Meniscus profile

Derive Eq. 11-14 from an exact analysis of the meniscus profile. Hint Start with Eq. II-12 and let p = y", where y" = pdpjdy. The total weight W is then given by W = lApgirj xydx. 

Equation 11-30 may be integrated to obtain the profile of a meniscus against a vertical plate the integrated form is given in Ref. 53. Calculate the meniscus profile for water at 20°C for (a) the case where water wets the plate and (b) the case where the contact angle is 40°. For (b) obtain from your plot the value of h, and compare with that calculated from Eq. 11-28. [Hint Obtain from 11-15.]... 

As an extension of Problem 11, integrate a second time to obtain the equation for the meniscus profile in the Neumann method. Plot this profile as y/a versus x/a, where y is the vertical elevation of a point on the meniscus (above the flat liquid surface), x is the distance of the point from the slide, and a is the capillary constant. (All meniscus profiles, regardless of contact angle, can be located on this plot.)... 

Effect of Micelle Volume Fraction on Meniscus Profile. 133... 

A much easier method has been developed by Padday et al. (1975) (Faraday Trans. I, 71, 1919), which only requires measurement of the maximum force or weight on the rod as it is pulled upwards. It has been shown by using the Laplace equation to generate meniscus profiles that this maximum is stable and quite separate from the critical pull-off force where the meniscus ruptures. A typical force-height curve is shown in Figure 2.24. 

Tallmadge, J. A. Lee, C. Y. Dynamic Meniscus Profiles in Free Coating III. Predictions Based on Two Dimensional Flow Fields. AIChE J. 1974, 20, 1079-1086. 

Therefore, by extrapolating the meniscus profile (in the meniscus, far from the film, the condition (3.39) holds) to its intersection with the extrapolated film surface at z = z = hi2, the macroscopic contact angle 6h is defined and... 

Obviously, Qh (x = r) is obtained again by an extrapolation of the meniscus profile at Pcl - const to the intersection with the extrapolated surface of the plane-parallel film. 

Padday et al, ) analyzed the meniscus profile for a vertical cylindrical rod, either standing in the fluid, or being moved upward without detachment. From the pull on the rod the surface tension cem be obtained. See also ref. 

The rise h of a liquid inside a partially wetted capillary (a < 90°), or the depression a > 90°) is related to ycosa, so that from h the contact angle can be obtained if y and the meniscus profile are known. In sec. 1.3a this method was used the other way around, i.e. to obtain y if a is known. Usually fully wetted cylinders are then used so that the contact angle does not enter the equations. What was said there about the profile remains applicable. This also applies to the deviations in the Laplace profile, incurred as a result of disjoining pressure 1). 

It may be noted that for the measurement of y narrow capillaries are preferred, whereas for the determination of a, wide capillaries have advantages in that the meniscus profile is more easily accessible optically. Assessing contact angle hysteresis is no problem, because measurements can be done with rising amd falling menisci. A problem with solutions of surfactants is that if upon capillary rise depletion takes place, which gives rise to a slow step in the establishment of the meniscus profile. 

The superscripts real and idealized in Equation 5.153 mean that the quantities in the respective parentheses must be calculated for the real and idealized meniscus profiles the latter coincide for r > Tg (Figure 5.16). Results for k and x calculated by means of Equations 5.153 and 5.154 can be found in Reference 180. 

In conclusion, it should be noted that the width of the transition region between a thin liquid film and Plateau border is usually very small — below 1 pm. That is why the optical measurements of the meniscus profile give information about the thickness of the Plateau border in the region r > (Figure 5.16). Then, if the data are processed by means of the Laplace equation (Equation 5.101), one determines the contact angle, a, as discussed above. Despite that it is a purely macroscopic quantity, a characterizes the magnitude of the surface forces inside the thin liquid film, as implied by Equation 5.148. This has been pointed out by Derjaguin and Princen and Mason. ... 

If the meniscus region is examined, it is seen that h as y 0. Consequently the average value of the velocity there is approximately Qlh, which is very small, as Q is a constant and A - < . Thus dynamic effects are very small and the profile is very close to that of a static meniscus. The method of determining static meniscus profiles is given in Chapter 1. Starting from the solution of L as a function of X (Landau and Lifshitz, 1959), one obtains... 

The system energy of Fig. 12 can be estimated by using the above solution for the meniscus profile. As stated in the previous section, we consider the potential energy E, the energy of the liquid vapor interfacial area Elv, and the work done by the three-phase contact line wetting the cone surface E f/. The dry cone surface and the horizontal liquid surface (z = 0) are taken for the reference state of system energy. E and which represent the work necessary to form the axisymmetric meniscus shown in Fig. 12, are calculated from... 

FIG. 13 Comparison of axisymmetric meniscus profile obtained numerically with that measured from a photograph. 

III. DIRECT CONTACT ANGLE MEASUREMENTS FROM THE MENISCUS PROFILES OF DROP-ON-FIBER SYSTEMS... 

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