A Water Droplet Pinning and Heat Transfer Characteristics on an Inclined Hydrophobic Surface

A water droplet pinning on inclined hydrophobic surface is considered and the droplet heat transfer characteristics are examined. Solution crystallization of polycarbonate is carried out to create hydrophobic characteristics on the surface. The pinning state of the water droplet on the extreme inclined hydrophobic surface (0° ≤ δ ≤ 180°, δ being the inclination angle) is assessed. Heat transfer from inclined hydrophobic surface to droplet is simulated for various droplet volumes and inclination angles in line with the experimental conditions. The findings revealed that the hydrophobic surface give rise to large amount of air being trapped within texture, which generates Magdeburg like forces between the droplet meniscus and the textured surface while contributing to droplet pinning at extreme inclination angles. Two counter rotating cells are developed for inclination angle in the range of 0° < δ < 20° and 135° < δ < 180°; however, a single circulation cell is formed inside the droplet for inclination angle of 25° ≤ δ ≤ 135°. The Nusselt number remains high for the range of inclination angle of 45° ≤ δ ≤ 135°. Convection and conduction heat transfer enhances when a single and large circulation cell is formed inside the droplet.


Analysis of Magdeburg Effects
Air is trapped in between the droplet meniscus when the water droplet is located on the textured hydrophobic surface. If some texture gaps (isolated gaps) are packed (not connected with air in the other texture gaps), pressure in the isolated gaps remains different than those of texture connected gaps. This, in turn, results in sealing of air trapped from atmospheric air because of droplet meniscus. As the droplet inclines, the meniscus arc changes slightly over the texture height giving rise change of the pressure in the trapped air while causing pressure force to be acting on the droplet meniscus. In case of expansion of the trapped air, due to slight volume change during the change of the geometric position of the droplet meniscus arc, a suction pressure is generated, which in turn results in Magdeburg like forces acting on the droplet meniscus. This contributes to the adhesion of the droplet on the surface. In the case of crystallized polycarbonate surface, texture composes of some closed packed gaps. Air volume inside the closed packed gap can be approximated by a half ellipsoid. The meniscus height across a single textured packed can be shown schematically in Fig. S1. Since micro-size spherules are formed on the crystallized polycarbonate surface, the spherules are presented as round textures in Figs S1a and S1b. The droplet meniscus height prior to bending can be formulated after incorporating the horizontal force balance. Consider Fig. S1a, the vertical force balance yields: The solution of eq. 3A yields the functional relation between the droplet meniscus height (  ) and the droplet height (h d ) in terms of fluid properties and lateral distance of the droplet meniscus across two consecutive texture pillars, where air is trapped. The value of  is in the order of 2.5 nm for 1 mm radius droplet and the spacing of closely spaced two spherules of 8 m apart in the lateral direction, which is the same as 2a in Fig. S1a.
Consider the droplet meniscus prior and after inclination (Fig. S1b), the air trapped volume in a single gap can be formulated after approximating it to a half of an ellipsoid. After considering air is an ideal gas, the pressure drop in the air trapped during the geometric change of droplet meniscus, due to hydrophobic surface inclination, can be formulated as follows: Consider the equation of state for air: where P is the pressure,  is the density, R is the gas constant for air, T is the temperature within the closed packed gap.
where  is the air volume, and m is the air mass within the closed packed gap. In the differential form: Now, consider the half of an ellipsoid resembling air trapped volume, differential form of this volume yields: The pressure force generated in single packed texture gap is: The number of closed packed gaps on the crystallized polycarbonate surface is to be incorporated finding the the total pressure force acting on the droplet meniscus. Therefore, the approximate total pressure force is: where n is the number of packed texture gaps on the crystallized polycarbonate surface.
Combining Eqs. 10A and 12A results in the total pressure force, which becomes: The area ratio of closed packed gaps sites over the total are of the crystallized polycarbonate surface is assessed using the texture height landscape image. In an averaged, the area ratio is estimated as in the order of 17%.
The pressure force (Magdeburg force) variation with the change of the height of the droplet meniscus within the air trapped in the closed packed is shown in Fig. S2. The pressure force increases significantly with small change of the droplet meniscus height in the closed packed texture gap. It should be noted that the area ratio estimated from the texture height landscape for the closed packed texture gaps is incorporated in Fig. S2. In addition, the average closed packed gap height is estimated as 3.4 m from AFM data.