Prandtl Tomlinson model

Figure 2 Illustration of an instability in the Prandtl-Tomlinson model. The sum of the substrate potential and the elastic energy of the spring is shown at various instances in time. The energy difference between the initial and the final point of the thick line will be the dissipated energy when temperature and sliding velocities are very small.

In the example shown in Figure 5, c is positive and the exponent y is unity however, neither of these statements are universal. For example, the Prandtl-Tomlinson model can best be described with y = 2/3 in certain regimes,26 27 whereas confined boundary lubricants are best fit with y = l.25 28 Moreover, the constant c can become negative, in particular when junction growth is important, where the local contact areas can grow with time as a result of slow plastic flow of the opposed solids or the presence of adhesive interactions that are mediated by water capillaries.29,30... 

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...

In this section, we give a brief overview of theoretical methods used to perform tribological simulations. We restrict the discussion to methods that are based on an atomic-level description of the system. We begin by discussing generic models, such as the Prandtl-Tomlinson model. Below we explore the use of force fields in MD simulations. Then we discuss the use of quantum chemical methods in tribological simulations. Finally, we briefly discuss multiscale methods that incorporate multiple levels of theory into a single calculation. 

The number of free parameters that define the athermal Prandtl Tomlinson model can be reduced to three by a convenient choice of units, b can be used to define the unit of the length scale, fob is the unit of the energy scale, and... 

Figure 8. Schematic representation of the time evolution of the potential energy in the Prandtl Tomlinson model (dashed lines) see Eq. (21). All curves are equidistant in time, separated by a time interval At. The circles denote mechanically stable positions, and the solid line indicates the motion of an overdamped point particle from left to right. Motion is linearly unstable on the thick portion of the line.

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... 

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. 

The top plate is connected to a laterally driven spring, of spring constant Xj, and to a spring X that is used to control the motion in the normal direction. f/(X, Z) is the effective potential experienced by the plate due to the presence of the embedded system, b is its periodicity, and ct characterizes the corrugation of the potential in the lateral direction. The parameters ri and are responsible for the dissipation of the plate kinetic energy due to the motion in the lateral and normal directions. In contrast to the traditional Prandtl Tomlinson model, here the dependence of U and ri on the distance Z between plates is taken into account. The detailed distance dependence is determined by the nature of the interaction between the plate and embedded system. As an example, we assume an exponential decrease of U and ri with a rate A as Z increases. The possibility of an external modulation of the normal load L (t) = X [Zp(f) — Z] is taken into account by introducing a time dependence into the position of the normal stage, Zp(f). 

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. 

Let us apply the Prandtl-Tomlinson model to the example of an AFM tip sliding along a one-dimensional periodic potential V x) with periodicity a (see Figure 9.15) ... 

Figure 9.15 Energyfor a sliding AFM tip in the one-dimensional Prandtl-Tomlinson model versus position. The position is given in units of the periodic surface potential (a = 0.4 run). Parameters were Vq = 0.5 eV and K = 1.5 N m The energy is plotted for a support moving to the right with a speed Vo = 20 nm s (i.e., xo = Vot is assumed)...

A fundamental question not yet resolved regarding stick-slip motion is the exact mechanism for the occurrence of stick-slip with a periodicity of the surface lattice. For interpretation of the stick-slip motion by the Prandtl-Tomlinson model, the tip is commonly treated as a single, pointlike entity without any additional internal degree of freedom. However, the contact area between AFM tip and crystal surface will typically contain some 10 unit cells. Atomic stick-slip was observed even for amorphous silicon and silicon nitride tips [1003] and for crystalline tips, atoms of tip and surface lattice will usually not be in registry, making the observation of stick-slip with a periodicity of the surface lattice surprising. 

Most theories of structural superlubricity are based on the Prandtl-Tomlinson model or the more advanced Frenkel-Kontorova model [1043, 1044], in which the single atom/tip is replaced by a chain of atoms coupled by springs. However, Friedel and de Gennes [1045] noted recently that correct description of relative sliding of crystalline surfaces should include the motion and interaction of dislocations at the surfaces. This concept was taken up by Merkle and Marks [1045] and generalized using the well-established coincident site lattice theory and dislocation drag from solid-state physics. 

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