Shape of a pendant drop

This technique is based on the determination of the shape of a pendant drop that is formed at the tip of a capillary. The classical form of the Young and Laplace equation relates the pressure drop (Ap) across an interface at a given point to the two principal radii of curvature, r( and r2, and the interfacial tension (Freud and Harkins, 1929) ... 

An alternative approach to obtaining the liquid-vapour or liquid-liquid interfacial tension is based on the shape of a pendant drop. In essence, the shape of a drop is determined by a combination of surface tension and gravity effects. Surface forces tend to make drops spherical whereas gravity tends to elongate a pendant drop. Fig. 4.14 shows the schematic of a pendent drop set-up and an example of the images one gets (see for details [186, 187]). 

Interfacial tension was measured using an axisymmetric drop shape analysis (ADSA) method [24]. In this method, interfacial tension is obtained by analyzing the change in the shape of a pendant drop of one liquid suspended in a second liqnid. The method is based on the Bashforth-Adams equation, which relates drop shape geometry to interfacial tension [25,26]. A schematic of a pendant drop with the appropriate dimensions for nse in the Bashforth-Adams equation is illustrated in fig. 13.6. 

Najmabadi, M., Tamm, F., Kiaiber, M., Baroud, Y., Drusch, S., Simon, S. (2013). Realtime determination of interfacial tension from the shape of a pendant drop based on embedded image processing. In 25th European Conference on Liquid Atomization Spray Systems, Chania, Greece, 2013. 

Shape of the liquid drop (Pendant drop method) The liquid drop forms as it flows through a tubing (Figure 2.11). At a stage just before it breaks off, the shape of the pendant drop has been used to estimate y. The drop shape is photographed and, from the diameter of the shape, y can be accurately determined. 

The pendant drop technique measures the shape of a liquid drop suspended from the tip of a capillary needle. The drop is optically observed and the surface tension is calculated from the shape of the drop. This method is not as precise as the force measurement method because it depends on the eye of the operator or the sophistication of detection hardware and analysis software. 

It should be noted diat the detachment of a pendant drop causes a disturbance at die bottom of die tube, which generates waves on die film above and to die side. At die moment die bridge breaks, the liquid remaining attached to the tube is typically shaped like a stretched triangle. The surface tension forces at the tip of diis shape furthest from the tube are very high and cause fast recoil of die liquid, which in turn leads to ripples diat propagate up the tube. These waves can disturb die formation of neighboring droplets and cause some side-to-side motion of droplet formation sites. 

In some cases, the amount of liquid available for surface tension measurement is very small, such as fluid from the eye, etc. Under these conditions, one finds that the following procedure is most suitable for the measurement of y. The liquid drop forms as it flows through a tubing. Figure 1.19. At a stage just before it breaks off, the shape of the pendant drop has been used to estimate y. The drop shape is photographed, and from the diameters of the shape, one can accurately determine y. Actually, if one has only a drop of fluid, then one can measure its y without the loss of sample volume (as in the case of eye fluid, etc.). 

The surface tension of polystyrene in supercritical carbon dioxide is determined experimentally by Axisymmetric Drop Shape Analysis-Profile (ADSA-PX where a high pressure and temperature cell is designed and constructed to facilitate the formation of a pendant drop of polystyrene melt. As pressures and temperatures increase, the surface tension of polystyrene decreases. A linear relationship is found between surface tension and temperature, and between surface tension and pressure. The slope of surface tension change with temperature is dependent on pressure. 

The usual experimental situation is that of a sessile drop and, as with the pendant drop, it is necessary to determine a shape parameter and some absolute length. Thus /3 may be determined by profile fitting, and Ze measured, where Ze is the distance from the plane at = 90 to the apex. If the drop rests with... 

Recently, the size and shape of a liquid droplet at the molten tip of an arc electrode have been studied,12151 and an iterative method for the shape of static drops has been proposed. 216 Shapes, stabilities and oscillations of pendant droplets in an electric field have also been addressed in some investigations. 217 218 The pendant drop process has found applications in determining surface tensions of molten substances. 152 However, the liquid dripping process is not an effective means for those practical applications that necessitate high liquid flow rates and fine droplets (typically 1-300 pm). For such fine droplets, gravitational forces become negligible in the droplet formation mechanism. 

