Friction on Mesoscopic Scale

Distribution of thresholds  

Let us discuss now physical origins of the distribution of threshold stretchings P_c(x). First of all, it can be coupled with the distribution P_cf (f_s) of (static) friction force thresholds of the contacts. If a given contact has an area A, then it is characterized by the force threshold  f_s∝A and the elastic constant k ~ ρc²√A. The displacement threshold for the given contact is x_s= f_s/k, so that f_s∝ x_s², or df_s/dx_s∝ x_s. Thus, using the relationship P_c(x_s)dx_s=P_cf (f_s)df_s, we obtain

Rough surfaces

For a contact of two hard rough surfaces (the multi-contact interface, or MCI in notations due to Baumberger and Caroli), the problem reduces to the statistics of asperities. Let the rough surface is characterized by hills of heights {h_i} distributed with a probability P_h(h).  Following the Greenwood and Williamson model of the interface, let us assume that all hills have the spherical shape of the same radius of curvature r. When this surface is pressed with another rigid flat surface, which takes a position at the level h₀, then the hills of heights h > h₀ will form contacts, or asperities. If the contacts are elastic (the Hertz contacts), then the contact of height h has the compression  (h−h₀), its area is πr(h−h₀), and it bears the normal force  f_l(h) ≈ (4π/3)E^*r^1/2(h−h₀)^3/2,  where E^* is the effective Young modulus (if both substrates have the same Young modulus E, then E^*= E/2(1−ν²), where ν is the Poisson modulus). If we assume that the shear static threshold for the given contact is proportional to the load force,  f_s(h) ≈ μf_l(h), or

where μ < 1 is a constant, then the distribution of static thresholds P_cf (f_s) can be coupled with the distribution of asperities heights P_h(h) with the help of relation P_cf (f_s)df_s ∝ P_h(h)dh, or P_cf (f_s) ∝ (dh/df_s)P_h[h(f_s)], where dh/df_s ∝ f_s^–1/3 according to Eq.(33).

For a strong load, when the local stress exceeds the yield threshold Y, the contacts begin to deform plastically. When all contacts are plastic, the local pressure on contacts is  p_load= H, where H is the hardness [H ≈ 3Y for the spherical geometry of asperities; for metals H ≈ (10^–3−10^–2)E ~ 10⁹ Pa]. Then the normal force at the contact is  f_l (h) ≈ πr(h−h₀)H. Assuming again that f_s (h) ≈ μf_l(h), we obtain

so that P_cf (f_s) ∝ P_h[h(f_s)]. For example, if the distribution of heights is exponential,  P_h(h) = h₀^–1exp(−h/h₀) Θ(h),  where h₀ is the average height, then for the elastic contacts we obtain (f, x > 0)

while for the plastic contacts

The distributions P_c(x) for the elastic and plastic contacts are presented in Fig.14. The force F(X) (almost) monotonically increases with X, approaching the kinetic value F_k. Thus, in the case of contact of hard rough surfaces, a relatively large concentration of low-threshold contacts prevents from appearance of the stick-slip motion even for a very soft driving spring (however, the elastic interaction between the contacts may strongly increase the probability thr stick-slip to appear, see ). The kinetic friction force F_k depends on the parameter B according to the power law (see Fig.15), F_k ∝ B^–3/4 ∝ h₀^3/4  for elastic contacts, and F_k ∝ B^–1/2 ∝ h₀^1/2 for plastic contacts.

Note: Typical values for rough surfaces are of the order  r ~ 10 − 100 μm,  h₀ ~ 1 μm, so that the average size of the contact is  a ≈ rh₀^1/2 ~ 3 − 10 μm. Are the contacts in the plastic or elastic regime, depends on the dimensionless parameter Ψ = (E/Y)(h₀/r)^1/2: the former case operates for Ψ >> 1 (as is typical for metals), while the latter, for Ψ < 1 (it corresponds, e.g., to the case of rubber friction); the polymeric glasses belong to an intermediate situation, Ψ > 1, where only a fraction of contacts is plastic.

