Friction on Mesoscopic Scale

Kinetic friction: dependence on velocity  

Here we consider the velocity dependence of kinetic friction with a model which makes minimal assumptions on the actual mechanism of friction so that it can be applied at many scales provided the system involves multi-contact interface (MCI). Using the master equation approach, we investigate the influence of two concurrent processes. First, at a nonzero temperature thermal fluctuations allow an activated breaking of contacts which are still below the threshold. As a result, the friction force monotonically increases with velocity. Second, the aging of contacts leads to a decrease of the friction force with velocity. Aging effects include two aspects: the delay in contact formation and aging of a contact itself, i.e., the change of its characteristics with the duration of stationary contact. All these processes are considered simultaneously with the master equation approach, giving a complete dependence of the kinetic friction force on the driving velocity and system temperature, provided the interface parameters are known.

Introduction

Above we described the master equation (ME) approach to describe the breaking and attachment events. This method allows us to calculate the velocity dependence of friction, which results from the interplay of two concurrent processes. First, at a nonzero temperature thermal fluctuations allow an activated breaking of contacts which are still below their mechanical breaking threshold. This phenomenon leads to a monotonic increase of the friction force F with the velocity v. Second, the aging of contacts leads to a decrease of the friction force with velocity. It includes two processes: the delay in contact formation, i.e., time lag between contact breaking and remaking, and the aging of a contact itself, i.e., the change of its characteristics with the time of stationary contact. To incorporate the latter effect, the master equation must be completed by an equation for the evolution of static thresholds. Above we considered thermal and aging effects separately; to relate the results to experiments, however, both contributions must be taken into account simultaneously.

Steady-state solution

We use the master equation for the distribution Q(x;X):

For the normalized distribution of static thresholds given by Eq.(K8) with n=1,

Pc(x) = (2/x*) u3e–u2,    where     u = x/x* ,

(K9)

we can express the steady-state solution of the master equation analytically. In this case

Jc(x) = (1+u2) e–u2,

(K10)

so that at zero temperature we have

P(x) = (2/x*) u3/(1+u2) ,

(K11)

U(x) = u2 − ln(1+u2) ,

(K12)

EP(x) = Jc(x) = (1+u2) e–u2,

(K13)

C[P] = x*/C0 ,  where  C0 = (4/3√π) ≈ 0.752 ,

(K14)

Q(x) = (C0/x*) (1+u2) e–u2,      u ≥ 0

(K15)

and the kinetic friction is

fk = Fk /(Nck) = fk0 = C0x* .

(K16)

The ME formalism described above can be extended to take into account various generalizations of the EQ model, such as temperature effects and contact aging, which are examined in the following sections.

Nonzero temperature

For concreteness, in what follows we assume that the contacts are soft, Eq.(K18), and we select n=1 in Eq.(K8), so that P_c(x) is given by Eq.(K9). At a nonzero temperature the total rate of contact breaking, Eq.(K17), is equal to P_T(x) = P(x) + (ω/v) h(x), where

h(x) = 1+ (1+b) u2 (1+b)2 e–u2

(K20)

with

b(T) = kx*2 2kBT  .

(K21)

The condition b = 1 defines a crossover temperature

kBT* = 1 2 kx*2.

(K22)

Then, a straightforward integration gives the function U_T(x) = ∫₀^xdξ P_T (ξ) = U(x) + ΔU(x), where

ΔU(x) = S0(v,T) [erf (u) − S1(T) u e–u2]

(K23)

with

S0(v,T) = ωx* C0v   (1 + b/3) (1 + b)2

(K24)

and

S1(T) = C0 2   (1 + b) (1 + b/3)  .

(K25)

The coefficient S₁(T) weakly changes with temperature from S₁(0) = 2/√π ≈ 1.128 to S₁(∞) = C₀/2 ≈ 0.376. On the other hand, the coefficient S₀(v,T) determines whether the effect of temperature is essential or not. The temperature-induced breaking of contacts is essential at low driving velocities only, when S₀(v,T) >> 1. Thus, the equation S₀(v_*,T) = 1 defines the crossover velocity:

v*(T) = ωx* C0   [1 + b(T)/3] [1 + b(T)]2 .

(K26)

We see that v_* monotonically increases with temperature as v_*(T) ≈ 0.443ωx_*T/T_*  at T << T_* and approaches the maximal value v_*≈ 1.33ωx_* at T >> T_*.

Then, E_PT(x) = e^{–U_T (x)} = (1+u²) e^–u2e^–ΔU(x), and we can find the kinetic friction force:

fk(v,T) = ⌠ ⌡ ∞ x  dx x EPT (x) /   ⌠ ⌡ ∞ x  dx EPT (x) .

