Capillary adhesion between elastically hard rough surfaces

The adhesion versus vapor pressure (p/p(s)) trend between two elastically hard rough surfaces is modeled and compared with experimental results. The experimental samples were hydrophilic surface-micromachined cantilevers, in which the nanometer-scale surface roughness is on the order of the Kelvin radius. The experimental results indicated that adhesion increases exponentially from p/p(s)=0.3 to 0.95, with values from 1 mJ/m(2) to 50 mJ/m(2). Using the Kelvin equation to determine the force-displacement curves, the mechanics of a wetted rough interface are treated in two ways. First, the characteristics of a surface with rigid asperities of uniform height are derived. At low p/p(s), menisci surrounding individual asperities do not interact. Beyond a transition value, [p/ps] (tr), a given meniscus grows beyond the asperity it is associated with, and liquid fills the interface. Capillary adhesion in each realm is found according to the integrated work of adhesion. Second, a more general approach allowing an arbitrary height distribution of Hertzian asperities subject to capillary forces is justified and developed. To compare with experimental results, a Gaussian height distribution is first assumed but significantly underestimates the measured adhesion. This is because equilibrium is found far into the Gaussian tail, where asperities likely do not exist. It is shown that by bounding the tail to more likely limits, the measured adhesion trend is more closely followed but is still not satisfactorily matched by the model. The uniform summit height model fits the data very well with a single free parameter. These results can be rationalized if the upper and lower surfaces are geometrically correlated.