From superhydrophobicity to icephobicity: forces and interaction analysis

The term “icephobicity” has emerged in the literature recently. An extensive discussion took place on whether the icephobicity is related to the superhydrophobicity, and the consensus is that there is no direct correlation. Besides the parallel between the icephobicity and superhydrophobicity for water/ice repellency, there are similarities on other levels including the hydrophobic effect/hydrophobic interactions, mechanisms of protein folding and ice crystal formation. In this paper, we report how ice adhesion is different from water using force balance analysis, and why superhydrophobic surfaces are not necessary icephobic. We also present experimental data on anti-icing of various surfaces and suggest a definition of icephobicity, which is broad enough to cover a variety of situations relevant to de-icing including low adhesion strength and delayed ice crystallization and bouncing.


CA hysteresis
The receding and advancing CAs on the tilted surface are given by substituting and into equation (1), respectively (e) (f)

Force and moment balance
Considering the free body diagram of the water droplet placed on the surface, the external forces F x , F y , and F z applied in x, y, and z direction, the surface tension force of water droplet acting upon the triple line, and the reaction force of the surface to the Laplace pressure inside water droplet applied to the solid-liquid wetting area are present. One can assume that F x , F y , and F z are the gravity forces applied to the center of gravity of the water droplet, G, as , and . The gravity force has no component in y direction. is the mass of the droplet and is the acceleration of gravity. The Laplace pressure has no components in x and y directions because it is applied perpendicular to the surface, therefore it should be considered only in z direction equation of force balance. Equations of balance of the forces and the moments per unit length can be presented as is the Laplace pressure applied to the solid-liquid interface. Laplace pressure is inversely proportional to the local droplet curvature. The difference in the local curvature of the droplet induced by tilting the surface, results in change in internal Laplace pressure at each point at the solid-liquid interface. Since the droplet is in an equilibrium configuration, the forces are balanced. Substituting equation (1)  For the water droplet placed on the tilted surface, considering equations (4a), (4c) and (4d) and assuming , the balance of forces and the moment in matrix form can be given as (q)

Ice force and moment balance
To write the balance equations of forces, one has to consider the shear force applied to the solid-ice interface. Considering the shear force applied to the solid-ice interface, the balance equations of force are given as is the shear stresses between ice and the solid surface applied in y direction. The value of is zero due to symmetry. and are the coefficients which determine the phase of deposited object so that for water , and for ice , . Equations (r), (s) and (t) can be applied to both water and ice deposited on the solid surface.
The balance equation of the moment of ice about the point O at the center of the solid-ice interface (Fig. 2d) is given by (u) Equations of the balance of forces and the moment for ice placed on vertical surfaces in matrix form can be given as (v)

Methods
The most common method to measure ice adhesion to the surface is applying a compressive or tensile force resulting in shear stress on the ice confined between two surfaces. Cylindrical or rectangular ice samples can be used. In order to measure the adhesion force of ice to different materials, we used a PASCO stress/strain apparatus 750 interface equipped with an economy force sensor (PASCO CI-6746, Supplementary Fig. a). The DataStudio software was used to record and analyze the data. We used various samples as the substrates and let the water being frozen on them using a plastic cylindrical mold. The sample was placed in the apparatus and the horizontal shear force was applied to the base of the ice column through a ring set around the ice and by rotating the apparatus handle with uniform velocity until the ice was separated from the substrate (Supplementary Fig. b). The dependency of the force vs. the time of deformation (approximately proportional to the displacement) was recorded by the computer (Supplementary Fig. c).
We prepared eight samples. The metallic samples were polished with a soft cloth impregnated with 1 μm silica and then were washed and cleaned with deionized water and finally were air dried. In order to freeze water column on substrates surface, thin tubes of plastic cut from a common drinking straw with 5 mm inside diameter and 10 mm height were used as molds. The plastic molds were placed vertically on substrates surface and a Permatex black silicone sealant was gently applied to the outside surface of mold's base on substrates to prevent water leakage. The surface roughness was measured with Phase II Surface Roughness Tester SRG-4500) The molds were then filled with water using a syringe and left inside the freezing room at -20 o C until the water was entirely frozen, and then the sealant was removed.
In order to measure the force applied to the ice column, each sample was transferred separately to the other freezing room with the temperature of -5 to -1 o C where the stress/strain apparatus was located. Then the horizontal shear force was applied to the base of the ice column until it was separated from the substrate (Supplementary Fig. b).
The magnitude of the applied force was recorded by a computer located outside the freezing rooms using the DataStudio software.
In order to investigate the wetting behavior of above-mentioned samples, the static advancing and receding water CAs were measured using a ramé-hart 250 goniometer/tensiometer (Supplementary table) Supplementary figure 1. Schematic of the apparatus (a) PASCO stress/strain apparatus (b) Horizontal force applied to the ice column (c) Force versus the time of deformation / distance as recorded by the computer (these dependences essentially constitute stress-strain curves). Color lines show the applied forces to the ice on various substrates versus time

Water droplet impact tests
A hydrophobic surface was produced by coating glass with soot. The water CA with the sample was 127. The sample was kept for 5 minutes in the freezing room at -22 C. A syringe filled with the tap water at 3 C was used to drop the water on the substrate from the height of 5 cm. The volume of each droplet was about 7 µl. It was observed that the droplet did not stick to the substrate and bounced off the surface.