invited paper for International Conference on Fracture, to be held Dec. 3-7, 2001, Honolulu, Hawaii, Hilton Hotel contacting area at the one asperity, will dominate the adhesion in this case. For example, at av D =10 nm, G=13 mJ/m2 is expected as seen in Fig. 3. At the other extreme, the surfaces are rough, and av D is large. Only the single point of contact contributes to the adhesion. In this case, we would expect 2 )/ ~ ( /(6 c oc AR D L G ), where R is the radius of the contacting asperities and 2 cL is the area of adhesion that is being probed (the term ) /(6 oc AR D is the van der Waals adhesion energy between two contacting spheres). With R=50 nm as a typical value for the polysilicon asperities in these experiments and 2 cL =100 mm2 as discussed above, we expect a lower bound for adhesion to be 0.014 mJ/m2. This latter extreme is a simplified expression of the Maugis model of rough surfaces [2], which takes van der Waals forces into account, but only at contacting asperities. That model is more appropriate here than the Fuller-Tabor approach because of the large E and small R of these surfaces. Note that the values of adhesion in Fig. 3 are much closer to the former than the latter extreme, implying that van der Waals forces over non-contacting areas dominate the adhesion. Negligible adhesion hysteresis measured in other experiments corroborates this notion [14]. To quantitatively understand the results, consider Fig. 4, where a histogram of the relative contributions from the range of oD values is plotted for different surface roughnesses. At small av D , most of the contribution to adhesion comes from non-contacting surfaces, whereas at large av D , the contribution from surfaces nearly in contact begins to become the largest contributor. We can now address the questions posed in the Introduction. (1) Typically, MEMS surfaces exhibit av D ~10-30 nm. Therefore, most adhesion in MEMS is due to van der Waals forces between non-contacting areas. Even for large av D in Fig. 3, this remains true - the reason for the small reduction in G is that nearby noncontacting asperities begin to contribute significantly to adhesion. However, as av D grows above 60 nm, the Maugis model will adequately describe the adhesion. (2) Adhesion as low as 0.01 mJ/m2 should be possible by making surfaces rough. However, because of the weak dependence of G on av D , extremely large roughness would be required. Given that MEMS structures are often used for optical reflection in mirror applications, this would be an unpopular choice. (3) There is a deviation in the calculated curve from ) /(12 2 av A Dp G= beginning at av D ~25 nm. We suggest this is a near-optimal separation value. For lower values of roughness, adhesion begins to increase significantly because of the close proximity of the surfaces, while for large values, optical reflectivity is significantly compromised. Using the Greenwood-Williamson model [3], which applies reasonably well for these surfaces, the ratio of real to apparent contact area is found to be approximately 10-8 for the smoothest surfaces. The real contact area is greatly overestimated in the finite element formulation because of the pixel size limitation. Depending on the lateral resolution used in the AFM measurements, the smallest possible ratio is (1/256)2=1.5·10-5 or (1/512)2=3.8·10-6. Because the contribution to the total adhesion of the contacting point is still small, this causes only a small error in the adhesion calculation. However, this is further evidence that van der Waals forces in the vast area between contacts dominates the adhesion of these surfaces, especially when the average separation is small. 0% 10% 20% 30% 40% 50% 60% 10 100 G contribution (%) D 0 (nm) D av 16.8 nm 28.0 nm 57.3 nm 37.8 nm 57.3 nm 37.8 nm 28.0 nm 16.8 nm
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