Models asperity contact

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... 

While the results and conclusions are consistent with the asperity contact model discussed earlier, the data does not unambiguously demonstrate the connection to asperity deformation. One of the complicating assumptions in Ref. [14] was that the shear modulus used in the comparison was a composite modulus calculated from the bulk material properties of each component in a two-pad stack. If asperity deformation is a dominant factor, a more appropriate value is the shear modulus of the contacting member. 

The problem of wear when the fluid film lubricant is no longer intact is associated with the asperity contact of structured surfaces. The contact behavior of such surfaces was discussed in Chapter 12 wear models governed by asperity contact were described in Chapter 13. Theoretically the laws controlling fluid film thickness can be coupled with asperity contact models to yield quantitative descriptions of the course of wear. In this section we shall deal with those cases in which the function of the lubricant is only to provide a fluid film separating the two rubbing bodies, and the events at the contact, once it is established, are determined by the interaction of mechanical parameters such as load and rubbing speed with the properties of the contacting interface. 

Elastic Contact Conductance Models of Mikic and Greenwood and Williamson. Sridhar and Yovanovich [106] reviewed the elastic contact models proposed by Greenwood and Williamson [26] and Mikic [66] and compared the correlation equation with data obtained for five different metals. The models were developed for conforming rough surfaces they differ in the description of the surface metrology and the contact mechanics. The thermal model developed by Cooper et al. [14] was used. The details of the development of the models and the correlation equations are reviewed by Sridhar and Yovanovich [106]. The correlation equation derived from the Mikic [66] surface and asperity contact models is... 

The correlation equation derived from the Greenwood and Williamson [26] surface and asperity contact models is... 

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. 

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. 

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. 

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-... 

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. 

Various continuum models have been developed to describe contact phenomena between solids. Over the years there has been much disagreement as to the appropriateness of these models (Derjaguin et al. [2 ] and Tabor [5-7]). Experimental verification can be complex due to uncertainties over the effects of contaminants and asperities dominating the contact. Models trying to include these effects are no longer solvable analytically. A range of models describing contact between both nondeformable and deformable solids in various environments are discussed in more detail later. In all cases, the system of a sphere on a plane is considered, for this is the most relevant to the experimental techniques used to measure nanoscale adhesion. 

Extensive in situ observations of failure processes within model single asperity contacts demonstrate the relevance of fracture mechanics approaches to the analysis of cracking processes within sliding contacts involving brit-... 

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. 

Figure 12-8. Simple model of deformable spherical asperities contacting a non-deformable flat surface. After J. F. Archard [4].

A statistical approach to the mechanics of asperity contact is that of Greenwood and Williamson [7], which may not be exact in all respects but does present a readily apprehensible physical model. Figure 12-14 is a diagram of the encounter of the smooth surface with the rough profile. 

Figure 12-17. The observed dependence of degree of contact on load for the rough surface of a constrained deformable specimen against a rigid smooth flat, compared with the relations obtained by (a) assuming noninteraction of asperities (broken line) and (b) the asperity interaction model (solid line). Data by Pullen and Williamson [14].

A first approach to the quantitative modeling of wear is the postulate that the governing parameter is the number of effective contacts as one surface, slides upon the other. In the modern realistic view these are asperity contacts, and hence the sizes and the distribution of the asperities are an essential part of the model. If W is the total number of asperity contacts in a given sliding process and H is the number of... 

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