Friction on Mesoscopic Scale

Stick-slip kinetics  

Above we described the rigid motion of the top substrate, when the ME solution always leads to smooth sliding. In order to describe a real dynamics of a tribosystem, we have to take into account the following three issues: (i) the slider inertia, (ii) the elastic instability, and (iii) the delay in contact formation or/and their aging.

Motion equation

First, we have to involve the motion equation for the slider; let it be modeled as a rigid entity (a nonrigid slider will be considered below) of mass M (see figure ). The slider is driven through a spring of the elastic constant K with the velocity v_d ,  so that the (experimentally measured) spring force is F_d= K (X_d−X),  X_d= v_dt.  Also the slider experiences the force  −F  from all contacts as described above with the ME or EQ approach. Every contact at the interface moves rigidly with the slider so long as the total force  f_i< f_si , where  f_si is a threshold value. Above the threshold the contact detaches from the slider and slips relative to it for a delay time τ_d after which it stops and attaches again to the slider; at this moment its stretching is determined from the condition that the total force  f_i on the contact is equal to the "backward" force  f_b~ 0. The force F from the interface acting on the slider is the sum of the forces from the pinned contacts,  f_i^(sub)= k_ix_i  (recall that x_i is the stretchings of the contact and k_i is its elastic constant), plus additionally the sum of drag forces from the detached/sliding contacts,  f_i^(drag) = −m_iη_i [v−dx_i(t)/dt ],  where  v(t) = dX(t)/dt  is the slider velocity, m_i is the mass of the contact and η_i is the corresponding damping coefficient (e.g., the "viscosity" of the molten lubricant).

Because the slider is rigid, at any abrupt change of its motion, e.g., at the onset or stopping of motion, it will undergo vibrations with the setup frequency Ω_S= (K/M )^1/2,  which cannot decay and therefore will disturb the results. In a real system, these oscillations will be attenuated because of internal friction inside the setup (i.e., due to phonon damping in the slider, when its bottom surface oscillates relative the top one). To incorporate such a damping, we introduce a viscous damping coefficient η_S for the slider motion relative its average velocity, so that finally the slider motion equation is

Model parameters. In simulation we used the EQ algorithm (the ME approach leads to the same results). For the sake of quantitative results, we extracted model parameters from the experiment of Klein (J. Klein, Phys.Rev.Lett. 98 (2007) 056101), where the OMCTS (octamethylcyclotetrasiloxane) lubricant film of the thickness h = 3.5×10^–9 m (four molecular layers) between two atomically smooth mica surfaces was studied with the surface force balance (SFB) technique. Thus, we took M = 1.47 g  and K = 97 N/m, that gives the setup frequency Ω_S= 257 s^–1 and the period τ_S= 2π/Ω_S = 0.0245 sec. Also we used A = 10^–10 m² for the total contact area and F_s= 18 μN for the static friction force. The damping coefficient for the drag force may be found from the bulk viscosity of OMCTS; this gives η = 2×10¹¹ s^–1. Assuming that there are N contacts at the sliding interface (N = 4080  in simulation), we can find the parameters for individual contacts: A_i= A/N for the contact area, a_i= √(A_i /π) for its radius, m = ρ_OMCTSA_ih for its mass,  f_s= F_s/N for its static threshold, and a = (A/N)^1/2 for the average distance between neighboring contacts. The contact elastic constant can be found as  k ≈ ρc²a_i; in simulation we took kN = 2000 N/m (note that it should be Nk >> K to have stick-slip). The static thresholds  f_si for individual contacts take random values from the Gaussian distribution with the mean  f_s and standard deviation Δf_s; other contacts' parameters are coupled with  f_si by m_i= mf_si/f_s,  k_i= k (f_si /f_s)^1/2 and η_i= η (when a contact reborns, its parameters are refreshed). Finally, we used η_S= 0.2Ω_S , v_d= 0.1 μm/s,  f_b= 0.1f_s, Δf_s= 0.01f_s, and τ_d=5×10^–4 sec (all the parameters were also varied in wide intervals).

