Frictional forces

The coefficient of friction /x between two solids is defined as F/W, where F denotes the frictional force and W is the load or force normal to the surfaces, as illustrated in Fig. XII-1. There is a very simple law concerning the coefficient of friction /x, which is amazingly well obeyed. This law, known as Amontons law, states that /x is independent of the apparent area of contact it means that, as shown in the figure, with the same load W the frictional forces will be the same for a small sliding block as for a laige one. A corollary is that /x is independent of load. Thus if IVi = W2, then Fi = F2. 

Fig. XII-8. A schematic friction phase diagram showing the trends found in the friction forces of surfactant monolayers. (From Ref. 53.)...

The radioautographic work suggests another model illustrated in Fig. XII-11. The load is supported over area A, with metal contacts of shear strength s over a portion of the area ctA and film-film contacts of shear strength Sf over the rest of the area. In analogy to Eq. XII-9, one can write the total frictional force, F as... 

Calculate the friction force between the surfactant layers in air in Fig. XII-12 using the relationship in Eq. XII-19. How does this compare with the friction shown in Fig. XII-12 ... 

An alternative approach is to consider ions of charge z e accelerated by the electric field strengtii, E, being subject to a frictional force, Kj, that increases with velocity, v, and is given, for simple spherical ions of... 

Thus, the requirement that the Brownian particle becomes equilibrated with the surrounding fluid fixes the unknown value of, and provides an expression for it in tenns of the friction coefficient, the thennodynamic temperature of the fluid, and the mass of the Brownian particle. Equation (A3.1.63) is the simplest and best known example of a fluctuation-dissipation theorem, obtained by using an equilibrium condition to relate the strengtii of the fluctuations to the frictional forces acting on the particle [22]. 

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... 

Carpick et al [M] used AFM, with a Pt-coated tip on a mica substrate in ultraliigh vacuum, to show that if the defonnation of the substrate and the tip-substrate adhesion are taken into account (the so-called JKR model [175] of elastic adliesive contact), then the frictional force is indeed proportional to the contact area between tip and sample. Flowever, under these smgle-asperity conditions, Amontons law does not hold, since the statistical effect of more asperities coming into play no longer occurs, and the contact area is not simply proportional to the applied load. 

Figure Bl.19.34. Cantilever deflection and corresponding frictional force in the v-direction as a fiinction of sample position as a mica sample is scaimed back and forth under a tungsten tip. (Taken from [124], figure 2.)...

Figure Bl.19.36. Image of the frictional force distribution of a pattern consisting of areas of CH -tenuinated and areas of COOH-tenninated molecules attached to gold-coated silicon. The tip was also fiinctionalized in (a) with CH3 species and in (b) with COOH species. The bright regions correspond to the higher friction force, which in (a) is observed on the CH areas and in (b) on the COOH areas. (Taken from [187], figure 3.)...

Overney R and Meyer E 1993 Tribological investigations using friction force microscopy MRS Bulletin XVIII-5 20... 

The often-cited Amontons law [101. 102] describes friction in tenns of a friction coefiBcient, which is, a priori, a material constant, independent of contact area or dynamic parameters, such as sliding velocity, temperature or load. We know today that all of these parameters can have a significant influence on the magnitude of the measured friction force, especially in thin-film and boundary-lubricated systems. 

Klein J ef a/1994 Reduction of frictional forces between solid surfaces bearing polymer brushes Nature 370 634-7... 

The atomic force microscope (ATM) provides one approach to the measurement of friction in well defined systems. The ATM allows measurement of friction between a surface and a tip with a radius of the order of 5-10 nm figure C2.9.3 a)). It is the tme realization of a single asperity contact with a flat surface which, in its ultimate fonn, would measure friction between a single atom and a surface. The ATM allows friction measurements on surfaces that are well defined in tenns of both composition and stmcture. It is limited by the fact that the characteristics of the tip itself are often poorly understood. It is very difficult to detennine the radius, stmcture and composition of the tip however, these limitations are being resolved. The AFM has already allowed the spatial resolution of friction forces that exlribit atomic periodicity and chemical specificity [3, K), 13]. 

Ultra-high vacuum (UHV) surface science methods allow preparation and characterization of perfectly clean, well ordered surfaces of single crystalline materials. By preparing pairs of such surfaces it is possible to fonn interfaces under highly controlled conditions. Furthennore, thin films of adsorbed species can be produced and characterized using a wide variety of methods. Surface science methods have been coupled with UHV measurements of macroscopic friction forces. Such measurements have demonstrated that adsorbate film thicknesses of a few monolayers are sufficient to lubricate metal surfaces [12, 181. 

Lubricants are added to lower interfacial frictional forces between individual particles and/or between particles and fonning die surfaces to improve compaction and ejection (i.e. extraction of the pressed compact from the fonning die). Individual particle surfaces can be lubricated by an adsorbed film that produces a smoother surface and/or decreases interiDarticle attraction. Fonning (die) surfaces can be lubricated by coating with a film of low-viscosity liquid such as water or oil. 

Dissipation, however, imposes limits on how precisely the potential can be reconstructed. With the introduction of the work performed by the frictional force Wfr, the uncertainty in the reconstructed potential U(x) can be presented as (Balsera et ah, 1997)... 

Including solvent in a molecular dynamics simulation creates a frictional force that damps some motion of the solute. This affects in particular the motions of exposed side chain in proteins. 

Langevin dynamics simulates the effect of molecular collisions and the resulting dissipation of energy that occur in real solvents, without explicitly including solvent molecules. This is accomplished by adding a random force (to model the effect of collisions) and a frictional force (to model dissipative losses) to each atom at each time step. Mathematically, this is expressed by the Langevin equation of motion (compare to Equation (22) in the previous chapter) ... 

Now we must consider how the frictional force experienced by this segment is related to the tangential velocity whose components are given by these expressions. 

In Chap. 9 we shall discuss in considerable detail a parameter called the molecular friction factor f. For velocities that are not too great, the friction factor expresses the proportionality between the frictional force a particle experiences and its velocity ... 

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