Surface energy, elasticity and the homogenization of rough surfaces

The concept of surface energy is widely used to understand numerous aspects of material behavior: fracture, self-assembly, catalysis, void formation, microstructure evolution, and size-effect exhibited by nanostructures. Extensive work exists on deriving homogenized constitutive responses for macroscopic composites—relating effective properties to various microstructural details. In the present work, we focus on homogenization of surfaces. Indeed, elucidation of the effect of surface roughness on the surface energy, stress, and elastic behavior is relatively under-studied and quite relevant to the behavior of both nanostructures and bulk material where surfaces are involved in some form or fashion. We present derivations that relate both periodic and random roughness to the effective surface elastic behavior. We find that the residual surface stress is hardly affected by roughness while the superficial elastic properties are dramatically altered and, importantly, they may also change sign—this has significant ramifications in the interpretation of sensing based on frequency measurement changes. Interestingly, even if the bare surface has a zero surface elasticity modulus, roughness is seen to endow it with one. Using atomistic calculations, we verify the qualitative validity of the obtained theoretical insights. We show, through an illustrative example, that the square of resonance frequency of a cantilever beam with rough surface can decrease almost by a factor of two compared to a flat surface.