Friction coefficient examples

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]. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

One of the major advantages of the worm gear is low wear, which is due mostly to a full-fluid lubricant film. In addition, friction can be further reduced using metals having low coefficients of friction. For example, the wheel is typically made of bronze and the worm of highly finished hardened steel. 

More serious errors may result when the grain-size of a specimen is small compared with the size of an indentation. Then, since all crystals are elastically anisotropic a rigid indenter will produce differing amounts of elastic strain in the grains depending on their orientations. This will create an effective roughening of the surface and increase the friction coefficient. This may result in overestimates of hardnesses. For example, this may underlie reports of nanocrystalline materials being harder than diamond. 

There is disagreement in the literature about the role of friction. Compare, for example, Cai (1993) with Ishikawa et al. (2000) This has arisen in various ways. In the case of metals, where the Chin-Gilman parameter is small, friction is not important for relatively large indents. However, as the C-G parameter becomes much larger for covalent crystals, and as the indent size decreases friction becomes more important. Also, environmental factors, such as humidity, affect friction coefficients. In the regime of superhardness with dry specimens and small indents friction becomes very important. 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

Some narrowing of the spectrum does occur when there is a strong positional dependence in the frictional coefficient. For example, the reduced compliance is ... 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Example 11.4. McGuiggan et al. [492] measured the friction on mica surfaces coated with thin films of either perfluoropolyether (PFPE) or polydimethylsiloxane (PDMS) using three different methods The surface forces apparatus (radius of curvature of the contacting bodies R 1 cm) friction force microscopy with a sharp AFM tip (R 20 nm) and friction force microscopy with a colloidal probe (R 15 nm). In the surface force apparatus, friction coefficients of the two materials differed by a factor of 100 whereas for the AFM silicon nitride tip, the friction coefficient for both materials was the same. When the colloidal probe technique was used, the friction coefficients differed by a factor of 4. This can be explained by the fact that, in friction force experiments, the contact pressures are much higher. This leads to a complete penetration of the AFM tip through the lubrication layer, rendering the lubricants ineffective. In the case of the colloidal probe the contact pressure is reduced and the lubrication layer cannot be displaced completely. 

Example 2.2 Consider an impact between a polyethylene particle (dp = 1 cm) and a copper wall. The incident velocity is 2 m/s, and the incident angle is 30°. The friction coefficient of the interface is 0.2. The densities of polyethylene and copper are 950 and 8,900 kg/m3, respectively. What is the contact time duration for the collision Estimate the rebound velocity of the particle. Repeat the problem for a copper particle colliding with a polyethylene wall. 

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