How Compliance Compensates for Surface Roughness in Fibrillar Adhesion

Fibrillar interfaces play an important role in the ability of many small animals to adhere to surfaces. Surface roughness is generally deleterious to adhesion because it hinders the ability of mating surfaces to make contact, but fibrillar surfaces compensate for surface roughness by virtue of their enhanced compliance. We examine the relationship between roughness and compliance by analyzing the mechanics of detaching an array of fibrils from a substrate. The theory of Johnson, Kendall, and Roberts is used to describe the interfacial adhesion of each fibril, and roughness is modeled by making the fibril length a random variable subject to a probability distribution. We solve for the mean force response of a fibrillar array as a function of the displacement of the entire array. From these results we extract the mean fibrillar pull-off force and work to separate the fibrillar array and substrate. We show how the mean fibrillar pull-off force decreases with increasing roughness-height standard deviation: the relationship is linear for small height standard deviation, and the pull-off force trails off to zero for very rough surfaces. Conversely, the work of separation is shown to be unaffected by small roughness-height standard deviation, although it decreases toward zero for rougher surfaces. The effects of roughness may be offset by increasing fibrillar compliance; for small roughness-height standard deviation, we show that the reduction in pull-off force is inversely proportional to the normalized compliance. We also show that the work of separation increases linearly with the compliance when the compliance is large compared with the roughness-height standard deviation.