Asperity contacts

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]. 

Yield pressurep of the asperities is about three times the tensile yield strength for many materials. The real area of contact is frequently a minute fraction of the total area. With a typical bearing contact stress of 3 MPa and a bron2e bearing asperity yield pressure of 500 MPa, for instance, less than 1.0% of the nominal area would involve asperity contact. 

The existence of asperity contacts in mixed lubrication causes great many local events and significant consequences. For example, the parameters describing lubrication and contact conditions, such as film thickness, pressure, subsurface stress, and surface temperature, fluctuate violently and frequently over time and space domain. It is expected that these local events would have significant effects on the service life of machine elements, but experimental measurements are difficult because of the highly random and time-dependent nature of the signals. Only a few successes were reported so far in experimental studies of mixed lubrication, mostly limited to the artificially manufactured... 

The statistic models consider surface roughness as a stochastic process, and concern the averaged or statistic behavior of lubrication and contact. For instance, the average flow model, proposed by Patir and Cheng [2], combined with the Greenwood and Williamsons statistic model of asperity contact [3] has been one of widely accepted models for mixed lubrication in early times. 

Statistic ML Modeis Combining Lubrication and Asperity Contact... 

The average pressure, Pc, due to asperity contact can be obtained from Eq (5) by dehningp =FM, which gives... 

As a result of asperity contact, the nominal contact zone is split into a number of discrete areas that can be cataloged either to the lubrication region or asperity contact area (Fig. 2). The mean hydrodynamic pressure in the lubrication regions, pi, can be calculated by the average flow model, while contact pressure is estimated via Eq (7). Consequently, the film thickness is determined through numerical iterations to... 

The word deterministic" means that the model employs a specific surface geometry or prescribed roughness data as an input of the numerical procedure for solving the governing equations. The method was originally adopted in micro-EHL to predict local film thickness and pressure distributions over individual asperities, and it can be used to solve the mixed lubrication problems when properly combined with the solutions of asperity contacts. 

All the models of mixed lubrication developed previously were based on a traditional idea, as schematically shown in Fig. 2, that the nominal contact zone, O, has to be divided into two different t3q>es of areas the lubricated area, 0,1, where two surfaces are separated by a lubricant film and the asperity contact area, where two surfaces are assumed to be in direct contact. The present authors and Dr. Zhu [16,17] proposed a different strategy for modeling... 

Instead of dividing the computation domain into lubrication regions and asperity contact areas, the mixed lubrication model proposed by the present authors assumes that the pressure distribution over the entire domain follows the Reynolds equation ... 

In this model, there will be no asperity contacts in the traditional sense, but as the film thickness between the interacting asperities decreases below a certain level, the right-hand terms in Eq (9), which represent the lubricant flow caused by pressure gradient, become so insigniheant that the pressure can be predicted by a reduced Reynolds equation [16,17] ... 

The asperity contact areas in conventional models of mixed lubrication correspond here to the areas where Eq (11) is applied. More discussions about this reduced equation are left to the next section. 

A typical method for thermal analysis is to solve the energy equation in hydrodynamic films and the heat conduction equation in solids, simultaneously, along with the other governing equations. To apply this method to mixed lubrication, however, one has to deal with several problems. In addition to the great computational work required, the discontinuity of the hydrodynamic films due to asperity contacts presents a major difficulty to the application. As an alternative, the method of moving point heat source integration has been introduced to conduct thermal analysis in mixed lubrication. 

As described previously, two different strategies have been developed in the DML model to solve the pressure distributions for hy drod5mamic lubrication and asperity contacts, simultaneously. The advantage and disadvantage of the two methods deserve a further discussion. 

A major shortage of the method is that the border positions and boundary pressure distributions between the hydrodynamic and contact regions have to be calculated at every step of computation. It is a difficult and laborious procedure because the asperity contacts may produce many contact regions with irregular and time-dependent contours, which complicates the algorithm implementation, increases the computational work, and perhaps spoils the convergence of the solutions. 

The model validation in mixed lubrication should be made under the conditions when asperity contacts coexist with lubrication. Choo et al. [45] measured film thickness on the surface distributed with artificial asperities. The experi-... 

If the surfaces are damaged during sliding so that wear debris and multi-asperity contacts are involved in the process, the mechanism of friction will be substantially different from what we discussed for wearless friction. 

Finally, it deserves to be mentioned that considerable numbers of models of static friction based on continuum mechanics and asperity contact were proposed in the literature. For instance, the friction at individual asperity was calculated, and the total force of friction was then obtained through a statistical sum-up [35]. In the majority of such models, however, the friction on individual asperity was estimated in terms of a phenomenal shear stress without involving the origin of friction. 

In 1999, Luo and Domfeld [110] proposed that there are two typical contact modes in the CMP process, i.e., the hydro-dynamical contact mode and the solid-solid contact mode [110]. When the down pressure applied on the wafer surface is small and the relative velocity of the wafer is large, a thin fluid film with micro-scale thickness will be formed between the wafer and pad surface. The size of the abrasive particles is much smaller than the thickness of the slurry film, and therefore a lot of abrasive particles are inactive. Almost all material removals are due to three-body abrasion. When the down pressure applied on the wafer surface is large and the relative velocity of the wafer is small, the wafer and pad asperity contact each other and both two-body and three-body abrasion occurs, as is described as solid-solid contact mode in Fig. 44 [110]. In the two-body abrasion, the abrasive particles embedded in the pad asperities move to remove materials. Almost all effective material removals happen due to these abrasions. However, the abrasives not embedded in the pad are either inactive or act in three-body abrasion. Compared with the two-body abrasion happening in the wafer-pad contact area, the material removed by three-body abrasion is negligible. 

These problems have been improved in recent years by the microfabrication of sharp tips with radii less than 10 nm, the observation in an SEM or STEM of the exact radius before and after the experiment, the use of robust carbon-nanotube probes, and general improvements in control electronics. However, another method used initially was the attachment of a small colloid particle in place of the AFM tip. These particles were considered a reasonably good approximation to a single-asperity contact their radii were accurately known and remained the same for the duration of the experiment. Such probes have also been used to investigate colloids where surface roughness is an important aspect of the colloid interaction. 

The most generally satisfactory local-level model for CMP is an asperity contact model such as that described by Yu ef al. [4]. In this model, applied... 

Fig. 8, Schematic diagram of (top) asperity contact during CMP, (middle) die-scale asperity contact for patterned wafer with topography height x and (bottom) die-scale asperity contact for patterned wafer with topography height x/3.

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