Elastic instability

Second, we have to take into account the elasticity of the slider which may lead to the elastic instability (for more details see ). The master equation allows us to calculate the force F(X) when the bottom of the solid block is displaced by X, but actually we don't control X. The displacement is caused by a shearing force F_d= K (X_d− X)  applied on the top of the solid block which displaces the top surface by X_d. The total force applied to the bottom of the sliding block, which determines its displacement X, is the sum of the applied force and the force from the interface, F_tot= K (X_d−X) − F(X). This force can be viewed as deriving from the potential

A necessary condition for smooth sliding is that X_d and X grow together with  X_d− X = B, where B is a constant that measures the shear strain of the solid block during the sliding. It is determined by the condition  ∂V_eff/∂(X_d−X) = −∂V_eff/∂X = 0, which simply means that the total force on the interface vanishes. Smooth sliding also requires this state be stable,

If we start from relaxed asperities, in the early stage of the motion F(X) is a growing function of X (see Fig.S18), and then it passes by a maximum when some contacts start to break and reform at lower asperity stress. As a result F'(X) = dF/dX becomes negative. Depending on the value of K, two situations are possible. For large K  (K > K^*= − max F'(X), stiff block) F'(X) never falls below  −K and the smooth sliding is a stable steady state. For small K  (K < K^*, soft block) F'(X) can become smaller than  −K so that the stability condition (S43) is no longer valid (this is the well-known elastic instability, e.g., see T.Baumberger and C.Caroli, Advances in Physics 55 (2006) 279). The instability may cause X_d− X to change abruptly by a breaking of all contacts and a quick slip of the block before the contacts reform with relaxed asperities. If the process repeats itself, we have the stick-slip motion. The master equation, which gives F(X), can be used to compute the period of the stick-slip. However, the existence of a stick slip is not only determined by the stiffness K of the block, but also by the distribution P_c(x) of the static thresholds and the time of contact reforming, as is described below.

Thus, if the slider is soft enough, K < K^*, its motion becomes unstable at the point X_c, where X_c is the (lowest) solution of the equation F'(X) = −K (see Fig.18). Let this occur at the moment t_c. After the unstable point, t > t_c, the motion equation in terms of ΔX = X − X_c and Δt = t − t_c is

with the initial condition ΔX = 0 and  dX/dt = v_d at Δt = 0. In the EQ model with the continuous distribution of static thresholds, the function F(X) near the point X_c can be approximated as  F(X) ≈ F(X_c) + F'(X_c) ΔX + (1/2)F''(X_c) ΔX² + (1/6)F'''(X_c) ΔX³, so that

Introducing the dimensionless variables  x = K (X − X_c) / F_s and  τ = Ω_S(t − t_c), we obtain the following equation for the slider coordinate:

where

Solution of Eq.(S46) should be found with the initial condition x(0) = 0 and  dx(τ)/dτ|_τ=0= D₁. At short times, τ << 1, the solution has the form  x(τ) ≈ D₁τ + (1/6) D₁τ³. Thus, as X(t) grows at t > t_c, the spring force quickly drops during a slip time τ_s≈ α/Ω_S  in the form

where α = (6F_sΩ_S /Kv_d)^1/3  (α ≈ 2.21 for the chosen set of parameters). Numerical solution of Eq.(46) is shown in Fig.S19.

On the other hand, if the whole lubricant film melts instantly at t = t_c, then for t > t_c we have F(X_c+ΔX) = Mη d(ΔX)/dt, and the motion equation in dimensionless variables takes the form

where D₀= − F_d(X_c)/F_s. Its solution for τ <<1 is  x(τ) ≈ D₁τ + Bτ², so that the spring force drops as

with B = (1/2)(D₀− ηD₁/Ω_S). This gives the slip time τ_s≈ β/Ω_S  with  β = [2/(D₀− ηD₁/Ω_S)]^1/2. Numerical solution of Eq.(S47) is shown in Fig.S19. One can see that in this case the slip time is much shorter; however, using artificially huge values for η, one may make τ_s as large as desired (see discussion).