Dry contact

Next, let us consider the dry contact of two flat surfaces. In an exotic case when both surfaces have the ideal crystalline structure, we come to the singular case of delta-function distribution P_c(f). Such a situation, however, is exceptional. A real surface always consists of domains, which are characterized by different crystalline orientation and even, may be, different structure. MD simulations show a large variation of the friction with relative orientation of the two bare substrates. In order to estimate a shape of the function P_c(x) emerging due to domain structure of substrates, let us consider a simple model. Let a domain of the top rigid substrate be crystalline (e.g., of triangular symmetry) with the lattice constant a and consists of N atoms, while the bottom substrate be also rigid crystalline (e.g., with square symmetry) so that it produces a periodic potential for the motion of the top substrate. For some values of a and N, the activation energy for domain displacement is high, while for other values, the activation energy is small or even may vanish. Besides, if the domain is rotated on the misfit angle φ, the activation energy should achieve minima at some angles.

For a fixed misfit angle φ, one can calculate the total potential energy of the domain U(X,Y), where X and Y are the center of mass coordinates of the domain. The extrema of U(X,Y) are determined by the equations ∂U/∂X = 0, ∂U/∂Y = 0. An extremum may correspond either to a minimum U_m or to a saddle point U_s; then the activation energy is given by ε_a=U_s−U_m. Assuming that  f_s ~ ε_a/a ∝ ε_a ,  we can find the threshold force f_s as a function of the misfit angle φ and the domain size N. Then, calculating a histogram of the function ε_a(φ),  we obtain the distribution P_cf (f_s) if all domains have the same size N and all angles are equally presented. Averaging it over different domain sizes N, e.g., with a weight function w(N) = e^–N/‹N›,  where ‹N› is the average domain size, we obtain the distribution P_c(x) shown in Fig.16 (note that it is similar to that of Fig.14 characteristic for rough surfaces).

Thus, in this system one also has to expect smooth sliding generally. However, in the estimation we assumed that the domains are rigid, while real substrates are deformable. Also, we supposed that all angles are equally presented, while some angles should have preference due to their lower potential energy. Both these factors should lead to the increase of the threshold values.

Lubricated surfaces: Lifshitz-Slözov coalescence

The dry-friction system considered above, is also exceptional. In a real system, almost always there is a lubricant between the sliding surfaces (called "the third bodies" by tribologists) – either a specially chosen lubricant film, or a grease (oil), or dust, or wear debris produced by sliding, or water or/and a thin layer of hydrocarbons adsorbed from air, etc.

In the conventional melting-freezing mechanism of friction, the lubricant is melted during slip, and solidifies when the motion stops. The solidification process can be described by the Lifshitz-Slözov theory. At the beginning, grains of the solid phase emerge within the liquid lubricant film. Then the grains grow in size according to the law  ‹r›∝t^1/3. The distribution of grains sizes is described as follows: the number of grains with the radius from r to r+Δr is equal to P_LS(r/‹r›)Δr/‹r›, where the function P_LS(u) is zero for u ≥ 3/2  (i.e., the maximum size of the grains is 1.5‹r›),  while for lower sizes it is given by

Due to coalescence of grains, the total number of grains decreases with time as N(t) ∝ t^–1.  When the size of a grain exceeds the film thickness h (that will occur when ‹r›(t) ≥ h/3),  such grains will pin the surfaces. Using the relationship P_cf (f)df ∝ P_LS(r/‹r›)dr/‹r›, we obtain that P_cf (f) ∝ (dr/df)P_LS(r/‹r›)/‹r›.  Because one grain gives the static threshold proportional to the contacting area,  f_s ∝ π(r²−h²/4) for r > h/2, we have dr/df ∝ f ^–1/2. Combining all things together, we obtain that

where ρ(t) = 2‹r›(t)/h = αt^1/3 (the pinning begins when ρ > 2/3),  and B is determined by the system parameters (elasticity of the contacts, proportionality between the threshold  f_s and the contact area for a contact/domain/grain, and the thickness of the lubricant film). The distribution (38) is shown in Fig.17; it should lead to the conventional tribological behavior: the stick-slip motion at low driving velocity and smooth sliding at high velocities.

In a general case, we also have to take into account that the lubricant film may consist of grains (domains) of different orientation or even different structure. Indeed, the proportionality coefficient in the relation  f_s ∝ π(r²−h²/4)  used above, should depend on the misfit angle between the lubricant domain and the substrate, so that the distribution P_LS(u) introduced above, additionally should depend on the misfit angle φ, P_LS(u; φ).  Thus, if there are grains with different orientation, distributed according to a function  R_f(φ), then

Next: Kinetic friction

Back to main page or tribology page

Last updated on April 19, 2014 by O.Braun.  Translated from L^AT_EX by T_TH