Velocity, kinetic friction

At finite velocity kinetic friction behaves quite differently in the sense that the commensurability plays a less significant role. Besides, the system shows rich dynamic properties since Eq (16) may lead to periodic, quasi-periodic, or chaotic solutions, depending on damping coefficient y and interaction strength h. Based on numerical results of an incommensurate case [18,19], we outline a force curve of F in Fig. 23 asafunction ofv, in hopes of gaining a better understanding of dynamic behavior in the F-K model. 

STATISTICAL MECHANICS OF STATIC AND LOW-VELOCITY KINETIC FRICTION... 

Additional information conveyed by Fig. 4 is that at finite temperatures, the mobility remains finite, although it may become exponentially small as T tends to zero. Hence strictly speaking, the zero-velocity kinetic friction Fk is always zero as long as temperamre T is finite thus... 

Friction is the resistance against change in the relative positions of two bodies touching one another. If the area of contact is a plane, the relative motion will be a sliding one and the resistance will be called sliding or kinetic friction. If the material in the area of contact is loaded beyond its strength, abrasion or wear will take place. Both phenomena are affected by numerous factors such as the load, relative velocity, temperature, and type material. 

The dWi are Gaussian white noise processes, and their strength a is related to the kinetic friction y through the fluctuation-dissipation relation.72 When deriving integrators for these methods, one has to be careful to take into account the special character of the random forces employed in these simulations.73 A variant of the velocity Verlet method, including a stochastic dynamics treatment of constraints, can be found in Ref. 74. The stochastic... 

Once temperature comes into play, the jumps of atoms between minima may be invoked prematurely, i.e., before the formation of instabilities, via thermal fluctuations. These thermally activated jumps decrease the force that is required to pull the surface atom, which leads to a decrease in the kinetic friction. The probability that a jump will be thermally activated is exponentially related to the energetic barrier of the associated process, which can be understood in terms of Eyring theory. In general, the energetic barriers are lower when the system is not at its thermal equilibrium position, which is a scenario that is more prominent at higher sliding velocities. Overall, this renders Fk rate or velocity dependent, typically in the following form ... 

Figure 5 Typical velocity relationship of kinetic friction for a sliding contact in which friction is from adsorbed layers confined between two incommensurate walls. The kinetic friction F is normalized by the static friction Fs. At extremely small velocities v, the confined layer is close to thermal equilibrium and, consequently, F is linear in v, as to be expected from linear response theory. In an intermediate velocity regime, the velocity dependence of F is logarithmic. Instabilities or pops of the atoms can be thermally activated. At large velocities, the surface moves too quickly for thermal effects to play a role. Time-temperature superposition could be applied. All data were scaled to one reference temperature. Reprinted with permission from Ref. 25. 

In their study, Park et al.100 investigated the frictional properties of fluorine-terminated alkanethiol SAMs grafted to gold surfaces. The frictional properties of the system were investigated by sliding two SAMs past one another at velocities in the stick-slip regime under various external loads. The simulations yield the shear stress as and the kinetic friction coefficient pk can be estimated from the slope of a plot of as versus load, using the relationships contained in Eqs. [4] and [5]. 

Following ozone oxidation, the surface of a PU film was graft-polymerized with DMAA and the coefficient of kinetic friction (pk) for the fully hydrated, grafted films of two different graft densities was determined against a cleaned steel plate in distilled water as a function of the sliding velocity [ 174]. It was found that grafting of PDMAA effectively reduced the frictional force. 

The friction coefficient can be measured in two ways the static friction coefficient Qus) and the dynamic or kinetic friction coefficient (fikX The static friction coefficient is defined as the ratio of the force required to initiate relative movement and the normal force between the surfaces the dynamic or kinetic friction coefficient is defined as the ratio of the friction force to the normal force when the two surfaces are moving relative to each other. For simplicity, much of the research has focused on the dynamic friction coefficients wherein the two surfaces move at a relative constant velocity. Most of the friction studies on skin have dealt with the dynamic friction coefficient and the subscript k is usually dropped. This overview references the dynamic coefficient of friction unless otherwise noted. 

FIG. 22 Variation of the coefficient of kinetic friction (jik) with the sliding velocity for surface-grafted polyurethane films in water. Graft density //xg/cm2 (O) 0, ( ) 140, and (A) 350. 

If the static friction is greater than the kinetic friction, slip-stick motion may be the result. In rigid plastics the kinetic friction coefficient is normally lower than the static coefficient, in elastomers the reverse applies. At high velocities it is sometimes difficult to separate the effects of velocity and temperature. 

Pauchon and Banerjee (1988), in their analysis of bubbly flows, have shown that the kinematic wave velocity based on a constant interfacial friction is weakly stable. They have also obtained a functional dependence of the interfacial friction factor on the void fraction by assuming the kinetic wave velocity equal to the characteristic velocity (kinetic waves are neutrally stable). They have assumed that turbulence provides the stabilizing mechanism through axial dispersion of the void fraction. 

The kinetic friction force is independent of sliding velocity (Coulomb s law). 

Two types of friction are commonly measured and calculated. The static friction Fj is defined as the minimum lateral force needed to initiate sliding of one object over a second, while the kinetic friction Fk v) is the force needed to maintain sliding at a steady velocity v. Observation of static friction implies that the contacting solids have locked into a local free-energy minimum, and Fj represents the force needed to lift them out of this minimum. It is a threshold rather than an actual force acting on the system, and it limits lateral motion in any direction. No work is done by the static friction, since no motion occurs. The kinetic friction is intrinsically related to dissipation mechanisms, and it equals the work done on the system by external forces divided by the distance moved. 

Coulomb contributed what is often called the third law of friction, i.e. that is relatively independent of sliding velocity. The experiments discussed in Section I.D show that the actual dependence is logarithmic in many experimental systems and that often increases with decreasing velocity. Thus there is a fundamental difference between kinetic friction and viscous or drag forces that decrease to zero linearly with v. A nearly constant kinetic friction implies that motion does not become adiabatic even as the center-of-mass velocity decreases to zero, and the system is never in the linear response regime described by the fluctuation dissipation theorem. Why and how this behavior occurs is closely related to the second issue raised above. 

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 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 leading correction to zero-velocity, zero-temperature kinetic friction of the form (7 Inas described in Eq. (28) apparently also applies to more comphcated elastic manifolds. Charitat and Joanny [97] investigated a polymer that was dragged over a surface containing sparsely distributed, trapping sites. They analyzed the competition between the soft elastic, intramolecular interactions, thermal noise, and the tendency of some monomers in the chain... 

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. 

Physisorbed molecules also provide a natural explanation for the logarithmic increase in kinetic friction with sliding velocity that is observed for many materials and represented by the coefficient A in the rate-state model of Eq. (5). Figure 16 shows calculated values of tq and a as a function of log for a sub monolayer of chain molecules between incommensurate surfaces [195]. The value of To becomes independent of v at low velocities. The value of a, which... 

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