Slip details

A typical example of stick-slip motion of the EQ model for the chosen set of parameters is presented in Fig.S20. It is very similar to that observed experimentally, including the large initial stick spike and subsequent stick spikes of smaller amplitude (e.g., compare Fig.S20 with Figs. 1 and 2 in Klein's paper, see ).

Details of the slip are shown in Fig.S21. During the slip time τ_s, the slider velocity grows up to a value v ~ Ω_SF_s/K. As the slider accelerates, the contacts break, the number of detached contacts grows, and the force F(t) on the slider from the contacts drops from F_d(t_c) to zero. After the delay time τ_d , the contacts reborn, and F(t) grows as F(t) = Ct with the rate C = Nkvt  back to the value ~F_d(t). Therefore, the force F oscillates with a period τ_B≈ F_s/Nkv ≈ K/NkΩ_S. Such oscillations exist provided τ_B< τ_s,  or K < Nk.

Third, let us consider the role of delay time. One may think that the elastic instability will result in stick-slip motion. However, the condition  K < K^*  is the necessary but not sufficient one for stick-slip; the second necessary condition is a nonzero delay time, τ_d> 0.  This is clearly demonstrated in Fig.S22: if τ_d= 0,  the system still approaches the smooth sliding (for more details see ).

The subsequent (after the slip) motion of the slider depends on a value of the parameter τ_d. If τ_d > Ω_S^–1, the spring force drops to negative values and then oscillates around zero with the frequency Ω_S during the time τ_d (Fig.S22c). Otherwise, if τ_d<< Ω_S^–1, the spring force drops to a value higher than F_b and then begins to grow, oscillating with a high frequency Ω_L= (Nk/M)^1/2= 1.17×10³ sec^-1  (so that 2π/Ω_L= 5.39×10^–3 sec; see Figs.S22a and S22b). In all cases, the ringing vibrations decay as e^–η_St.

If we neglect ringing oscillations, then the dependence F_d versus v_dt  is "universal". Thus, if the velocity is so small that τ_ss >> η_S^–1, i.e., the oscillations are completely damped during the time per stick-slip cycle τ_ss,  the amplitude of stick-slip variation remains the same. But if the driving velocity is so high that τ_ss< η_S^–1,  then the oscillations will disturb the system dynamics and may lead to smooth sliding.

Role of threshold's dispersion. The system behavior (either stick-slip or smooth sliding) is controlled by the dispersion Δf_s. Indeed,

Thus, if Δf_s is so small that K^*> K, then the motion corresponds to stick-slip; otherwise the smooth sliding regime is achieved. In the stick-slip regime, an increase of Δf_s leads to the decrease of the period τ_ss of stick-slips as demonstrated in Fig.S23. The time τ_s of the F(t) drop during slip also increases with Δf_s, but this effect is rather small.

Finally, note that the ratio Δf_s/f_s should decrease with the time of stationary contact due to aging of contacts; namely this aging is responsible for the transition from stick-slip to smooth sliding with the increase of driving velocity as described in next section .

Role of slider elasticity. Above we have assumed that the slider is a rigid body; in a real setup, however, both the slider mass and its elasticity are distributed through the slider. One may think that in this case, only the most bottom atomic layer of the slider starts to move at the onset of slip. Therefore, a characteristic frequency will be determined by the mass M_l  of the layer and its elastic constant K_l ;  this could lead to a much higher (atomic scale) frequency. A similar question – which mass, either the total mass M or the layer mass M_l , defines the time scale of the system – appeared in the problem of minimal velocity for the atomic-scale smooth sliding. As was proven, when the slider velocity decreases, first the most bottom layer stops; then the stopping wave emerges and takes away the kinetic energy of the slider. In a result, the time scale of that problem is determined by the layer mass M_l , and the minimal velocity is of atomic-scale order (~m/s).

