Friction scale

Unlike the deformation models where the friction scales with the apparent contact area, the adhesion models assume that the friction is proportional to the real area of contact. The direct measurement of the real area of contact is generally impracticable and it is normal to resort to analytical models or computer simulations.Surfaces are invariably rough and the type and scale of the surface topography as well as the deformation characteristics of the solids control the real contact area. Some of the topographical factors were introduced in Section 23.3.1.2 for elastic bodies. At the simplest level the contact area for elastic contacts can be considered as arising from the contact of many spherical asperities of radius with a plane. For one asperity the contact area A is given by ... 

We have not analysed this system variable in detail but it is notable that the static friction scales quite accurately as the reciprocal of the square root of the tension. The data of Figure 8 have been plotted in this form to demonstrate the relationship. The fundamental resonance frequency v of a string under tension is given by -... 

It is interesting to note that, as expected, coupling stiffness (Fig. 9.27) varies mainly with the axial force and exhibits a work-hardening behavior. Also, the friction scaling (Fig. 9.30) was found to be very close to one which shows that the initial estimate of the value of was quite accurate. 

Fig. 9.30 Variation of friction scaling, with gearbox output velocity and axial force...

Friction can now be probed at the atomic scale by means of atomic force microscopy (AFM) (see Section VIII-2) and the surface forces apparatus (see Section VI-4) these approaches are leading to new interpretations of friction [1,1 a,lb]. The subject of friction and its related aspects are known as tribology, the study of surfaces in relative motion, from the Greek root tribos meaning mbbing. 

The surface forces apparatus of crossed mica cylinders (Section VI-4D) has provided a unique measurement of friction on molecular scales. The apparatus is depicted in Fig. VI-3, and the first experiments involved imposing a variation or pulsing in the sepa-... 

C. M. Mate, Atomic Scale Friction, in Handbook of MicrolNano Tribology, B. Bhushan, ed., CRC Press, 1995. 

Thus the average velocity decays exponentially to zero on a time scale detennined by the friction coefficient and the mass of the particle. This average behaviour is not very interesting, because it corresponds to tlie average of a quantity that may take values in all directions, due to the noise and friction, and so the decay of the average value tells us little about the details of the motion of the Brownian particle. A more interesting... 

Lateral force microscopy (LFM) has provided a new tool for the investigation of tribological (friction and wear) phenomena on a nanometre scale [110]. Alternatively known as friction force microscopy (FFM), this variant of AFM focuses on the lateral forces experienced by the tip as it traverses the sample surface, which... 

Mate C M, Erlandsson R, McClelland G M and Chiang S 1987 Atomic-scale friction of a tungsten tip on a graphite surface Phys. Rev. Lett. 59 1942... 

Erlandsson R, Hadzioannou G, Mate M, McClelland G and Chiang S 1988 Atomic scale friction between the muscovite mica cleavage plane and a tungsten tip J. Chem. Phys. 89 5190... 

Flu J, Xiao X-D, Ogletree D F and Salmeron M 1995 Atomic scale friction and wear of mica Surf. Sc/. 327 358... 

A valuable reference to anyone involved with friction at small scales. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

We assume that the unbinding reaction takes place on a time scale long ( ompared to the relaxation times of all other degrees of freedom of the system, so that the friction coefficient can be considered independent of time. This condition is difficult to satisfy on the time scales achievable in MD simulations. It is, however, the most favorable case for the reconstruction of energy landscapes without the assumption of thermodynamic reversibility, which is central in the majority of established methods for calculating free energies from simulations (McCammon and Harvey, 1987 Elber, 1996) (for applications and discussion of free energy calculation methods see also the chapters by Helms and McCammon, Hermans et al., and Mark et al. in this volume). 

When the friction coefficient is set to zero, HyperChem performs regular molecular dynamics, and one should use a time step that is appropriate for a molecular dynamics run. With larger values of the friction coefficient, larger time steps can be used. This is because the solution to the Langevin equation in effect separates the motions of the atoms into two time scales the short-time (fast) motions, like bond stretches, which are approximated, and longtime (slow) motions, such as torsional motions, which are accurately evaluated. As one increases the friction coefficient, the short-time motions become more approximate, and thus it is less important to have a small timestep. 

Dry nitrocellulose, which bums rapidly and furiously, may detonate if present in large quantities or if confined. Nitrocellulose is a dangerous material to handle in the dry state because of sensitivity to friction, static electricity, impact, and heat. Nitrocellulose is always shipped wet with water or alcohol. The higher the nitrogen content the more sensitive it tends to be. Even nitrocellulose having 40% water detonates if confined and sufftcientiy activated. AH large-scale processes use nitric—sulfuric acid mixtures for nitration (127—132). 

Ap Frictional pressure drop Pa CO Characteristic frequency or reciprocal time scale of flow 1/s... 

In a simulation it is not convenient to work with fluctuating time intervals. The real-variable formulation is therefore recommended. Hoover [26] showed that the equations derived by Nose can be further simplified. He derived a slightly different set of equations that dispense with the time-scaling parameter s. To simplify the equations, we can introduce the thermodynamic friction coefficient, = pJQ. The equations of motion then become... 

If we scale time as t = xr, then the frst term in (5.52) decreases as l/>/, while the other two are independent of friction. Therefore, at large rj the second derivative term in (5.52), as well as the kinetic energy term in the action, can be neglected, and the entire effect of friction is to change the timescale. That is, the solution to (5.52) is Q x) = Q x/ri) where Q is a function independent of rj. The instanton velocity is scaled as Q cc and the action (5.38) grows linearly with r, ... 

The instanton action behaves in accord with the scaling predictions and is independent of coq. Loosely speaking, the frequency ojq is replaced by the friction coefficient rj. Grabert et al. [1984b] have studied the energy loss A tunneling process and found that is... 

---