Surface effects on thin film wrinkling

A recent discussion here about the effect of surface stress on vibrations of microcantilever has gained some interest from our members. A few years ago, Zhigang and I looked at surface effect on buckling of a thin elastic film on a viscous layer (Huang and Suo, Thin Solid Films 429, 273-281, 2003). Although the physical phenomena (buckling vs vibrations) are different, the conclusion is quite consistent with Wei Hong and Pradeep's comments toward the end of the discussion. That is, surface stress only contributes as a residual stress and thus does not affect the buckling wavelength (frequency in space in analogy to frequency in time for vibrations).

More recently, a group at NIST has found that, for ultrathin polystyrene films (h < 40 nm) on PDMS, the buckle wavelength depends on the film thickness nonlinearly, different from a conventional theory commonly used for sandwich plates with stiff skin layers. Using the conventional theory, they found that the elastic modulus of polystyrene decreases by an order of magnitude when the film thickness changes from 100 nm to 5 nm. I worked with them and proposed an explanation based on surface effects (see Stafford et al., Macromolecules 39, 5095-5099, 2006). Here again, we found that surface stress alone does not help with the change of buckle wavelength. As pointed out in one of Zhigang's comments, we had to introduce a superficial elastic modulus. To include both surface stress and surface modulus, we assumed a surface layer of finite thickness, where the surface stress can be accounted for as a residual stress in this surface layer. By using a relatively soft surface layer of about 2 nm, we were able to reasonably reproduce the measured thickness-dependence of buckle wavelength.

It is intuitively plausible that, for ultrathin films (or cantilever beams), surface properties (not just surface stress) can affect the mechanical behavior. For polymer thin films, a 2 nm surface layer is possible based on molecular dynamics simulations of polymer density near a free surface. This leads to an observable change of buckling behavior for film thickness less than about 50 nm. For other materials (metals or Si), I expect some differences: the surface layer should be thinner (< 1 nm?); the surface modulus could be higher than bulk (hard surface), etc. For more details of our model and some general discussions about surface effects on thin film buckling, please see a pre-print here (Huang et al., J. Aerospace Engineering, to appear in January 2007); Update on December 28, 2006: now published as J. Aerospace Engineering 20, 38-44 (2007).