Prandtl Tomlinson friction model

Making assumptions regarding the dissipation of heat can also influence solid friction, although typically it is less of an issue. This can be explored most easily within the Prandtl-Tomlinson model however, the lessons to be learned... 

Figure 10 Friction velocity relationship Fk( o) in the Prandtl-Tomlinson model at...

Figure 9. Average kinetic friction F (independent of a) in the athermal Prandtl Tomlinson model at low velocities v for two different spring strengths k and various damping coefficients 7. The symbols at r o = 0 indicate the static friction force for k = 0.1k. All units are reduced units.

The first discussion of the effect of thermal fluctuations on friction forces in the Prandtl Tomlinson model was given by Prandtl in 1928 [18]. He considered a mass point attached to a single spring in a situation where the spring fei in Fig. 7 was compliant enough to exhibit elastic instabilities, but yet sufficiently strong to allow at most two mechanically stable positions see also Fig. 8, in which this scenario is shown. Prandtl argued that at finite temperatures, the atom... 

In the following, we focus on the application of the Prandtl Tomlinson model to the interpretation of AFM experiments. As mentioned in Chapter in.A.3, the potential bias is continuously ramped up as the support of an AFM tip is moved. This results in a different friction velocity relationship... 

For Vo below the second threshold denoted by Vq, the kinetic friction is zero in the limit of quasi-static sliding that is, for sliding velocity v Q. That is, for Vo < Vq" the kinetic friction behaves like a viscous drag. For Vo > the dynamics is determined by the Prandtl Tomlinson-like mechanism of elastic instability, which leads to a finite kinetic friction. The threshold amplitude Vq increases with k and is always larger than zero. Therefore, in the commensurate case, vanishing kinetic friction does not imply vanishing static friction just like in the PT model. The FKT model for Vj, < Vo < is an example of a dry-friction system that behaves dynamically like a viscous fluid under shear even though the static friction is not zero. 

The transition from zero to finite friction with increasing load for small tips can be understood from the Prandtl Tomlinson model. The control parameter k decreases with load because the interaction between surfaces is increased and the internal stiffness of the solid and tip is relatively unchanged. The pinning potential is an edge effect that grows more slowly than the area (Section II). Thus the transition to finite friction occurs at larger loads as the area increases. Tips that were only 5 atoms in diameter could exhibit friction at very small loads. However, for some starting positions of the tip, no friction was observed even at 7.3 GPa. When the diameter was increased to 19 atoms, no friction was observed for any position or load considered. 

Over a wide range of system parameters the dilatancy is smaller than the characteristic length A. Under this condition the generalized Prandtl Tomlinson model predicts a linear increase of the static friction with the normal load, which is in agreement with Amontons s law. It should be noted that, in contrast to the multi-asperity surfaces discussed in Section VII, here the contact area is independent of the load. The fulfillment of Amontons s law in the present model results from the enhancement of the potential corrugation, a2C7oexp(l — Z/K), experienced by the driven plate with an increase of the normal load. 

A paper by Prandtl [18] on the kinetic theory of solid bodies, which was published in 1928, one year prior to Tomlinson s paper [17], never achieved the recognition in the tribology community that it deserves. PrandtI s model is similar to the Tomlinson model and likewise focused on elastic hysteresis effects within the bulk. Nevertheless, Prandtl did emphasize the relevance of his work to dry friction between solid bodies. In particular, he formulated the condition that can be considered the Holy Grail of dry, elastic friction If the elastic coupling of the mass points is chosen such that at every instance of time a fraction of the mass points possesses several stable equilibrium positions, then the system shows hysteresis. In the context of friction, hysteresis translates to finite static friction or to a finite kinetic friction that does not vanish in the limit of small sliding velocities. Note that the dissipative term that is introduced ad hoc in Eq. (19) does vanish linearly with small velocities. 

---