User:Amirrost/Elastocapillarity

Elastocapillarity refers to the deformation of solid structures due to capillary pressure developed within the liquid droplets or films present in their proximity. From the viewpoint of mechanics, elastocapillarity phenomena essentially involve competition between the elastic strain energy in the bulk and the energy on the surfaces/interfaces. In the modeling of these phenomena, some challenging issues are, among others, the exact characterization of energies at the micro scale, the solution of strongly nonlinear problems of structures with large deformation and moving boundary conditions, and instability of either solid structures or droplets/films.The capillary forces are generally negligible in the analysis of macroscopic structures but often play a significant role in many phenomena at small scales.

Young-Laplace Equation
The capillary pressure developed within a liquid droplet/film can be calculated using the Young–Laplace equation (e.g. ):
 * $$\Delta p = -\gamma \nabla \cdot \hat n =\gamma (\cfrac{1}{R_1} + \cfrac{1}{R_2})$$

where:


 * $$\Delta p$$ is the difference between the pressure across the liquid interface (Pa),
 * $$\gamma$$ is the surface tension of the liquid (N/m),
 * $$\hat n$$ is the unit normal pointing out of surface,
 * $$R_1,R_2$$ are the principle radii of curvature at any point on the free surface of the liquid film or droplet (m).

If the liquid wets the contacting surfaces then this pressure difference is negative i.e. the pressure inside liquid is less than the ambient pressure, and if the liquid doesn't wet the contacting surfaces then the pressure difference is positive and liquid pressure is higher than the ambient pressure.

Examples of elastocapillarity
The coalescence happens in a brush after removing it from water is an example of elastocapillrity. Elastocapillary wrapping driven by drop impact is another example. Most of the small scale devices such as microelectromechanical systems (MEMS), magnetic head-disk interface (HDI), and the tip of atomic force microscopy (AFM) for which liquids are present in confined regions during fabrication or during operation can experience elastocapillary phenomena. In these devices, where the spacing between solid structures is small, intermolecular interactions become significant. The liquid can exist in these small scale devices due to contamination, condensation or lubrication. The liquid present in these devices can increase the adhesive forces drastically and cause device failure.

Elastocapillarity in contact between rough surfaces
Every surface though appears smooth at macro scale has roughness in micro scales which can be measured by a profilometer. The wetting liquid between contacting rough surfaces develops a sub-ambient pressure inside itself, which forces the surfaces toward more intimate contact. Since the pressure drop across the liquid is proportional to the curvature at the free surface and this curvature, in turn, is approximately inversely proportional to the local spacing, the thinner the liquid bridge, the greater is the pull effect.


 * $$\Delta p = -\gamma (\cos \theta_A+ \cos \theta_B)/h_{fs}$$

where:
 * $$\theta_A,\theta_B$$ are the liquid-solid contact angles for the lower and upper surfaces, respectively,
 * $$h_{fs}$$ is the gap between the two solids at the location of the free surface of the liquid.

These tensile stresses put the two surfaces into more contact while the compressive stresses due to the elastic deformation of the surfaces tend to resist them. Two scenarios could happen in this case: 1. The tensile and compressive stresses come into balance which in this case the gap between the two surfaces is in the order of Surface roughness|roughness of the surfaces, or, 2. The tensile stresses overcome the compressive stresses and the two surfaces come into near complete contact in which gap between surfaces is a small fraction of the Surface roughness|surface roughness. The latter case is the reason for failure of most microscale devices. An estimate of the tensile stresses exerted by the capillary film can be obtained by dividing the adhesion force, $$P_{liq}$$, between two surfaces to the area wetted by the liquid film, $$A_{wet}$$. Because for relative smooth surfaces, the magnitude of the capillary pressure is predicted to be large, it is anticipated that the capillary pressures will be of large magnitude. A lot of works have been done to ascertain whether there may be some practical limit to the development of such negative pressures (e.g. ).