Meniscus, shape

Both methods described in the previous section require the determination of the meniscus shape. We first describe the case of a finite meniscus bridging two surfaces and then the case of an unbounded meniscus obtained when dipping a tip in a liquid bath. 

Given the assumptions mentioned before, the pressure inside the meniscus is constant and related to the meniscus curvature by the Laplace pressure. In the axisymmetric situation, the interface shape r(z) thus verifies  

In order to facilitate the calculation of capillary forces, several approximations on the meniscus shape have been proposed. They are mainly applied for experimental conditions where the radius of curvature of the meniscus interface is much smaller than the radius of curvature of the solid surfaces. This is relevant for the surface force apparatus where the surface has centimetric radius, while the meniscus is typically tens of hundreds of nanometers. The most used approximation is the toroidal approximation assuming the liquid interface has a circular profile. Obviously, such a meniscus does not exhibit a constant curvature. Nevertheless, this approximation gave good results, in particular for small contact angles, and is therefore widespread (see Ref. 15 for its application in various geometries and section 9.3.1.1 for an example of its application in atomic force microscopy [AFM]). In the case of capillary condensation between a plane and a sphere with a large radius of curvature R, in contact, the tension term of the capillary force is negligible and the Laplace term leads to the simple formula F = AnRy cos 9 A parabolic profile is also sometimes used to eliminate some numerical difficulties inherent in circle approximation. 

For a liquid bridge with a given volume, the resolution of the meniscus shape is not trivial. A numerical resolution was proposed using a double-iterative method. It can handle any surface shapes and boundary conditions and compute the capillary force from the geometrical method.  

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The flow in a heated capillary depends on a number of parameters including the channel geometry, physical properties of the liquid and the heat flux. An immediate consequence of the liquid heating and evaporation is convective motion of both phases. The latter leads to a velocity and temperature field fransformation and a change in fhe meniscus shape. 

The first term on the right-hand side is due to the hydrostatic pressure (or suction) acting over the base of the rod and the second term is due to surface tension forces around the perimeter. (It should be noted that O is not equal to the equilibrium contact angle 0 but is determined by the meniscus shape.)... 

A variety of drop, bubble, and meniscus shapes have axial symmetry. As is the case in the capillary in Figure 6.3b, p varies with z and, in general, the two radii of curvature may vary from position to position on the surface also. With these ideas in mind, the Laplace equation becomes... 

Meniscus Shape. Figure 6 shows a schematic representation of the inner portion of a CZ furnace fonned by the melt, crystal, and crucible. The crystal is attached to the melt by the melt-ambient meniscus denoted by dDm, If the traction caused by hydrodynamic forces is neglected (a good... 

The dynamics of the Czochralski system can be described only by heat-transfer models that include the interaction of the shape of the meniscus, which are referred to as thermal-capillary models (TCM), because only these models give self-consistent determination of the meniscus shape, crystal radius, and heat transfer in each phase. 

Capillary Instability. The meniscus shape may become unstable in the sense that small perturbations to it cause the melt to separate from the... 

Copies of Boris Cahan s thesis were circulated among many working in the fundamentals of the field in the 1970s. The thesis gave its readers access to the details of the elegant experimental studies it reported on the interplay between meniscus shapes and the thickness of the boundary layer, i0, electrolyte conductivity, and the local heat developed in porous electrodes used in fuel cells (and which tends to dry out thin menisci). 

CHANGE OF MENISCUS SHAPE DURING IMMERSION AND EMERSION PROCESSES... 

Figure 26.10 A schematic representation of the meniscus shape and position of the three-phase contact line (solid/liquid/air) during immersion and emersion of a hydrophobic surface (e.g., TMS treated polymers) the dual arrows indicate which direction the beaker is moving, the small arrow on the plate indicates the direction the three-phase contact line is moving.

It should be noted that on the receding cycle the wet plate surface has previously interacted with water molecules for a different period of time depending on the immersion depth of the plate. Therefore, the bottom deeper immersed portions of the plate interact with the water molecules for a longer period than the shallow immersed portions closer to the top of the plate. This causes small but continuous changes in the meniscus shape even after the three-phase contact line starts to move in the advancing and receding processes. 

Dynamic hysteresis is caused largely by the change of meniscus shape during a transition stage. Therefore, a hydrophobic surface shows the larger separation of the immersion line and the emersion line than a hydrophilic surface as seen in Figure 26.16, which depicts dynamic hysteresis for untreated, TMS-treated, and (TMS + 02)-treated polymer films. However, dynamic hysteresis is probably not maximal with TMS treatment. 

Figure 1.7. Computation of the weight of the risen column for an arbitrary meniscus shape.

In addition, the required contact angles have also been obtained from the meniscus shape, from flotation and using a Wllhelmy thread technique. Bascom ) reviewed these methods and gave a number of results. Gu and Li ) extended their method for vertical cylinders, quoted in sec. 5.4f, to fibers crossing an oil-water interface. 

The existing theories for explaining criticalities of capillary condensation and hysteresis can be classified as meniscus theory based on a single idealized pore considering the difference in meniscus shape [1] networking, pore blocking... 

Here b is the radius of curvature at the particle apex, where the two principal curvatures are equal (e.g., the bottom of the bubble in Figure 5.7a). Unfortunately, Equation 5.99, along with Equation 5.106, has no closed analytical solution. The meniscus shape can be exactly determined by numerical integration of Equation 5.102. Alternatively, various approximate expressions are available. Eor example, if the meniscus slope is small, z 1, Equation 5.99 reduces to... 

The precision of surface tension measurements using the capillary rise method can be further increased if the deviation of the meniscus shape from the spherical is taken into account. This correction is especially important when capillaries of large radii are used. Corrections for non-spherical meniscus curvature are based on tabulated numerical solutions of the differential Laplace equation [6]. The capillary rise method yields a values with a precision of up to hundredths of mN/m. 

We conclude this discussion by alerting the reader to the concept of the dynamic contact angle (and line), which appears in the literature of flows governed by surface tension (Dussan V. 1979). In a flow field where the contact line moves, it is necessary to know the contact angle as a boundary condition for determining the meniscus shape. If this angle is a function of the speed of the contact line relative to the solid surface, then the force balance inherent in... 

In a narrow tube with a circular cross section, the shape of the liquid meniscus approximates a spherical segment with constant radius of curvature R = a/cos 0, where 0 is a static contact angle. The deviation of the meniscus shape from a sphere can be caused by the influence of gravity. The ratio of hydrostatic gravitational force to the surface tension force is characterized by a dimensionless pa-... 

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