Axisymmetric meniscus

In most cases a mechanical approach is more adequate. This is based on the concept that in mechanical equilibrium at each point on the interface, the curvature is adjusted such that the difference in the pressures between the two phases is balanced by the capillary pressure. This approach is particularly fruitful when applied to axisymmetric menisci, like drops and bubbles. In this ease, assuming a Cartesian coordinate system with the origin at the drop apex, O, and the vertical axes, z, in the symmetry axis and directed towards the interior of the drop, at any point, S, of the interface we have... 

It is usually called axisymmetric drop shape analysis.The interfacial tension and contact angles are determined from the shape of the axisymmetric menisci of both sessile and pendant drops. The employed strategy is to fit the shape of an experimental drop to the theoretical drop profile according to the Laplace equation, using surface tension as adjustable parameter. Details of the methodology together with a program to implement it can be found elsewhere. ... 

The technique of Axisymmetric Drop Shape Analysis-Profile ADSA-P) [10,11] was used for image analysis and experimental parameters. Surface or interfacial tensions were obtained by fitting the Laplace equation of capillarity from the shape and dimensions of the acquired axisymmetric menisci [12]. The value of surface tension was generated as a fitting parameter after a least square algorithm was employed to minimize the difference between experimental drop profiles and theoretical ones [13]. During this procedure, the density difference between polystyrene and carbon dioxide was an input parameter [14, 15, 16], which was determined by the Sanchez and Lacombe (S-L) equation of state (EOS). 

In the special case of spherical interface H= HR, with R the sphere radius, and Eqnation 5.95 takes its most popular form, 2o// = AP. In the case of axisymmetric meniscus (z axis of symmetry. Figure 5.7) the Laplace equation rednces to either of the following two equivalent forms " ... 

In this section, let us eonsider the macroscopic wetting behavior of an axisymmetric meniscus from a thermodynamic viewpoint and discuss the possibility of the measurement of the contact angle and interfacial tension [61]. As in the analysis stated above, the theoretical consideration is based on the assumption described by Eq. (9). 

The axisymmetric meniscus under a conical surface is chosen as the subject, as shown in Fig. 12. The cylindrical coordinates r and z are taken to be the radial and horizontal directions, respectively. If some relations of differential geometry are inserted into the radii of curvature in Laplace equation (2), the profile of the axisymmetric meniscus can be determined by the following differential equation [31,62]. 

FIG. 12 Schematic of axisymmetric meniscus attached to a downward cone surface. 

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. 

FIG. 14 System energy change with (a) the apparent contact angle 6 and (b) non-dimensional meniscus height H, when axisymmetric meniscus attaches to a circular cylinder. The system is stable at states C and A for advance and retreat of meniscus, respectively. 

A similar result can be obtained for the axisymmetric meniscus attached to an upward cone, as shown in Fig. 18. The coordinate z is taken to be the downward direction, while the other variables are the same as in Fig. 12. indicates the depth of the cone vertex. The system energy, instead of E-, can be calculated in the same manner as for the downward cone. Here we consider the advance of the three-phase contact line. If we take the dry surface as a... 

As stated above, we theoretically discussed the unstable wetting behavior of an axisymmetric meniscus attached to a cone surface. It was suggested that... 

Transitions from steady-state to time-dependent surface-tension-driven motions are well known also and are important in meniscus-defined crystal growth systems. For example, the experiments of Preisser et al. (51) indicate the development of an azimuthal traveling wave on the axisymmetric base flow in a small-scale floating zone. 

The second justification for the angular condition is that this condition is necessary for the determination of the radius of the crystal at the trijunction as a function of heat-transfer conditions and pull rate. This argument is simple. The dimensionless Young-Laplace equation of capillary statics gives the shape of an axisymmetric melt-ambient meniscus as... 

For axisymmetrical Interfaces, as in cylindrical capillaries, pendent and sessile drops, this is often achieved by introducing the radius of curvature b at the apex or lowest point as the measure of the meniscus size, anticipated in table 1.1. At that point = b, hence J = 2/b and at the top Ap = 2y/b. If z is now... 

G, Mason N.R. Morrow, Meniscus configurations and curvatures in non-axisymmetric pores of... 

Similar calculations can be done for families of unbounded sessile profiles. An unbounded sessile profile can be visualized physically as the configuration of an axisymmetric dry patch, or that of a meniscus around a cylindrical rod with its axis perpendicular... 

The methods used in this study are mainly the drop and bubble shape technique and the torsion pendulum rheometry. The drop and bubble shape technique, which is based on fitting the shape of an axisymmetric liquid meniscus to the respective Gauss Laplace equation, allows to continuously monitor the... 

The role of the Maxwell pressure residting from a normal gas phase interfacial electric field that scales as /R in elongating the Uquid meniscus into a cylindrical microjet stracture can also be verified through a dynamic simulation in which the equations governing the coupled interactions between the hydrodynamics (Eqs. 1-3) and electrodynamics (Eq. 9) are solved simultaneously for a constant potential liquid meniscus in the longwave limit in axisymmetric polar coordinates (r,0,z), subject to the boundary conditions... 

Interfacial Electrokinetic Flow, Fig. 4 Spatiotemporal evolution profiles of the electrospray meniscus height R showing the initial stages of microjet formation obtained through an axisymmetric longwave model [11]... 

A similar axisymmetric model to that described above, assuming the jet to be slender such that the longwave approximation holds (Ro < L, in which Rq is the initial meniscus height and L the characteristic length scale of... 

For polymers interfacial and surface tensions are more practically obtainable from analysing the shapes of pendant or sessile drops or bubbles, all of which are examples of axisymmetrical drops. Bubbles may be used to obtain surface tensions at liquid/vapour interfaces over a range of temperatures and for vapours other than air. Drops can also be used to obtain vapour/liquid surface tensions but they are particularly suited to determination of liquid/liquid interfacial tensions, for example for polymer/polymer interfaces. All the methods are based on the application of equation (2.2.1). The principles are illustrated in figure 2.4, in which a sessile drop is used as the specific example. Just like for the capillary meniscus, the drop has two principal radii of curvature, R in the plane of the axis of symmetry and / 2 normal to the plane of the paper. At the apex, O, the drop is spherically symmetrical and R = Rz = b and equation (2.2.12) becomes... 

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