(K27)

At a low driving velocity, v << v_*, we may put ΔU(x) ≈ S₂u, where

S2(v,T) = ωx* v (1+b)2  ,

(K28)

and Eq.(K27) leads to

fk ≈ x*/S2 = (v/ω) (1 + b)2 .

(K29)

A linear dependence of the kinetic friction on the driving velocity at low velocities corresponds to the creep motion due to temperature activated breaking of contacts and was predicted in several earlier studies, although our approach allowed us to derive it rigorously. The dependence (K29) may be interpreted as an effective "viscosity" of the confined interface:

η =  fk v = 1 ω   [ 1 + kx*2 2kBT ] 2   .

(K30)

At a high velocity, v >> v_*, when  e^–ΔU(x) ≈ 1 − S₂u + ¹/₂(S₂u)², we obtain C ≈ 3√π/4 − S₂ + (5√π/16) S₂² so that

fk ≈ fk0 ( 1 − C1S2 + C2S22 ) ,

(K31)

where C₁ = 5√π/8 − C₀ ≈ 0.356 and C₂ = 16/9π − 1/3 ≈ 0.233. Equation (K31) agrees qualitatively with that found by Persson in the case of b >> 1.

Figure K1: The friction force fk as a function of the driving velocity. Dash-dotted blue line shows the low-v approximation (K29), dashed red line shows the high-v approximation (K31), and dotted magenta line shows a logarithmic fitting. k = 1, ω = 1, x*= 1, kBT = 1.

Approximate expressions (K29, K31) together with the numerical integration of Eq.(K27) are presented in Fig.K1. Also we showed a logarithmic fitting which operates in a narrow interval of velocities near the crossover velocity only. Persson showed that the logarithmic dependence may be obtained analytically, only if the P_c(x) distribution has a sharp cutoff at some x = x_s as, e.g., in simplified versions of the EQ model with P_c(x) = δ(x−x_s).

Although we cannot obtain analytical results for the stiff contacts, we calculated the f_k(v) dependences numerically (see Fig.K2), which shows that the effect remains qualitatively the same.

Aging of contacts

where D_v≡ D/v, and v = dX(t)/dt is the driving velocity. Finally, we come to the set of equations (K1−K3, K34). For the steady-state regime, Eq.(K34) reduces to

It was shown above that the kinetic friction monotonically decreases with the driving velocity as F_k(v) − F_k(0) ∝ −v/D in the low-velocity limit and F_k(v) − F_k(∞) ∝ D/v in the high-velocity case. One may expect that at low velocities this decreasing will compensate the friction increasing due to temperature induced jumps. The problem, however, is more involved.

When the temperature effects are incorporated, Eq.(K36) for the function P_c(x) in the steady state takes the form

Dv C[PT] L Pc(x) = Pc(x) − Pci(x) .

(K37)

Figure K3: The kinetic friction force fk as a function of the driving velocity v for different values of the aging rate: D = ∞ (red dotted), 1 (blue short-dashed), 10–1 (black solid), 10–2 (magenta dash-dotted), 10–3 (blue short-dotted), and D = 0 (red dotted curve). k = 1, ω = 1, kBT = 1; the initial and final Pc(x) distributions are given by Eq.(K9) with x*i = 0.1 and x*f  = 1 correspondingly. Dashed curves shows the dependences at T = 0.

Figure K4: Same as in Fig.K3 in log-log scale. (b) The dependences of the effective "viscosity" η = fk/v on the velocity v (dashed lines show power-law fits). .

Numerical solutions of Eq.(K37) are presented in Figs.K3 and K4a: the initial increase of the kinetic friction F with the driving velocity v due to the temperature activated breaking of contacts is followed by the decrease of F due to contacts aging. Figure K4b shows also the dependence of the effective "viscosity" η = f_k /v on the driving velocity v. It is constant at low velocity and then decreases; the latter may be approximately fitted by a power law η(v) ∝ v^–α with the exponent α changing from 1.5 to 1 as D decreases.

Using the definition (K32) of the operator L and Eq.(K33) for the function B(x), the l.h.s. of Eq.(K37) may be rewritten as

DvC L Pc(x) = DvC   d dx   [ Pcf (x) d dx   Pc(x) Pcf (x) ] ,

(K38)

while the r.h.s. of Eq.(K37) may be presented as

Pc(x) − Pci(x) = − d dx [ Jc(x) − Jci ] ,

(K39)

where J_ci(x) = ∫_x^∞dξ P_ci(ξ). Using Eqs.(K38) and (K39), we can find the first integral of Eq.(K37):

Dv C[PT] Pcf (x)   d dx   [ Pc(x) Pcf (x) ] = Jci(x) − Jc(x) .

(K40)

Integration of Eq.(K40) leads to an integral equation for the function P_c(x):

Pc(x) = Pcf (x) [ 1 + v DC[PT]   ⌠ ⌡ x 0    Jci(ξ) − Jc(ξ) Pcf (ξ)  dξ ] .

