Two rough surfaces

Detennining the contact area between two rough surfaces is much more difficult than the sphere-on-flat problem and depends upon the moriDhology of the surfaces [9]. One can show, for instance, that for certain distributions of asperity heights the contact can be completely elastic. However, for realistic moriDhologies and macroscopic nonnal forces, the contact region includes areas of both plastic and elastic contact with plastic contact dominating. 

Figure 3 Microscopic contact between surfaces. Note that the two rough surfaces only make intimate contact at a small number of distinct points. The sum of the areas at these points corresponds to the microscopic area of contact Area].

When chemical reactions are involved in a process, it is important to know the reaction temperature. In the model described here, reactions at a point on the wafer surface are assumed to be driven by temperature excursions due to contact by passing pad asperities. This is known as flash heating. In systems in which there is dry sliding contact between two rough surfaces, it is known that flash temperatures at asperity contacts can be much higher than the average temperatures of the workpieces involved. In CMP, however, the contact is lubricated and cooled by the slurry, and this needs to be taken into account. In the case of polishing on a rotary tool, it is possible to derive a simple estimate of the mean reaction temperature, and it is this that we use in the chemical part of the two-step model. 

A strong dilatation occurs in clay subject to the clay smear process. Previously, dilatation has typically been perceived as a characteristic of strong rock e.g., during shear of two rough surfaces in strongly lithified or cemented sediments. 

Fig. 14 Typical flow curves for a concentrated emulsion sheared between two rough surfaces closed circles the line is a fit to the Herschel-Bulkley equation), and between rough and smooth surfaces open circles hydrophilic glass surface open squares hydrophobic polymer surface). 7a denotes the apparent shear rate, and is the value of the apparent shear rate at the yield point (cr = CJy). CJs is the sticking yield stress below which the emulsion adheres to the surface...

Figure 6.34 Friction between two rough surfaces The forces act at the contact sites.

Equations 12-15 through 12-18 are the Hertzian compliance relations for a sphere pressed against a sphere. Figure 12-14 depicts the contact of a flat surface with a surface whose asperity summits are spheres. Later it will be shown that the contact of two rough surfaces can be reduced to a combination of their individual contacts with a rigid plane. 

The contact of two rough surfaces with each other can be reduced to the contact of a single rough surface with a smooth, hard flat. This problem is treated in detail by Kimura [12] and by Nayak [13]. If each rough surface can be represented by a function of the orthogonal Cartesian coordinates x and y,... 

Friction transforms mechanical energy into heat by different mechanisms. To understand these we must remember that real surfaces are not flat but always contain some rugosity (Chapter 3). When two rough surfaces rub on each other a number of phenomena may contribute to friction (Figure 10.4) ... 

Figure 10.5 Contact of two rough surfaces of ductile materials.

Figure 3. Apparent contact between two rough surfaces at a low (left) and a high pressure.

One does not find two rough surfaces which are identical and even those which were obtained under the same processing parameters may differ from each other in an uncontrollable way. Such surfaces can be treated in a similar... 

The mean height of summits is located above the mean height of the surface as a whole, as indicated in fig.2. The summit heights, zs, are assumed to follow a gaussian or normal distribution with standard deviation Os. If two rough surfaces are brought in contact, their asperities will start to interact, and a pressure, Pasp. caused by asperities deformation will appear. Pasp can be calculated by ... 

The analysis of the lubricated contact between gear teeth is very complex. It involves two rough surfaces in relative motion separated by a lubricant film. This film when submitted to high contact pressures and to sliding, due to the differences between the surfaces velocities, increases its temperature, modifying significantly its physical properties, namely its viscosity. 

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