Deformation of an elastic thin solid induced by a liquid droplet

Abstract

The aim of this study is to formulate and solve the equilibrium shape of a liquid droplet of contact radius r 0 in the middle of an elastic isotropic thin solid sheet of radius l, by using the different surface concepts such as surface free energy and surface stress. This is achieved by determining the form of the interfaces as a result of minimization of the total free energy of the system. The profiles are obtained from integration of Euler's differential equations. An interpretation of these equations is given in terms of mechanical forces. In the domain 0 < r < r 0, normal to the sheet there act two shearing forces. One of them derives from Laplace's overpressure inside the drop. It is uniformly distributed over the area π r 2 0. The other one, coming from the total surface stress of the sheet, leads to shearing forces proportional to the local slope. Outside the drop ( r 0 < r < l), there are at the same two types of normal shearing forces, one due to the surface stress, the other type acting on the triple line of length 2π r 0. Each of these mechanical forces acts on distinct domains of the sheet. Therefore it is not allowed to superpose these forces acting on the entire sheet 0 < r < l. If such a superposition is unduly made, a ridge appears at the triple line. The liquid drop is found to be spherical as expected in the absence of gravity. If gravity acts, qualitative modifications are discussed. The profile of the sheet is pseudo-parabolic with an inflection circle of radius r i changing within 0 < r i < l according to the fastening ratio l r 0 of the sheet. The sag under the drop depends on the Laplace overpressure and the flexure rigidity D of the sheet. It varies therefore with the contact angle as sin a and as ( r 0 e ) 3 , e being the sheet thickness. The sag approaches the value of e for radius r 0 e > 10 when considering usual solids of Young-modulus E = 10 12 erg cm −3. Earth gravity does not influence markedly the results for drops with r 0 < 0.1 cm. The effect of surface stress s depends on the characteristic length | s|/ D. It becomes numerically important only for very thin sheets. The total surface stress in 0 < r < r 0 may enhance or decrease the sag according its sign. Instabilities (buckling) may occur above some limiting value | s|/ D. For negative surface stress, the profile outside the droplet ( r 0 < r < l) shows oscillations which represent a stable situation. A sheet without droplet with negative surface stress shows buckling if its size l exceeds some critical value. The presence of a droplet is able to stabilize this situation. The final equilibrium problem is only solved by optimizing also the size r 0 of the droplet so that the total free energy is minimal. From this, the wetting angle α and the deformation angle β at the triple line are given analytically. The elastic properties of the sheet determine not only β but also influences slightly α. Finally, according to a tradition going back to Young-Maxwell and for the purpose in hands, we consider average thermodynamic forces acting along the triple line. We show that Young-Maxwell's construction based on Neumann's triangle is possible. Our equations show that Young's famous equation of wetting is true up to the second order of the deformation angle β. Calculating this order an extended Young's equation is given. Our general result is easily brought to the special cases where the droplet lies either on a membrane or on another liquid surface.