invited paper for International Conference on Fracture, to be held Dec. 3-7, 2001, Honolulu, Hawaii, Hilton Hotel zone d are insignificant. However, externally applied loads, van der Waals forces between the surfaces and contact at asperities must be considered to analyze our results. To model the interfacial forces, we measured the topography of the top and bottom surfaces by AFM (double-sided tape applied to the cantilevers allows them to be removed from the substrate and placed upside down for imaging). A question arises as to the area of the contacting region that should be modeled. By considering the free body diagram of the loaded cantilever, there must be a short region of compressive contact just beyond the crack tip. From simple beam mechanics, a point reaction force exists, but from elastic considerations, this region has length ~2t, and therefore the contact area should be considered is ~2tw=80 mm2. In fact, 10x10 mm2 AFM images with 256 or 512 pixels in each direction (e.g., 40 or 20 nm lateral resolution) were used in our analysis. The AFM topograph data was read into a finite element program, and the top and bottom surfaces were placed in contact in various ways as will be described below. An elastic-plastic model was created to describe the silicon material with E=165 GPa and hardness H=12 GPa. However, it was soon found that for any reasonable pressure as applied by the external modulation, only the first contacting asperity in the contact zone deforms, and then by less than 0.5 nm. At each pixel, a parallel plate law for van der Waals forces was used to model the adhesive forces, similar to the equation posed in the Introduction, e.g., ) /(12 2 o A Dp G= . However, oD now replaces do, where oD represents the gap at each individual pixel, and the adhesion energy is summed up over the individual pixels and divided by the total area. For the few pixels where there is actual contact, a cutoff value of oc D =0.3 nm [16] was used. With A=5·10-20 J for a fluorocarbon surface, a surface energy of 15 mJ/m2 is calculated in these regions. For comparison, values of 7 and 28 mJ/m2 for advancing and receding surface energies respectively were recently determined by surface force apparatus measurements for a fluorocarbon surfactant (TAFC, (C8F17C2H4)2-L-Glu-Ac-N+-(CH3)3-Cl -) applied to a mica surface. The surfaces were placed in contact in various combinations. This included the original top and bottom measured layers with various random shifts in alignment, and pairs of the bottom layers including mating of the bottom layer to itself. The circles in Fig. 3 are the calculated adhesion results for the various combinations of the data. The solid line represents data from an individual placement combination, but with the roughness scaled to both lower and higher scales. The value of av D » 2do in Fig. 3 is determined by the finite element analysis for a given placement of the top and bottom surfaces. DISCUSSION The abscissa av D in Fig. 3 is better used than do because it takes into account the actual alignment of the two surfaces, thereby reflecting the separation of the associated highest asperity pair. At small av D values, ) /(12 2 av A Dp G= is a good approximation for the calculated adhesion (circles in Fig. 3) . Of course, this equation will always be a lower bound for the adhesion values because of the non-linearity in this equation. However, as indicated by both the data and the model, the adhesion does not fall off with 2 1/ av D at large av D values. It is important to realize that for these deposited layers, there is no long range waviness to the surfaces. Therefore, surfaces can be near each other over large distances without contacting. To qualitatively understand the results, we consider two conceptual extremes in the adhesion between rough surfaces. In one, the surfaces are relatively smooth and contact is at only one asperity point. Van der Waals forces across non-contacting portions of the surfaces, whose area is far greater than the
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