighbors around the "starting" junction  i, the probability to have an avalanche of size  s ≥ 2  is Ps(2) = 6p. Then, the j-th junction may stimulate its own (five) nearest neighbors to relax, thus the probability to have an avalanche of size  s ≥ 3  is  Ps(3) = Ps(2)×5p.  Iterating, we have  Ps(s) = νpPs(s–1),  or  Ps(s) ~ (νp)s,  where 3 ≤ ν ≤ 5  [ν = 5  if the avalanche forms a one-dimensional non-intersecting curve and  ν ≈ 3  when the avalanche is compact (2D)]. Using  Ps(s) = ∫s∞ds′ P(s′),  we obtain for the avalanche distribution  P(s) ~ exp(–s/s),  where the average size of the avalanche is  s = –1/ ln(νp).  Thus, if static frictional force does not depend on time, we always have  νp < 1,  so that  s < ∞,  so the avalanche cannot occupy the whole system. Taking  ν = 5  and   p ≈ 1/7,  we obtain  s ≈ 3  for the triangular lattice. This is in agreement with simulation which yields  s ≈ 4  for the  v = 3  case. The fluctuations of the total frictional force scale as  ‹f(t) – ‹f(t)» ~ N–1/2 ~ 1/√A  with the number of junctions  N  or the contact area  A.  Note that similar considerations for the one-dimensional system lead to  ν=1  (the avalanche can expand in one direction only) and  p ≈ 1/3,  so that  s ≈ 1.

 

Avalanche at stick-slip goto top

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At a low velocity, when the time dependence of the static frictional force is important, the slipping of junctions becomes synchronized and an avalanche can occupy the whole system. The distribution of forces  P(fi)  now has a more complicated form, it is constant for forces  0 <  fi < fs  (here we put  fb= 0  for simplicity) and monotonically decreases to zero for forces  fs< fi< fsm.  For purposes of an estimation let us assume that this decrease may be described by a simple linear dependence,  P(fi) = 2(fsm– fi) / (fsm2– fs2)  (see figure to the right). The condition that after the relaxation of the i-th junction, the nearest neighboring junction  j-th will relax too, now takes the form  fj(t–0) ≥ (1–b) fsi(τi).  The probability that the  j-th junction will relax, is equal to  p = ∫f′fsm df P (f) = ( fsm– f′)2 / (fsm2– fs2),  where  f′=‹(1–b) fsi(τi)›. With the parameters used above, if we take  ‹fsi(τi)› ≈ 0.5(fsm+ fs),  we obtain  p > 1/7. Moreover, if  ‹fsi(τi)› < 1.43,  we obtain  p > 1/5,  so that  νp > 1,  and the avalanche will expand over the whole system.

 

Conclusion goto top

  • The macroscopically observed “smooth” sliding corresponds to the atomic-scale stick-slip motion (not resolved in experiment?).

  • The macroscopic-scale stick-slip behavior emerges due to the concerted motion of the many junctions because of their interaction.

  • The transition itself is smooth.

  • The “transition” takes place at  vc~ a/τ,  where  a  is the average distance between junctions and  τ  is an “aging” time of a single junction. Reasonable values for these parameters (e.g.,  a ~10–6 to 10–3 m  and  τ ~ 1 to 103 s)  lead to experimentally observed values of  vc (see however )..

Still open questions:

  • •nature of the junctions (asperities? “solid islands”?)

  • mechanism of the  fs(t)  dependence (squeezing out of the lubricant? aging of individual contacts? gradual increasing of the contact area? coalescing of contacts?)

See O.M. Braun and J. Röder, Phys. Rev. Lett. 88 (2002) 096102 "Transition from stick-slip to smooth sliding: An earthquakelike model"  (pdf files may be found here)

 

NB: Thus, we came to the conclusion that in order to describe the stick-slip to smooth sliding transition in agreement with experiments, the system must (i) be 2D, (ii) include interaction between the contacts, (iii) incorporate some chaos (in positions of contacts and in the initial configuration), and (iv) include contact's aging. However, above we used the model where all contacts are identical, while in a real tribosystem, the contacts should be characterized by different parameters, e.g., different values for the static thresholds. In the next Chapter we show that the model with a distribution of thresholds  Pc(fs)  naturally describes meso- and macroscopic tribosystems. In such a model, the items (i, ii) become not crucial, and the items (i, ii, iii) just act to create the distribution of thresholds. As for aging, it still is important to explain the dependence of system dynamics on the driving velocity  vd.

 

Next: Friction on mesoscopic scale

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Last updated on October 8, 2008 by Oleg Braun.         Translated from LATEX by TTH