Tomlinson model

It has been found from MD simulations that friction of SAMs on diamond decreases with the increasing chain length of hydrocarbon molecules, but it remains relatively constant when the number of carbon atoms in the molecule chain exceeds a certain threshold [44], which confirmed the experimental observations. In simulations of sliding friction of L-B films, Glosli and McClelland [45] identified two different mechanisms of energy dissipation, namely, the viscous mechanism, similar to that in viscous liquid under shear, and the plucking mechanism related to the system instability that transfers the mechanical energy into heat, similar to that proposed in the Tomlinson model (see Chapter 9). On the basis of a series work of simulations performed in the similar... 

The example demonstrates that the instability and consequent energy dissipation, similar to those in the Tomlinson model, do exist in a real molecule system. Keep in mind, however, that it is observed only in a commensurate system in which the lattice constants of two monolayers are in a ratio of rational value. For incommensurate sliding, the situation is totally different. Results shown in Fig. 21(b) were obtained under the same conditions as those in Fig. 21 (a), but from an incommensurate system. The lateral force and tilt angle in Fig. 21(b) fluctuate randomly and no stick-slip motion is observed. In addition, the average lateral force is found much smaller, about one-fifth of the commensurate one. 

Tomlinson model that results in a finite kinetic friction. 

The FK model accounts for the effects that have been ignored in the Tomlinson model, resulting from the interactions of neighboring atoms. For a more realistic friction model of solid bodies in relative sliding, the particles in the harmonic chain have to be connected to a substrate. This motivates the idea of combining the two models into a new system, as schematically shown in Fig. 24, which is known as the Frenkel-Kontorova-Tomlinson model. Static and dynamic behavior of the combined system can be studied through a similar approach presented in this section. 

Weiss, M. and Elmer, F. J., "A Simple Model for Wearless Friction The Frenkel-Kontorova-Tomlinson Model, Physics of Friction, B. N. J. Persson and E. Tosatti, Eds., Kluwer Academic Publishers, 1996,pp. 163-178. 

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. 

Example 11.2. In Fig. 11.9 the experimental results from a friction force microscope experiment is compared to simulations based on an extended two-dimensional Tomlinson model [481], The tip was assumed to be connected elastically to the holder (coordinates (xo, yo) that is scanned with the velocity v relative to the sample surface. The path (x(t),y(t)) of the tip was calculated using effective masses m. , my, spring constants Kx, Ky, and damping constants jx, 7y. The equation of motion for this system is ... 

Figure 11.9 Friction force microscope pictures (a, b) of a graphite(OOOl) surface as obtained experimentally with FFM and results of simulations (c, d) of the stick-slip friction using a two-dimensional equivalent of the Tomlinson model. The friction force parallel to the scan direction (a, c) and the lateral force perpendicular to the scan direction (b, d) are shown. The scan size is 20 Ax 20 A. Pictures taken from Ref. [481] with kind permission from R. Wiesendanger.

The Prandtl Tomlinson (PT) model [17] (usually only referred to as the Tomlinson model) is the simplest model that allows for elastic instability and hence for pinning between two incommensurate solids. In its original version (see Fig. 7), atoms in the upper wall are coupled harmonically to their ideal... 

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. 

Figure 7. Schematic representation of the one dimensional Frenkel Kontorova Tomlinson model, a and b denote the lattice constant of the upper sohd and the substrate, respectively. The substrate is considered rigid, and its center of mass is kept fixed. In the shder, each atom is coupled with a spring of lateral stiffness to its ideal lattice site and with a spring of stiffness 2 to its neighbor. The PT model is obtained for 2 0, while the Frenkel Kontorova model corresponds to k = 0. We will drop the subscripts for these two cases since a single spring is relevant.

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... 

This result describes well the numerical calculations according using a modified Tomlinson model [88] and, moreover, is in a good agreement with experiment... 

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. 

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