Now, however, the situation is just opposite. To show this, let us consider the slider as consisting of N_l layers, each of the mass M_l= M/N_l, elastically coupled by springs of the elastic constant K_l= KN_l  (see inset in Fig.S24). Indeed, if we fix the bottom layer and apply a force F to the top layer, then the latter will shift on the displacement ΔX = ∑_l=1^N_l ΔX_l ,  where ΔX_l = F/K_l , so that ΔX = F/K  as before. Now let the top layer be driven with the velocity v_d, while the bottom layer be in frictional contact with the bottom substrate. The dependence of the elastic force in the slider on time, obtained with simulation for two different values of N_l , is shown in Fig.S24 (note that now we may not use the artificial damping η_S ,  because internal degrees of freedom are included). As seen, the slip kinetics is almost independent on the number of layers N_l  and is determined by the minimal slider mechanical frequency Ω_S.

The frequency Ω_S can be found with the help of elastic theory. Let the slider have a cylinder shape of height L and radius r, and let it be characterized by the section S = πr², inertial momentum I = πr⁴/4, mass density ρ and Young modulus E. If the cylinder foot is fixed and a force F is applied to its top, the latter will be shifted on the distance ΔX = FL³/3EI  (the problem of bending pivot, see Sec.20, example 3 in the Landau and Lifshitz (LL) textbook). Thus, the effective elastic constant of the slider is K = 3EI/L³.  The minimal frequency of bending vibration of the pivot with one fixed end and one free end, is given by Ω_S= (3.52/L²)(EI/ρS)^1/2  (see Sec.25, example 6 in LL). Taking M = ρSL, we finally obtain Ω_S≈ 2.03√(K/M).

Interacting contacts

Finally, let us consider the role of interaction between the contacts (here we present a qualitative analyses only; for an accurate consideration of the problem see ). A concerted, or synchronized breaking (triggering) of contacts may be studied numerically only within the earthquakelike model, it cannot be included accurately in the ME approach (although it may be incorporated indirectly in a mean-field fashion by renormalization of the distributions P_c(x) and R(x), see ).

The elastic interaction between contacts separated by a distance r is described by the pairwise potential V(r) = g/r³, where g is a constant. Using ∂V/∂x = V'(r) ∂r/∂x,  V'(r) = dV/dr = −3g/r⁴ and ∂r/∂x = x/r, we obtain that the force acting on the ith contact from its nearest neighbors (NN), is equal to

The interaction becomes important when  g/a³ ~ f_sa,  where a is the average distance between NN contacts; therefore, it is interesting to check, how the system behavior changes with the dimensionless parameter ξ = g/(f_sa⁴).

The interaction between the contacts works roughly in the same way as the dispersion Δf_s:  the stronger is the interaction, the wider is the range of model parameters where stick-slip operates. To demonstrate this effect, we calculated the system kinetics with increasing strenght of the interaction  (ξ = 0 to 0.3)  for softer contacts (kN = 200 N/m) and a larger dispersion of the thresholds (Δf_s/f_s= 0.3). The results are presented in Fig.S25: for these parameters, the system quickly goes to smooth sliding for noninteracting contacts (Fig.S25a), but demonstrates stick-slip for a strong interaction ξ = 0.3  (Fig.S25d).

Details of the slip are shown in Fig.S27: without interaction, contact's detaching is sequential, while for a strong interaction, all contacts detach simultaneously, and the force F abruptly drops to zero. This is clearly demonstrated in Fig.S27 (compare panels (a) and (d)).

The interaction may lead to avalanches as was observed in the EQ simulation: for ξ = 0 the contacts break sequentially, one after another, while for ξ > 0 one contact may stimulate the breaking of nearest contacts, and the average size of avalanche increases with ξ as demonstrated in Fig.S26. For a large strength of interaction, ξ = 0.3, the avalanche occupies the whole system (Fig.S26d).

Discussion

In tribological community there exists an opinion that the viscosity of a thin confined film is many orders of magnitude higher than the bulk viscosity (P.A.Thompson, M.O.Robbins, G.S.Grest, Israel J.Chem. 35 (1995) 93;  H.-W.Hu, G.A.Carson, S.Granick, Phys.Rev.Lett. 66 (1991) 2758). For example, considering the drop of the spring force during slip, Klein (J.Klein, Phys.Rev.Lett. 98 (2007) 056101) came to the conclusion of "high viscosity" of the molten lubricant film, η ≈10⁴η_OMCTS. So large viscosity of a thin film, however, has no reasonable explanation. The cases of a single or two molecular layers confined between two planar surfaces are exceptional in the sense that properties of such films are far from their bulk ones. But already a three-layer film, if it is melted (liquidized) during slips, should exhibit properties not very different from the bulk ones, as follows from almost all MD simulations.

