Retention forces and contact angles for critical liquid drops on non-horizontal surfaces
Graphical abstract
Predictions of retentive forces, based on a force-balance model are compared to measurements from the literature [C.W. Extrand, Y. Kumagai, J. Colloid Interface Sci. 170 (1995) 515] for critical drops on several surfaces. The proposed (Eq. (13)) fits the data with ().
Introduction
Liquid drops can adhere to vertical or inclined surfaces by surface tension. The retention of liquid drops on non-horizontal surfaces is manifested in applications like spraying of paint or pesticide, or condensation of vapor on fins. The geometry of a retained drop influences other physical quantities such as its volume, the forces acting on it, and the heat transferred through it. Of particular significance are static drops on the verge of sliding on inclined surfaces. The retention of critical-size drops is the focus of this study.
One of the important quantities in the study of critical drops on inclined surfaces is the force retaining a drop at given conditions. Understanding the retention forces can lead to better control of the maximum sizes of retained drops. Extrand and co-workers [1], [2] measured retentive forces for different liquid–surface combinations and related them to drop geometry. The surface-tension force, , holding a critical drop can be related to the contact angles by where k is a constant, γ is the liquid–vapor surface tension, R is a length scale representing the size of the drop contour, and and are the advancing and receding contact angles, respectively. The factor k depends on the geometry of the drop. Disagreements in the literature regarding contact-angle distributions in drops and the shapes of their contours have resulted in predictions of k that differ by more than 300%, as will be explained later.
Previous studies of drops on non-horizontal surfaces have also disagreed on the values of contact angles at critical conditions. (In this work, we will refer to the maximum and minimum contact angles in a drop at general conditions as and , respectively. The advancing and receding contact angles, and respectively, are those characteristic of the surface–liquid combination.) Brown et al. [3] and Milinazzo and Shinbrot [4] calculated critical contact angles based on 3-dimensional drop geometry and allowed them to take any values between 0 and 180°, without considering limitations imposed by the type of liquid–surface combination. Both studies predicted and of a drop at incipient motion to depend on the size of the drop and the inclination angle of the surface. On the other hand, the experiments of MacDougall and Ockrent [5] showed critical contact angles to be almost constant for a given liquid–surface combination. In some cases, measurements of on inclined plates slightly exceeded of the horizontal surface. Recently, Krasovitski and Marmur [6] presented a theoretical analysis to show that and of a drop on an inclined plate are not always equal to and , respectively, of the liquid–surface. They imposed theoretical heterogeneity on the surface and calculated contact angles based on 2-dimensional drop geometry. The results indicated that can be lower than for hydrophobic surfaces, whereas can be higher than for hydrophilic surfaces. However, it is not clear the analysis presented by Krasovitski and Marmur [6] should be interpreted to apply at critical conditions. The maximum and minimum contact angles, and , of drops at critical conditions will be explored in this work.
The maximum drop volume, , is another important variable for drop retention on solid surfaces. Reliable prediction of depends on knowing the geometry of the drop and the forces acting on it at critical conditions. Dussan [7] derived an equation to approximate the maximum volume for the case of small drops with small contact angle hysteresis, (), where ρ is the liquid density, and α is the surface inclination angle. Goodwin et al. [8] tested static drops on rotating surfaces and measured critical rotational speeds. They reported good agreement between their rotational data and the model of Dussan and Chow [9] for small drops and small contact angels. Briscoe and Galvin [10] measured the sizes of critical drops on inclined surfaces and found Dussan's equation to under-estimate their data by less than 23%, with better agreement at small inclination angles.
In two recent articles on the shapes of drops under general conditions on vertical and inclined surfaces, we described an experimental study of drop profiles, contours, and contact angles for a wide range of drop sizes and surface conditions [11] and a method for approximating volumes of drops on non-horizontal surfaces [12]. In this new article, we build on our earlier results to study retention forces and drop parameters at critical conditions. A critical condition for a drop is reached when the gravitational force overtakes surface tension—the point of incipient motion. Such a condition can be caused by an increase in drop size or surface inclination. In this manuscript we examine the contact angles and forces at critical conditions and consider the relations between them.
Section snippets
Apparatus
The experimental setup was explained in detail in a previous paper [11]. A camera and microscope are connected to an arm and rotated to record profile images of drops at various azimuthal angles. The profiles are analyzed to obtain the contact angle variation along the circumference of the drop. The drop profile is also recorded. Several surfaces and liquids were used to test drops of various sizes at different surface-inclination angles.
Contact angles
Measurements of contact angles for drops of different sizes on surfaces with various inclinations were presented earlier [11] for general conditions. This study focuses on drops at critical conditions. The data, which covered a range of 49° to 112°, showed that the ratio of the maximum contact angle in a drop to the advancing angle of the surface () is almost equal to unity for all tested values of the Bond number, Bo. The Bond number is defined as where ρ is the liquid
Conclusions
The parameters of critical-size drops resting on vertical and inclined surfaces have been studied. Measurements indicated that the minimum contact angle of a drop at incipient motion is the characteristic receding angle of the liquid–surface combination. Although this conclusion is empirical, it is confirmed by further analysis and by data from other investigators. A reliable prediction of the surface-tension force holding a critical drop is provided based on observations of geometric
Acknowledgement
The authors acknowledge the support of the Air Conditioning and Refrigeration Center, ACRC, of the University of Illinois at Urbana-Champaign.
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