Two-dimensional meniscus

One can observe that a meniscus attached to a horizontal plate spontaneously falls off at a certain critical height of the plate. On the other hand, if the plate is immersed into a liquid bath, the liquid spontaneously spreads and wets the entire plate at a critical depth. In this section, we first discuss the unstable phenomenon of a two-dimensional meniscus under a horizontal plate from a thermodynamic viewpoint based on Eq. (10) above. Then, in order to verify... 

The energy of the system illustrated in Fig. 4 is calculated when the two-dimensional meniscus attaches to the horizontal plate at macroscopic contact angle 6. S, V, and L in the figure indicate the solid, vapor, and liquid phases, respectively. The coordinates x and z are taken to be the horizontal and vertical directions, respectively. The geometry of the meniscus can be obtained from the solution of Laplace equation (2). The radius of curvature is estimated from differential geometry. Considering the static pressure due to gravitation as AP, Eq. (2) is rewritten for the two-dimensional case as [2]... 

The contact line does not move since the energy increases in the directions of both advance and retreat. This means that the two-dimensional meniscus is stable for macroscopic contact angles between 0a and 0r. 

The above equation can be derived by the differentiation of Eq. (21) and (dO/ d5) = 0 (the shape of a two-dimensional meniscus does not change with S, as seen in Fig. 4). As shown in Fig. 6, the system becomes neutral stable when... 

FIG. 7 Two-dimensional meniscus attached to a horizontal plate with small sawtooth roughness microscopic view of Fig. 4. 

Eick et al. [8] analyzed the wetting behavior of a two-dimensional meniscus attached to a vertical plate with roughness similar to that depicted in Fig. 7. They suggested the following relations for the macroscopically observed contact angles. 

It seems possible to measure contact angles by applying the thermodynamic instability of a two-dimensional meniscus, as stated above. However, there are many metastable positions on the horizontal plate as shown in Fig. 11,... 

TABLE 1 Wetting Behavior of Two-Dimensional Meniscus Under a Horizontal Plate The Behavior is Analyzed from a Microscopic Viewpoint... 

Fig. 17b, which is different from the two-dimensional meniscus under a horizontal plate where there are many metastable positions on the plate surface. This could make it easy to recognize when instability occurs. It may be possible to obtain 0r from the measured critical height of 77b-... 

III. METHOD OF APPLYING GEOMETRICAL INSTABILITY OF THE TWO-DIMENSIONAL MENISCUS... 

Fig. 23 schematically shows the principle of the method of using a circular cylinder as the test solid. S, L, and V in the figure indicate solid, liquid, and vapor phases, respectively. The cylinder held horizontally is first immersed and then slowly drawn from the liquid bath. We can see a pair of two-dimensional menisci formed under the cylinder, as shown in Fig. 23a. As the cylinder is raised to a certain critical height, the waists of the two meniscus curves contact each other and the liquid breaks off from the solid surface. The geometry of the two-dimensional meniscus can be determined from the Laplace equation (11) and the contact angle as a boundary condition, as mentioned in Section I. Hence we could calculate the contact angle using the...

FIG. 23 Principle of the method for the measurement of contact angle using circular cylinder based on geometrical instability of two-dimensional meniscus (a) receding contact angle (b) advancing contact angle. 

FIG. 25 Schematic of two-dimensional meniscus attached to a circular cylinder. 

---