In our model with a distribution of static thresholds, the time τ_s of F(t) drop is almost solely determined by the setup frequency Ω_S.  This time remains approximately unchanged even if we increase the model parameter η (which corresponds to the film "viscosity") in 10⁵ times as demonstrated in Fig.S28. 

The conclusion about a huge viscosity of a thin confined film is based on the assumption that the lubricant film is (ideally) homogeneous and, at the onset of slip, the whole film melts simultaneously so that the further slip proceeds with a liquid film. When the film instantly melts, the force F on the slider from the interface abruptly drops to the viscous force (which starts from zero being proportional to the slider velocity), while the force from the driving spring remains unchanged at the beginning. Therefore, if the slip is viewed as a uniform, massive lubricant melting event, then the limiting factor would be the lubricant's viscosity, and in that case very large viscosity values for the lubricant's viscosity are required, 10⁴ to 10⁷ times higher than that of the bulk lubricant, to explain experimental results.

However, this assumption has neither experimental nor theoretical support. From surface science physics it is known that a thin film may hardly be homogeneous on a meso- and even nanoscale, as well as the surfaces themselves are not ideally crystalline on these scales. Generally, a film consists of domains with different orientation or even different structure and, therefore, are characterized by different threshold stress yields. Thus, it is reasonable to assume that the film does not melt and begin to slide all at ones, but different domains start to slide one by one, as described by the EQ model. In this case, the decrease of F is gradual owing to consequent breaking (melting) of different contacts (domains), and this approach describes the X(t) dependences observed experimentally.

The mechanism of slip onset (either sequential breaking of the contacts or the instant melting of the lubricant film) determines details of the slip kinetics which in principle may be resolved with SFB experiments. The order of magnitude of the slip time, however, in both cases is determined by the setup mechanical frequency Ω_S. In our model adapted to the SBF setup, the sliding mass M is concentrated at the end of the spring, i.e., the slider plus spring form a usual pendulum. A pendulum is characterized by two characteristic times – the inverse of its frequency, Ω^–1, and the inverse of its damping coefficient, η^–1; whichever of these times is shorter, it determines the system kinetics.

The frequency of ringing vibrations after the slip is either Ω_S (if F(t) oscillates around zero in the case of τ_d> Ω_s^–1) or Ω_L, i.e., much higher in the case of  τ_d< Ω_s^–1. However, the question is more involved: in the model we assumed that the bottom substrate is rigid and fixed. In a real setup, only the bottom of the base may be fixed, while the base has its own mass M_B and elasticity K_B. When the sliding stops and the two substrates are pinned together, the whole system of the mass M_S= M_B+ M + Nm  should oscillate with a frequency ~(K_B/M_S)^1/2. Unfortunately, usually one does not know the experimental values for these parameters.

Notice also that, in the case of dF(v)/dv < 0 which emerges because of contact's aging, the regimes of stick-slip and smooth sliding may be separated by a regime of irregular (chaotic) motion due to inertia effects.

See also:

O.M. Braun, Europhys. Lett. 109 (2015) 48004 "Stick-slip vs. smooth sliding in the multicontact interface"

O.M. Braun and E. Tosatti, Philosophical Magazine 91 (2011) 3253-3275 "Kinetics and dynamics of frictional stick-slip in mesoscopic boundary lubrication"

O.M. Braun and E. Tosatti, Europhys. Lett. 88 (2009) 48003 "Kinetics of stick-slip friction in boundary lubrication"

Next: Stick-slip versus smooth sliding

Back to main page or tribology page

Last updated on March 3, 2015 by O.Braun.  Translated from L^AT_EX by T_TH