Back to Table of Contents

42323-GB9
Decoupling the Causes for Drop Adhesion

Rafael Tadmor, Lamar University

Normal 0 false false false MicrosoftInternetExplorer4

The relation between drop radius, r, the force to push the three phase contact line and the advancing and receding contact angles is studied. To keep the line energy (also named line tension) independent of r, the modified Young equation predicts that the advancing and receding contact angles, change considerably with r. As shown by many investigators, the actual change is negligibly, if at all is there. We show that the line energy is a function of the Laplace pressure and demonstrate that this way the modified Young equation is correct and still the advancing and receding contact angles should hardly change with r.

Recent experimental evidences suggest that the unsatisfied normal component of the Young equation slowly deforms the surfaces. Such deformation, though it may be small in size, is shown experimentally to be strong enough to break chemical bonds. Shanahan and de Gennes showed that the deformation due to the unsatisfied normal component of the Young equation is proportional to the Laplace pressure inside the drop, P = 2Gamma/R, where Gamma is the surface tension and R is the radius of curvature of the drop (r = R·sin(Theta)). Since wetting properties are sensitive to the specific functional groups on the surface, so surfaces of identical chemical nature that orient different functional groups on the surface differ considerably in their wetting properties. Thus, even a sub-nanomatric distortion could induce a significant effect on the wetting properties, let alone if chemical bonds are broaken.

If the surface deformation associated with the Laplace pressure is a main factor for line energy effects then as the drop volume, V, varies, the line energy is also expected to vary. This line energy, however, is not the classic one (noted as k) which was derived for a perfectly flat rigid surface (and under constant volume constrain!) and hence it can not describe the change of the contact angle with r which results from a distortion of the surface. To describe this, a whole new energy minimization which takes into account the surface deformation needs to be done. We take a simpler approach and add a correction function to the line energy. We assume that the total line energy, L, is the line energy of a flat surface, k, multiplied by an enhancing factor, e, that corresponds to the degree of surface deformation or alteration induced by the Laplace pressure. The total line energy, L, can be viewed as a more comprehensive line-pinning-energy-per-unit-length function that includes a part derived for constant r, (i.e. k), and a correction part derived for constant angle, e.

L = k e                                                                                                                         (1)

We therefore write the e functionality for varying drop size as:

e(R) = (2Gamma/R)/G                                                                                               (2)

Where Gamma is the air-liquid surface tension, and G is some property of the substrate with dimensions of stress (analogous to some nonometric related yield stress) and corresponds to a characteristic substrate deformation (e is dimensionless). From geometrical considerations we write Eq. 8 as:

Hence

L = k e = 2Gamma(Cos(theta) – Cos(thetaY))(Sin(Theta))/G                                  (3)

Where Theta is the apparent contact angle, and ThetaY is the Young equilibrium contact angle. We see that L finally has no r dependence.

Eq. 3 agrees with the experimental observation that the advancing and receding contact angles hardly change with r. Thus we finally have a possible explanation for the common behavior observed experimentally: Eq. 3 predicts no variation of contact angle with drop size. Note that LG are coupled and similarly to k they obtain opposite signs for the advancing and the receding cases.

Back to top