ABSTRACT
As mentioned above, it is, in principle, possible to obtain contact angles by measuring the critical height or depth at which the geometrical instability of a two-dimensional meniscus occurs. Here the relationship between the critical height of cylinder or plate and the contact angles is discussed theoretically. First, let us consider the circular cylinder shown in Fig. 25. In the figure, the x-and z-axes are taken to be the direction of the stationary liquid surface and the vertical direction through the cylinder center, respectively. The height of the cylinder bottom HB from the stationary liquid surface shown in Fig. 25 is used as the measuring height of the cylinder. In this section, we use the nondimensional solution of the Laplace equation, i.e., Eqs. (14) and (15), which determines the geometry of the two-dimensional meniscus. The unknown integral constant C in Eq. (14) should be determined from the boundary condition. Since the meniscus is in contact with the cylinder of radius R at the receding contact angle θR, as shown in Fig. 25, Eq. (14) should satisfy the boundary condition
where x0 and indicate the coordinate at the attachment point of the meniscus and the angle between the z-axis and the cylinder radius at the attachment point, respectively, as seen in Fig. 25. ψ is the inclination of the
meniscus curve from the x-axis. Using the capillary constant defined by Eq. (13), the above boundary condition is rewritten in the nondimensional form as
Equation (14) can be rewritten as below if the constant C is calculated from the boundary condition.
