Equilibrium drop shapes

The equilibrium drop shape is related to the surface tension of the drop through its curvature. In other words, the tendency for the drop to assume a curved interfacial shape is due to its surface tension. A consequence of this is the existence of a pressure difference across the interface... 

The pendant drop method involves a drop of the more dense polymeric component immersed in a matrix of the lower density polymer component (both above the Tg and T ). The equilibrium drop shape is a balance between gravitational and surface forces. The interfacial tension can be determined from the drop shape, with the diameters de and ds at equilibrium from the expression ... 

The time required to conduct an interfacial tension experiment depends largely on the properties of the surfactants and less on the chosen measurement method. A notable exception is the drop volume technique, which, due to the measurement principle, requires substantial ly more time than the drop shape analysis method. Regardless of the method used, 1 day or more may be required to accurately determine, e.g., the adsorption isotherm (unit D3.s) of a protein. This is because, at low protein concentrations, it can take several hours to reach full equilibrium between proteins in the bulk phase and those at the surface due to structural rearrangement processes. This is especially important for static interfacial tension measurements (see Basic Protocol 1 and Alternate Protocols 1 and 2). If the interfacial tension is measured before the exchange of molecules... 

Sitting or pendent drop. Both methods involve the determination of the shape of the drop in mechanical equilibrium. The shape is determined by the balance between gravitation and surface tensional forces. If gravitation is negligible the shape is always spherical irrespective of the surface tension. 

Of the various other methods we mention a few of a more dynamic nature. From wave damping yiw) can in principle be obtained co is the frequency of the applied wave. See sec. 3.6g. Guido and Villone ) proposed a procedure to obtain interfacial tensions from the rate at which shear-deformed droplets retracted to their equilibrium spherical shape. De Hoog and Lekkerkerker ) determined very low interfacial tensions by following the initial state of the Rayleigh break-up of elongated drops. Although these methods are unlikely to develop into routine procedures, they demonstrate how wide the methodical spectrum is. 

The substitution of the above equation into the generalized Laplace equation (with the dependence of capillary pressure on the z coordinate accounted for) yields the Laplace equation in the differential form, the numerical integration of which leads to the exact mathematical description of the drop or bubble surface shape in the gravitational field [6,14]. The exact description of the equilibrium surface shape is of importance in the evaluation of surface tension from the experimental data at interfaces with high mobility, such as liquid-gas and liquid-liquid ones (See Chapter 1,4). 

The sessile drop method is similar to the pendant drop one [Sakai, 1965]. The same scheme is used, but in this case the droplet is resting on a plane surface immersed in the second component (see Figure 4.8) — Vj is calculated from analysis of the drop shape at equilibrium (characterized by the relative magnitude of the shape parameters, X and Z, dehned in Figure 4.8), knowing the densities of the polymeric huids at the temperature of measurement. 

To describe a liquid meniscus and hence to obtain the interfacial tension from the drop shape the Laplace equation is used as the mechanical equilibrium condition for two homogeneous fluids separated by an interface [188]. It relates the pressure difference across a curved interface to the surface tension and the curvature of the interface... 

To quantify the effects of mixed waste con sition on wettability and interfacial tension equilibrium, aqueous phase receding contact angle and interfacial tension were measured. Inter cial tension was measured ida a spinning drop tensiometer Model 500 (University of Texas, Austin, TX) and contact angles were obtained using axisymmetric drop shape analysis (17) on quartz slides. Contact angles are reported through the aqueous phase. 

Figure 8.1. Schematic representation of the advancing emulsification process of two immiscible liquids during the input of mechanical energy (e or 14 ), where the white areas within the frames represent the continuous phase and the shaded areas the dispersed phase (not to scale). The formation of locally different curvatures is clear to see. The equilibrium state (d) is characterized by the lowest average curvature and a spherical drop shape with a certain drop size distribution...

Interfacial tension measurement techniques can be divided into two categories equilibrium and transient methods [41]. The pendent-drop method is the most commonly applied method to measure interfadal tension under pressure and involves the measurement of density differences between two fluids and the equilibrium drop profile shape. In the following section, examples of interfacial tension reduction are presented for binary polymer/C02 systems and for polymer blends. 

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. 

Today, thanks to the fast development of computer enhanced imaging techniques and numerical fitting procedures, the accuracy and the sampling rate of drop shape methods are substantially increased. Thus, this technique is an important tool for the investigation of adsorption dynamics, and it is particularly suitable for studying processes with characteristic times from a few seconds up to hours and even longer. In fact, there is a large number of experimental studies in which the drop shape technique is used to evaluate the adsorption equilibrium properties, like adsorption isotherms and the dynamic surface tension behaviour. The method is also extensively utilised in the study of surfactants and proteins both in liquid/liquid and liquid/air systems. 

The drop shape method is possibly the most useful one for the investigation of the adsorptive transfer, i.e. the adsorption kinetics at the interface between two liquid phases containing the surfactant from the partition equilibrium. This phenomenon is particularly significant when situations far from the partition equilibrium are considered, in systems characterised by a high solubility of the surfactant in the recipient phase or by a large solubility of the surfactant in both phases. The latter case represents a typical situation for many types of ionic surfactants in water-oil and water-alkane systems, as demonstrated by the partition coefficients measured for various solvents [52, 53, 54, 55, 56]. 

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