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42323-GB9
Decoupling the Causes for Drop Adhesion

Rafael Tadmor, Lamar University

The equilibrium contact angle, which is a function of surface properties[1, 2] is difficult to determine within the advancing and receding angles. The as-placed contact angle considered here and the advancing and receding angles are more commonly reported.

What is the true value of the advancing and receding angles?

The advancing and receding angles obtained by tilting the surface differ from those of planar surfaces[3]. Yet, studies show that horizontal the advancing and receding angles are functions of drop size. Thus there is a difficulty in determining one value of the advancing and receding angles for a given system. Here we obtain a unique value of the advancing and receding angles which is independent of drop size. We do this using a drop that is placed gently on a surface.

For very small drops the as placed contact angle approaches the advancing one and for big drops it approaches the receding and at some size it matches the equilibrium contact angle.

Drops' contact angles differ from the equilibrium contact angle due to the pinning of the three phase contact line which induces a force resisting drop motion. The force per radius associated with this pinning, k/r, is given by equation 2. k has opposite values for the advancing (kA) and receding (kR) contacts:

kA/ rA = y(cos(advancing angle) – cos(equilibrium angle)) (2a)

kR/ rR = y(cos(receding angle) – cos(equilibrium angle)) (2b)

From this, the relation between the advancing and receding angles and the equilibrium one can be calculated in reference [4].

Without hydrostatic force, the line pinning force equals the capillary force:

Line pinning force = y (cos(apparent contact angle) – cos(equilibrium contact angle)) (4)

In the case of zero hydrostatic force the maximal line pinning force resistance is:

Max. Advancing Line pin force = y (cos(advancing angle) – cos(equilibrium angle)) (5)

We now introduce gravity, then the hydrostatic pressure is tgh (t is density, h is drop height) and the capillary force is

Capillary related force = y (cos(equilibrium angle) – cos(as placed angle)) (6)

Three forces hold the drop in position when their sum equals zero:

Hydrostatic force + capillary force + line pinning (friction) force = 0 (7)

We denote hydrostatic force by tghd where d is a length proportional to the ability of the interface to hold the pressure.

Simplifying Equation 7 we get:

Tghd= y (cos(advancing angle) – cos(as placed angle)) (8)

Eq. 8 shows that the effect of hydrostatic force is to move the advancing to the as placed angle and d can be readily calculated.

We also measured the advancing and receding contact angles by tilting the substrate and found advancing(tilt) = 44.8 degrees and receding(tilt) = 30.3 degrees. We note:

(1) The receding angle obtained from the tilt stage method is within the spectrum and thus higher than what one gets by extrapolating the as placed angle to infinite drop diameters (specifically, 28.9 degrees < 30.3 degrees). (2) The advancing angle obtained from the tilt stage method is not inside the available as placed spectrum, yet, it is clearly lower than what one gets by extrapolating it to zero drop diameters (specifically, 47.2 degrees > 44.8 degrees).

These observations are surprising: the extremes obtained by placing the drops on a surface are more extreme than those obtained by tilting the surface.

The as placed angle decreases with drop size to reduce the hydrostatic pressure. This involves d.

Experimentally we observe that d is constant. The other unknown parameter in equation 9 is the advancing angle. However since d is constant the advancing contact angle can be obtained. The corresponding advancing angle obtained for the experimental system of hexadecane drop on octadecyle trimethyl ammonium surfactant covered mica surface is: 47.1 degrees.

Thus a unique advancing contact angle value, is deduced from the as placed contact angle.

References:

[1] R. Tadmor and P. S. Yadav, J Coll Interf Sci doi: 10.1016/j.jcis.2007.09.029 (2007).

[2] P. S. Yadav, D. Dupre, R. Tadmor, J. S. Park, and D. Katoshevski, Surface Science 601, 4582 (2007).

[3] B. Krasovitski and A. Marmur, Langmuir 21, 3881 (2005).

[4] R. Tadmor, Langmuir 20, 7659 (2004).

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