(K41)

The limit D_v≡ D/v → 0 may be studied with the help of Eq.(K37) by substituting P_c(x) ≈ P_ci(x) into its left-hand side. For the function (K9), this approach gives

Delay in contact formation

Finally, let us take into account the delay in contact formation following the work of Schallamach. Let τ be the delay time, N  be the total number of contacts, N_c be the number of coupled (pinned) contacts, and N_f = N−N_c be the number of detached (sliding) contacts. The fraction of contacts that detach per unit displacement of the sliding block is Γ(v,T) = ∫dx P(x) Q(x), i.e., when the slider shifts by ΔX, the number of detached contacts changes by N_cΓ ΔX, so that N_f = ΓvτN_c. Using N_c+N_f = N, we obtain N_c = N / (1 + Γvτ) and N_f = N Γvτ/ (1 + Γvτ). If we define x = 1/Γ and v_d = x/τ, we can write

The coupled contacts produce the force  f_k defined above by the steady-state solution of the master equation. The combined dependence which incorporates temperature effects, aging and delay in contact formation, is shown in Fig.K6 for different values of the parameter v_d.

However, above we assumed that the sliding contacts experience zero friction, while these contacts may experience a viscous friction force f_l = η_lv, where η_l corresponds to the (bulk) viscosity of the liquid lubricant. In this case the kinetic friction should be additionally multiplied by a factor β(v) = 1 + v²/v_hv_d , where v_h= f_k /η_l  (v_h>> v_d). Such a correction may be expected at huge velocities only, e.g., for v ~ 1 m/s. In this case the function F_k(v), after decreasing, reaches a minimum at a velocity v₀ ≈ (v_hv_d)^1/2, and then increases according to a law  f_k(v) ∝η_lv. Note that the viscous friction which comes from the excitation of phonons in the substrates, as shown in MD simulation, may also depend on the velocity, e.g., as η_l ∝ v⁴.

Making link with experiments

The contact parameters k and ω may be estimated with the help of elastic theory. Let us assume that a contact has a cylinder shape of height h (the thickness of the interface) and radius r_c, so that it is characterized by the section A_i= πr_c², the (geometrical) inertial momentum I = πr_c⁴/4, a mass density ρ and a Young modulus E. If the cylinder foot is fixed and a force Δf is applied to its top, the latter will be shifted on the distance Δx = Δf h³/3EI (the problem of bending pivot, see Sec.20, example 3 in [LL]). Thus, the effective elastic constant of the contact is k = Δf/Δx = 3EI/h³. The minimal frequency of bending vibration of the pivot with one fixed end and one free end, is given by ω ≈ (3.52/h²) (EI/ρA_i)^1/2 (see Sec.25, example 6 in [LL]).

Let us consider a contact of two rough surfaces and assume that a = h = r_c. Then we obtain

ω ≈ 1.76 √ E/ρ rc

(K49)

for the attempt frequency,

k = (3π/4)Erc

(K50)

for the contact elasticity, and

x* = rc2σ*/k

(K51)

for the threshold distance. For steel substrates we may take ρ = 10⁴ kg/m³ for the mass density, E = 2×10¹¹ N/m² for the Young modulus, and σ_c = 10⁹ N/m² for the plasticity threshold. Assuming that σ_*= σ_c and r_c ≈ 1 μm, we find that ω ≈ 7.9×10⁹ s^–1, k ≈ 4.7×10⁵ N/m, x_* ≈ 2.1×10^–9 m, b ≈ 2.7×10⁸ for room temperature (i.e., b >> 1), so that the crossover velocity is quite low, v_*≈ 0.03 μm/s.

Now let us consider a lubricated system, e.g., the one with a few OMCTS layers as studied by Klein [J. Klein, Phys.Rev.Lett. 98, 056101 (2007)]  and Bureau [L. Bureau, Phys.Rev.Lett. 104, 218302 (2010)], and assume that the lubricant consists of solidified islands which melt under stress as proposed by Persson. In this case, instead of using the Young modulus, let us assume that x_*= r_c; this allows us to find the parameter EI = ah³σ_*/3γ_c. Then, the elastic constant is k = aσ_*/γ_c, the attempt frequency is

ω ≈ 1.15 √ σ*/ahργc3 ,

(K52)

the parameter b is given by

b ≈ γca3σ* /2kBT ,

(K53)

and in the case of b >> 1 the crossover velocity is

v* ≈ γcωa /3C0b ≈ kBT / √ a5hρs*γc3 .

