ABSTRACT
Equation (12) can be solved analytically. Using the inclination angle of the meniscus ψ shown in Fig. 4, the meniscus geometry can be determined by [2]
(14)
(15)
where C is the integral constant. The infinitesimal energy change is considered when the three-phase contact line shifts
by an apparent length ∆S from D to D1, as seen in Fig. 4. It is noted that ∆S includes roughness or heterogeneity in itself. The contributions to the energy increment are classified into the following three items [8, 17]: (1) potential energy of the meniscus ∆EP, (2) work necessary to increase the liquid-vapor interfacial area ∆ELV, and (3) energy change due to contact line movement ∆EW. Before the energy of the system in Fig. 4 is discussed, let us calculate energies (1) and (2) for the meniscus attached to an inclined plate, as shown in Fig. 5, for the purpose of generality. Both can be obtained from
(16)
(17)
where and H indicate the angle of plate inclination and the attachment height of the meniscus, as shown in Fig. 5, respectively. EP and ELV are the potential and liquid-vapor interface energy of the meniscus as a whole, respectively. In the above equations, the
horizontal liquid surface is taken as a reference state of energy EP and ELV. Referring to Li and Neumann [17],
the above calculation can be easily obtained by the use of meniscus geometry, i.e., Eqs. (14) and (15). The sum of Eqs. (16) and (17) is written as
∆EP+∆ELV−σLV cos θ∆S. (18)
Equation (18) can also be applied to the meniscus under a horizontal plate shown in Fig. 4. It is noted that in Eq. (18), we neglect the contribution from the liquid in a trough of small roughness with higher order than ∆S. The energy ∆EW is dependent on the direction of contact line movement (i.e., the sign of ∆S), as stated by Eq. (10). The values of ∆EW for liquid advancement and retreat are again written here as
∆EW=−σLVcos θA∆S (advancing, ∆S>0) (19a)
∆EW=−σLVcos θR ∆S (receding, ∆S<0). (19b)
The total energy change of the system, ∆E=∆EP+∆ELV+∆EW,
(20)
can be obtained from Eqs. (18), (19a), and (19b), and (dE/dS) is written as
(21a)
(21b)
Now the behavior of meniscus as shown in Fig. 4 is discussed using Eqs. (21a) and (21b). The meniscus height shown in Fig. 4 is obtained from Eq. (15) as
(22)
First, we discuss the case of meniscus height i.e., θ>θA. The following relation can be written for the differential coefficient of energy from Eq. (21a).
