Figure: the transition from sliding to the locked state as the force decreases with time for the 1D top substrate of N = 2048  layers with the constant damping ηl= 0.1 in the substrate (g = 10, η = 0.1) (enlarged view)	Figure: the averaged velocity of the top substrate versus the dc force for the 1D model with the constant damping ηl= 0.1 in the top substrate (g = 10, η = 0.1) (enlarged view)

Analytics

A general approach: let us present the total force as  F = F₁ + F₂ + F₃,  where

F₁  is the force because of the flux of energy into the top substrate,  F₁ = E_loss /a,  while the sum

F₂+F₃ = mηv  is due to the external damping of the atoms in the lowest (contact) layer of the top substrate when it moves with respect to the bottom substrate, and consists of the "trivial" contribution F₂ = mη‹v›  and the "fluctuating" contribution

Then we can use the following technique.

Green-function technique

1. The (top) substrate with internal (excitable) degrees of freedom is described by the causal (phonon) Green function  G(ω),  (ω₂–iωη_l–D)G(ω) = 1,  where D is the elastic matrix of the semi-infinite (top) substrate and  η_l describes the damping inside the sliding block. Then the generalized susceptibility is  α(ω) = –G(ω)/m  (here ω must be real).
2. When the top block moves with an average velocity  <v>,  its atoms in the lowest layer oscillate with the  washboard frequency  ω₀ = (2π/a)<v> + higher harmonics.
3. Linear response theory: when a system is excited by a small periodic force,  f(t) = Re f₀ exp(iω₀t)  (here  f₀  is real), then its velocity will oscillate with the amplitude  v₀=iω₀x₀,  where  x₀=α(ω₀)f₀  (let  f₀=f_s).
4. The rate of energy losses (the energy absorbed by the top block per one time unit) is

   R = 1 2 f02ω0 Im α(ω0) ,   or

   R = π 4   f02 m ρ(ω0) ,

   where  ρ(ω)  is the density of phonon modes in the top substrate ( ∫₀^∞ dω ρ(ω) = 1 ),

   ρ(ω) = – 2 π ω Im G(ω) .

5. The energy absorbed by the top substrate during its motion for one period of the external potential is equal to E_loss⁽¹⁾ =Rτ =2πR/ω₀=Ra /‹v›,  and the contribution  F₁  is  F₁= E_loss⁽¹⁾/a.
6. If we take into account the lowest harmonic of the washboard frequency only,  v(t) = ‹v› + |v₀| cos(ω₀t + φ),  then  |v₀| = ω₀|a(ω₀)| f₀,

   F1= π 4   f02 m‹v› ρ(ω0) ,  and

   (F1)

   F3= 1 2   mη ‹v› |v0|2 = η f02 2m   æ è 2π a ö ø 2   |G(ω0)|2 ‹v›

   Note that for a single atom, the backward threshold force is determined by the F₃ contribution only. However, when the moving block has internal degrees of freedom, then  F₁> 0, and the backward velocity may be nonzero, so that  F₂> 0  too.
7. In a general case, one has to take into account all harmonics ω_n= (n+1)ω₀,  n = 0,...,∞,  and perform the self-consistent calculation.  Namely, for a given velocity ‹v› we have to start with some approximate shape of  x(t)  [e.g.,  x(t)=‹v›t]  and then calculate step by step:
   1. the force acting on the atoms of the top block,

      f(t) = – sin x(t) – m η dx(t)/dt ,

   2. make its Fourier transform,

      fn = ⌠ ⌡ τ 0  dt eiωnt f(t) ,

   3. find the velocity in response to this force,

      vn= –iωn fnG(ωn) ,

   4. then perform the backward Fourier transform,

      v(t) = (2π)–1ω0 ∞ ∑ n=0  e–iωnt vn ,

   5. and calculate x(t) as the integral of v(t) over time; the output trajectory must coincide with the input one;

   6. finally, when self-consistency has being achieved, calculate

      F1 = – 1 2m‹v›   ∞ ∑ n=0  ωn |fn|2 Im G(ωn) = – π am   ∞ ∑ n=0  (n+1) |fn|2 Im G(ωn)     and

      F3 = mη 2‹v›   ∞ ∑ n=0  |vn|2 =  πmη‹v› a   ∞ ∑ n=0  (n+1)2 |fnG(ωn)|2.

One-dimensional model

Figure: the frictional force (solid curve and solid diamonds) as a function of the velocity of the 1D top substrate with  g = 10 (ωm≈ 6.32)  and  η = 0.1, the contribution  F1 due to radiation into the top substrate (dotted curve and open diamonds), the trivial contribution  F2= mηv (solid line), and the fluctuating contribution F3 (dashed curve and crosses).

The function  f(v)  has a minimum at  v = v_b  where  f(v_b) = f_b. Thus, if the dc force applied to the top substrate decreases, then the transition from the sliding regime to the locked state takes place at   f = f_b  when the average velocity of the top block is nonzero, v_b> 0.

The comparison of the analytical and simulation results was presented above in Fig.1D. One can see that the agreement is quite good (recall that in analytical approach we took into account the lowest harmonic of the washboard frequency only). Emphasize also that  the analytical approach corresponds to the constant-velocity algorithm of MD simulation, while the numerical results were obtained with the constant-force algorithm; both approaches lead to the same result.

Figure: (a) the threshold force  fb  and (b) the velocity  vb  for the sliding-to-locked transition as functions of the external damping  η  in the 1D model of the top substrate for two values of the substrate elastic constant  g = 10 (ωm≈ 6.32) and  g = 100 (ωm= 20).   The threshold values do not depend on the total mass of the sliding (top) block, but do depend on the elasticity of the block: a larger is  g, the lower are both threshold values  fb  and  vb.

Three-dimensional model

For the semi-infinite 3D substrate the Green function can be approximated by (Braun 1989)

where  ξ = ω/ω_m,  |ξ| ≤ 1. The real part of Green function may be found with the help of Kramers-Kronig relation,

The "surface" density of phonon states is equal to

The substitution of this expression into Eq.(F1) yields

In the limit  ‹v›→ 0  we obtain

Thus, in the approximation when only the main harmonic of the washboard frequency is taken into account, in the limit  ‹v›→ 0  we obtain  F₁=(8/ω_m³)‹v›,  F₂=η‹v›  and  F₃=(18η/ω_m⁴)‹v›,  which all together gives  F_b= 0  and  v_b= 0.  However at low velocities of the top block, the motion of the atoms is highly anharmonic, and the higher-order harmonics of the washboard frequency must be taken into account. The self-consistent calculation leads to the results presented in the figures below.

Conclusion

In all cases, the microscopic transition from stick-slip motion to smooth sliding occurs at  v ~ atomic scale,  i.e. v ~10 m/s, while “macroscopic” experiments show the transition at spring velocities as low as  v ~1 μm/s  (more than six orders lower). A large mass of the sliding block does not inhibit the freezing transition, because the block as a whole never stop abruptly – only the bottom surface of the block stops. Thus, nothing in common: the experimentally observed macroscopic smooth sliding corresponds in fact to the microscopic (atomic-scale) stick-slip motion, while the experimental (macroscopic) stick–slip motion is due to the f_s(t) dependence (aging of contacts). The corresponding theory is similar to that used in earthquakes-like models.

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Last updated on September 30, 2008 by Oleg Braun.             Translated from L^AT_EX by T_TH