(K54)

For a four-layer OMCTS film one may take ρ = 956 kg/m³, h ≈ 3.5×10^–9 m, F_*≈ 2×10^–5 N and A ≈ 10^–10 m² so that σ_*≈ 2×10⁵ Pa. Assuming γ_c= 0.5 and a ≈ 1 μm, we obtain for room temperature, k_BT = 4×10^–21 J, that ω ≈ 8×10⁸ s^-1 and b ≈ 1.25×10⁷, i.e. this system is in the low-temperature limit too, although the crossover velocity is much higher than for rough surfaces, v_* ≈ 16 μm/s.

Figure K7: The shear stress σ as a function of the shear rate dγ/dt for different values of the OMCTS film thickness from nl=2 to 6 monolayers.   Inset: the crossover velocity v* as a function of the number of layers.

Moreover, we may calculate the dependence f_k(v) for different thicknesses of the lubricant film. If the film consists of n_l layers, then the film thickness is h = n_ld, where d ≈ 8.75 Å  is the diameter of the OMCTS molecule. Let us assume that the maximal shear stress exponentially decreases with the number of layers according to the results of MD simulation, σ_*= σ₀ e^–βn_l , where β ~ 1 is a numerical constant. Taking σ₀= 4×10⁶ N/m² and β = 1.5, we obtain the dependences of the shear stress σ = F_k/A on the shear rate  dγ/dt = v/h  shown in Fig.K7, which may be compared with the experimental dependences (Fig.2a) of Bureau.

Note that our approach may overestimate the value of the crossover velocity v_*. First, the crossover will occur earlier if the delay and/or aging effects play a significant role. Besides, at low temperatures the stiff contacts lead to higher "viscosity" and lower values of v_* than the soft contacts considered above (see Fig.K2). Second, we completely ignored the elastic interaction between the contacts. If the latter would be incorporated, a breaking of one contact may stimulate neighboring contacts to break as well, i.e., the value of the parameter a should describe such a cooperative "contact" size which may be much larger than those of individual ones (see ).

Giving a quantitative evaluation of the influence of aging on the velocity dependence of the friction coefficient is harder than for the temperature dependence due to insufficient experimental data. Aging appears to cause a decrease of friction as velocity increases, and thus, when such a behavior is observed experimentally, it can be considered as a strong indication of the presence of aging. Our analysis indicates that the combined effect of temperature and aging leads to a maximum in the friction coefficient versus velocity. Therefore, when aging is manifested by a decreasing friction versus velocity, extending the experiments to lower velocities and temperatures might detect the maximum and thus provide some quantitative data to evaluate the aging parameters.

Although the aim of our work was to find the dependence of the kinetic friction on the driving velocity, our approach allows us to find the dependence on temperature as well. However, the behavior of a real tribological system is more involved, because all parameters may depend on temperature T in a general case. For example, the delay time τ may exponentially depend on T if the formation of a new contact is an activated process; the same may be true for the aging rate D. In this case one may obtain a nonmonotonic temperature dependence of friction with, e.g., a peak at cryogenic temperatures.

Conclusion

We emphasize that our ME approach is only valid for a system with many contacts, for example, N > 20 at least. When the contact is due to a single atom as it may occur in the AFM/FFM devices, the friction can be accurately described by the Prandtl-Tomlinson model and should follow the logarithmic  f_k(v) dependence,  f_k (v) ∝ ( lnv/v₀ )^2/3. But if the AFM/FFM tip is not too sharp so that the contact is due to more than one atom, the logarithmic dependence is only approximate.

In this work we had in mind that contacts correspond to real asperities in the case of the contact of rough surfaces or to "solid islands" for the lubricated interface. However, the ME approach also operates when the contact is due to long molecules which are attached by their ends to both substrates. Such a system was first studied by Schallamach and then further investigated by several authors. Note that when all molecules are identical, they are characterized by the same static threshold, i.e., this system is close to the singular one, where the logarithmic f_k(v) dependence has to have a wide interval of operation.

Finally, let us discuss restrictions of our approach. First of all, we assumed the somehow idealized case of wearless friction; wearing may mask the predicted dependences. Besides, the interface is heated during sliding; this effect is hard to describe analytically as well as to control experimentally. Then, we did not estimated the delay/aging parameters; moreover, these parameters, e.g., the delay time τ, may depend on the driving velocity v. Besides, we assumed the simplest mechanism of aging described by the Smoluchowsky equation, while the real situation may be more involved, e.g., it may correspond to the Lifshitz-Slözov mechanism. Also, we assumed that the reformed contacts appear in the unstressed state, R(x) = δ(x) in Eq.(K1), which may not be the case in real systems. The most important issue, however, is the incorporation of the elastic interaction between the contacts as well as elastic deformation of substrates at sliding (see ).

Published in: O.M. Braun and M. Peyrard, Phys. Rev. E 83 (2011) 046129 "Dependence of kinetic friction on velocity: Master equation approach"

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Last updated on April 22, 2014 by O.Braun.  Copyright © by O.Braun.  Translated from L^AT_EX by T_TH