The shape of a drop forming slowly at a submerged orifice is the basis for the hanging-drop (pendant-drop) method for determining inter-... 

Recently, surfactant adsorption and y have been measured at C02-water and C02-organic interfaces with a tandem variable-volume tensiometer (Harrison, 1996). A pendant drop of an aqueous or organic phase, saturated with C02, may be suspended in C02 or a C02-surfactant mixture and equilibrated. From the digitized droplet shape and density difference between the phases, y may be calculated from the Laplace equation. In Figure 8.1, y of the binary C02-water (da Rocha et al., 1999), -polyethylene... 

Figure 6.2 Liquid surface tension determination by the drop shape method a. A pendant drop is formed by suspending the liquid from the tip of a thin tube. b. A sessile air (or vapor) bubble is formed in a liquid by injecting the gas from the tip of a needle connected to a syringe.

A liquid interface is the first requirement for all the many forms of capillarity. At least one phase must be sufficiently fluid. The shape of a liquid with an interface to air or another liquid is determined by the surface or interface tension. The shape of a liquid surface or interface at rest changes only when the surface tension, forces of gravity, and in some cases the electric field forces (Lorentz forces) are altered, provided the solid boundaries have constant dimensions and certain angles. The shape of pendant drops, sessile drops on a solid substrate, a meniscus against a solid wall and the length of so-called surface waves are well-known examples of capillarity. These phenomena need separate examination of liquid interfaces, as the surface state between two phases cannot be deduced from their bulk properties. 

Pendant Drop Method A method for determining surface or interfacial tension based on measuring the shape of a droplet hanging from the tip of a capillary (in interfacial tension the droplet may alternatively hang upward from the tip of an inverted capillary). Also termed the hanging drop (or bubble) method. 

In recent years, several theoretical and experimental attempts have been performed to develop methods based on oscillations of supported drops or bubbles. For example, Tian et al. used quadrupole shape oscillations in order to estimate the equilibrium surface tension, Gibbs elasticity, and surface dilational viscosity [203]. Pratt and Thoraval [204] used a pulsed drop rheometer for measurements of the interfacial tension relaxation process of some oil soluble surfactants. The pulsed drop rheometer is based on an instantaneous expansion of a pendant water drop formed at the tip of a capillary in oil. After perturbation an interfacial relaxation sets in. The interfacial pressure decay is followed as a function of time. The oscillating bubble system uses oscillations of a bubble formed at the tip of a capillary. The amplitudes of the bubble area and pressure oscillations are measured to determine the dilational elasticity while the frequency dependence of the phase shift yields the exchange of matter mechanism at the bubble surface [205,206]. 

Pendant or Sessile Drop Method The surface tension can be easily measured by analyzing the shape of a drop. This is often done by optical means. Assuming that the drop is axially symmetric and in equilibrium (no viscous and inertial effects), the only effective forces are gravity and surface or interfacial forces. In this case, the Young-Laplace equation relates the shape of the droplet to the pressure jump across the interface. Surface tension is, then, measured by fitting the drop shape to the Young-Laplace equation. Either a pendant or a sessile drop can be used for surface tension measurement. The pendant drop approach is often more accurate than the sessile drop approach since it is easier to satisfy the axisymmetric assumption. Similar techniques can be used for measuring surface tension in a bubble. 

Modern methods of measuring the surface tension include the pendant drop method, the sessile drop method, and others (7,8,15). These methods depend on the shape of a drop of the polymer or a bubble in it, and on the balance of surface tension and gravitational forces see Figure 12.3 (8). 

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 differential equation, Eq. (5.9), can be applied to the solution of such problems as the equilibrium configuration of a liquid drop (Fig. 5.2(a)) a pendant drop (Fig. 5.2(b)) and the shape of the meniscus formed in a capillary tube (Fig. 5.2(c)). All of these problems have an excess pressure across the boundary surface of the fluid that is non zero. The excess pressure in these examples is due to the gravitational field. For example in Fig. 5.2(c) this is produced by the height of fluid above the general level of the fluid in the bath. So the excess pressure, p, will typically be of the form,... 

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