Sommerfeld number

Another type of surface-treated pads is the low-shear surface-engineered pads. When used in copper CMP, the pads exhibited vastly improved tribological performance in comparison to the untreated pads. More specifically, these pads gave a much lower COF over a larger range of Sommerfeld numbers compared to untreated pads. The reduction in shear-induced stress and its accumulation within the copper films can significantly... 

Figure 13.2 Variation of Friction with Sommerfeld Number for a Series of Dispersions of Molybdenum Disulphide in Mineral Oil (Ref.455)...

Herzig" also studied the effect of dispersed molybdenum disulphide in a hydrodynamic bearing, and showed that, where full fluid film lubrication Is not present, dispersed molybdenum disulphide can decrease friction. The coefficient of friction in an oil-lubricated journal bearing is related to the Sommerfeld Number ZN/P where Z Is the viscosity, N the rate of rotation, and P the bearing pressure or specific load. Figure 13.2 shows some empirical relationships between coefficient of friction and the Sommerfeld Number for an oil with different concentrations of dispersed molybdenum disulphide in a foil bearing . 

It can be seen that at low Sommerfeld Number, that is in the mixed or boundary region with high load, low viscosity or low speed, the molybdenum disulphide dispersions give a decrease in friction varying from 40% with a 1 % dispersion to 60% with a 10% dispersion. At higher Sommerfeld Number with full... 

However, the assumption which gave us Eqn 2-47 is physically unrealistic. A more rigorous approach, details of which can be found in the monograph by Pinkus and Sternlicht, enables the computation of the attitude angle g. Numerical methods are used for the most part. Once a value has been obtained for g the value of e is easily calculated. It is customary to insert the value of e thus found into Eqn 2-51 to calculate the Sommerfeld number, although, as Pinkus and Sternlicht show, a more realistic relation is... 

Some examples of measurements of eccentricity-ratio plotted against Sommerfeld number are shewn in Fig. 5. These were in qualitative agreement but there were quantitative differences for the various values of clearance. 

In Figure 8, the eccentricity ratio Is predicted to be sensibly constant for a given value of (Bjj/B). This may be Interpreted by considering the Sommerfeld number which Is a function of the eccentricity ratio. For a given load and bearing geometry, therefore, the eccentricity ratio would be constant If an Increase In shaft speed produced a decrease In... 

Sommerfeld Number that is useful for estimating the minimum film thickness in journal bearing design. 

Figure 5.12. Variation of hic with Sommerfeld Number (.S ) for a journal bearing of essentially infinite length (i.e., dH = 0).

More detailed analysis of journal bearing friction reveals that the Petroff equation gives satisfactory results only above a Sommerfeld Number (5 ) of 0.15. For lower values of S the coefficient of friction will be higher than the lightly loaded approximation due to Petroff. The curve labeled Reynolds (based on a complete solution due to Reynolds) in Fig. 5.13 enables the coefficient of friction (/) to be estimated when the eccentricity of the journal in the bearing plays a significant role (that is, when S <0.15). 

Figure 5.13. Variation of d/c vs Sommerfeld Number (-S ) for a journal bearing operating under a wide range of loads. The Petroff solution based on concentricity of the journal is seen to hold only for values of S >0.15. For lower values of S, a solution is required that takes journal eccentricity into account (labeled Reynolds).

Couples increasing Sommerfeld number Lubrication regimes... 

Somewhat contrary results were obtained when a dimpled-surface thrust bearing was tested at low speeds (below 1 m/s) and loads of up to 3.45 MPa [67]. The bearing was of a bending-pad type with three pads rotated in an oil bath with a stationary collar. The measured pressure profiles had bumps in the converging part of each dimple. It was concluded that they could improve load capacity. The study also revealed that depending on Sommerfeld number the friction coefficients of the dimpled bearing were up to 7.3% lower than those associated with a plain surface. Unfortunately, dimple dimensions were not given. 

---