Experimentally determined friction force

Figure 12. Experimentally determined friction force vs. relative scan angle for PTFE. The corresponding fits are based on equations 1 and 2. For details see text.

The calculated energetics also allow one to estimate the normal load, thereby providing access to the friction coefficient once the friction force is known. This method, albeit crude, has been shown to yield good results when compared with experiments in cases such as graphite sheets sliding past one another.68 However, one should realize that approximations in the determination of the normal load, the assumption that friction only depends on energy barriers, and the lack of a consideration of dynamical aspects of system may lead to significant deviations from experimental results for many other systems. 

The constitutive equation for a dry powder is a governing equation for the stress tensor, t, in terms of the time derivative of the displacement in the material, e (= v == dK/dt). This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature of the constitutive equation for the three-dimensional case for dry powders. The normal stress-strain relationship and the shear stress-strain relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion. 

Experimentally, the static friction force is determined a.s a maximal force needed to initiate sliding motion. The question arises whether the static friction obtained in this way is a unique (inherent) property of the system, or whether it depends also on the conditions of the measurement. Section 11.C.2 only discussed the maximum possible shear force between a tip and a substrate, but the effect of thermal fluctuations and the effect of the driving device for instance, the stiffness of the driving device or the tip itself was not included. 

It should be noted that the binomial friction law of Derjagin states that the adhesion bonds of two bodies and the friction forces determine a tangential load. The aggregate of these values is represented by the concept of static friction. Experimental validation [392, 393] of the binomial friction law allows the determination of this sum, but not of each item separately. As has been noted [392], this method serves only as an indirect estimation of adhesive joint strength. 

The approximate experimental determination of xl), is based on measurement of the velocity of a charged particle in a solvent subjected to an applied voltage. Such a particle experiences an electrical force that initiates motion. Since a hydrodynamic frictional force acts on the particle as it moves, a steady state is reached, with the particle moving with a constant velocity U. To calculate this electrophoretic velocity U theoretically, it is, in general, necessary to solve Poisson s equation (Equation 3.19) and the governing equations for ion transport subject to the condition that the electric field is constant far away from the particle. The appropriate viscous drag on the particle can be calculated from the velocity field and the electrical force on the particle from the electrical potential distribution. The fact that the sum of the two is zero provides the electrophoretic velocity U. Actual solutions are complex, and the electrical properties of the particle (e.g., polarizability, conductivity, surface conductivity, etc.) come into play. Details are given by Levich (1962) (see also Problem 7.8). 

When the areas of actual and nominal contact are equal, the linear plot of frictional force Ff j. against the normal load F does not pass through the coordinate origin Ff =/(F ). In this situation, we can extrapolate the straight-line plot to the case in which F = 0 and thus find the magnitude of the adhesive force. Further, F d can be determined experimentally by detachment of two steel surfaces from each other [35, p. 275]. Below we have listed comparative values for the magnitude of adhesive force between steel surfaces with a Class 14 finish, separated by a lubricant layer. The force of adhesion was determined experimentally (by measurement in a triboadhesiometer) and by extrapolation of the lines for Ff =/(F ) ... 

The experimental results in Figure 1 display a single line scan over a range of 100 mn for an applied load of 9.5 nN and a pH of 4. The upper panel shows the topographical signal, while the lower panel shows the signal due to lateral forces. To determine the frictional force, we have calculated the difference between a forward and a backward scan. Since the samples are very smooth (rou ness below 0.1 nm over a scan range of 100 mn),... 

Figure 3 shows the total load dependence of the friction force measured for modulation amplitudes of 50nm and lOOnm. The contact location was arbitrarily chosen on the same surface. Both curves are described by the same Amontons law. The friction coefficient defined by the slope of the linear fit is )li=0.087 0.001. When plotted as the friction force versus the total load, the intercept is zero. It is important at this point to specify that the error associated to the friction coefficient arises from the fitting analysis of our data, which therefore, determines the precision of the experiment and not the overall acuracy of the experiment. Indeed, the main source of uncertainty in our measurements originates in the precision in the cantilever metrics measured by optical microscopy and SEM which is of the order of 3% to 5%. Some other sources (19,28), like the position of the laser spot on the backside of the cantilever affects the absolute accuracy of the friction measurements to an extend that is difficult to evaluate. We expect the overall accuracy on the friction measurement to be less than 60%(25). Nevertheless, since the crucial experimental conditions were optimized and kept constant from an experiment to the other, the comparison remains valid. 

N = 0 value. This is an unfortunate drawback of the experimental technique the estimate for the Nq is essential for an understanding of the mechanisms of the influence of surfactants and polymers on the friction between fibers and on the pulp rheology. This estimate can be obtained from direct measurements of cohesive forces in contacts between crossed fibers. The molecular component of the cohesion, p, in the contact between two fibers can be obtained as the force necessary to rupture the fiber-fiber contact. The shear friction force can then be determined as the product of the previously determined friction coefficient, p, and the normal force Nq. In the absence of an external load, the value of Nq is solely the result of the molecular attraction forces, that is, F = iNq = pp. 

The coefficient /x depends on the dimensions of the particles. This coefficient, which is equal to the slope of the straight line characterizing the relationship between the adhesive and frictional forces and the mass of the powder on rotating the dust-laden surface, may be determined experimentally (Fig. 1.6, p. 20). 

The value of Fg may be determined experimentally. For this purpose we may use a tribometer to determine the relation between the shear force (which is equal and opposite to the frictional force) and the value of the normal load (in the absence of which the shear stress is simply employed in overcoming the forces of adhesion) (Fig. XI.1). 

In the chain-parallel direction, only the HOPE showed a periodic stick slip behavior with a repeat distance of ca. 2.5 A. This distance is equal to the repeat unit of polyethylene (16, 20). For PTFE the LFM friction loops in our experiments did not reveal any stick slip behavior. Thus, in this case, we cannot determine the value of do in the chain-parallel direction. In this case we can, however, assume that the value of d is close to 0. Based on equation 1, the friction anisotropy is therefore expected to be larger for PTFE, than that for HDPE. For a semi-quantitative comparison of friction forces predicted by equation (1) on one hand and experimental results obtained on highly oriented polymer surfaces on the other hand, one should be... 

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