Axisymmetric drops

In this case, expression for the excess free energy takes the following form  

FIGURE 2.17 The drop profile inside the transition zone. 

Unfortunately, Equation 2.60 cannot be integrated as is done in the case of two-dimensional menisci and drops. However, the latter equation can be rewritten 

The first term on the left-hand side of the preceding equation is due to the first curvature (similar to the case of the two-dimensional menisci or drops in Equation 2.23), and the second term is due to the second curvature, which is shown in the following text to be small as compared to the first term. The characteristic length of the transition region, L, is given by Equation 2.49 L - heH. The latter expression shows that L H. Let us estimate the ratio of the second term to the first term on the left-hand side of Equation 2.61  

The latter estimation shows that the second term on the left-hand side of Equation 2.61 is small as compared to the first term and can be neglected in the transition region. After that. Equation 2.61 can be integrated, and in a similar manner as Equation 2.23, Equation 2.26 can be recovered. Outside the region of the action surface forces. Equation 2.60 can be easily solved. This solution is the outer solution, whereas the solutions obtained in the previously mentioned method all are inner solutions. The matching of these two asymptotic solutions gives the real profile in the case of axial symmetry (see this procedure in the case of complete wetting in Section 2.4). 

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Neumann has adapted the pendant drop experiment (see Section II-7) to measure the surface pressure of insoluble monolayers [70]. By varying the droplet volume with a motor-driven syringe, they measure the surface pressure as a function of area in both expansion and compression. In tests with octadecanol monolayers, they found excellent agreement between axisymmetric drop shape analysis and a conventional film balance. Unlike the Wilhelmy plate and film balance, the pendant drop experiment can be readily adapted to studies in a pressure cell [70]. In studies of the rate dependence of the molecular area at collapse, Neumann and co-workers found more consistent and reproducible results with the actual area at collapse rather than that determined by conventional extrapolation to zero surface pressure [71]. The collapse pressure and shape of the pressure-area isotherm change with the compression rate [72]. 

The axisymmetric drop shape analysis (see Section II-7B) developed by Neumann and co-workers has been applied to the evaluation of sessile drops or bubbles to determine contact angles between 50° and 180° [98]. In two such studies, Li, Neumann, and co-workers [99, 100] deduced the line tension from the drop size dependence of the contact angle and a modified Young equation... 

Illustration Stretching of low-viscosity-ratio elongated drops. For the case p < 1 and Ca/Cacnt = 0(1), the dynamics of a nearly axisymmetric drop with pointed ends, characterized by an orientation m ( m = 1) and a length L(t), is given by (Khakhar and Ottino, 1986b,c)... 

Khakhar, D. V., and Ottino, J. M., A note on the linear vector model of Olbricht, Rallison, and Leal as applied to the breakup of slender axisymmetric drops. J. Non-Newtonian Fluid Mech 21,127-131 (1986b). 

Contact Angle Measurements. Axisymmetric drop shape analysis - profile (ADSA-P) The hydrophobicity/ hydrophilicity of a solid surface is usually expressed in terms of wettability, which can be quantified by contact angle measurements. ADSA-P is a technique to determine liquid-fluid interfacial tensions and contact angles... 

Kwok, D.Y., Gietzelt, T., Grundke, K., Jacobasch, H.J., and Neumann, A.W. (1997). Contact angle measurements and contact angle interpretation. 1. Contact angle measurements by axisymmetric drop shape analysis and a goniometer sessile drop technique. Langmuir, 13, 2880-2894. 

S. Lahooti, O.I. del Rio, A.W. Neumeinn and P. Cheng, Axisymmetric Drop Shape Analysis (ADSA), ch. 10 in the book edited by Neumann and Spelt, mentioned in sec. 1.17c. 

D. Li, P. Cheng and A.W. Neumann, Contact Angle Measurement by Axisymmetric Drop Shape Analysis (ADSA), Adv. Colloid Interface Set 39 (1992) 347, and references to the Neumann group in sec. 5.12 D.Y. Kwok, A.W. Neumann, Contact Angle Measurement and Contact Angle Interpretation, Adv. Colloid Interface Set 81 (1999) 167-249. 

When the point of zero slope lies on the axis of rotational symmetry, as is the case with axisymmetric drops and bubbles, the solution for points in the immediate vicinity of the axis is... 

Chen, P., del Rio, O.I. and Neumann, A.W. (2000) Axisymmetric drop shape analysis, in Physical Chemistry of Biological Interfaces. Marcel Dekker, New York... 

Bashforth F and Adams JC (1883) An attempt to test the capillary action, Cambridge University Press and Deighton Bell Co., Cambridge Chen P, Kwork DY, Prokop RM, del-Rio 01, Susnar SS and Neumann AW (1998) Axisymmetric drop shape analysis (ADSA) and its applications , in Drops and bubbles in interfacial research, D. Moebius and R. Miller Eds., Studies in Interface Science Series, Vol. 6, Elsevier, Amsterdam Dukhin SS, Kretzschmar G and R. Miller R (1995) Dynamic of adsorption at liquid interfaces. Theory, experiments, applications, D. Moebius and R. Miller Eds., Studies in Interface Science Series, Vol. 1, Elsevier, Amsterdam Joos P (1999) Dynamic Surface Phenomena, VSP, Utrecht, 1999 Kovalchuk VI, Zholkovskij EK, Kragel J, Miller R, Fainerman VB, Wiistneck R, Loglio G and Dukhin SS (2000) Bubble Oscillations in a Closed Cell. J Colloid Interface Sci 224 245-254... 

Figure 6-2. A sketch of the initial configuration of spreading drop on a solid substrate. The drop may be either a two-dimensional ridge of fluid (with the sketch showing the cross-sectional view), or it may be an axisymmetric drop. The thin film approximation requires that the radial breadth of the drop be much larger than its height. The initial values of these quantities are denoted as Rq and H0.

We assume that we have a thin film. A fixed amount of surfactant is deposited onto the surface of this thin film, either as a 2D strip or as an axisymmetric drop. We assume that this surfactant is insoluble in the bulk fluid of fhe thin film. We denote the total (fixed) mass of surfactant as M, which may be related to fhe surface concentration F (measured in units of mass per unit surface area) in a form that encompasses both the 2D and axisymmetric geometries ... 

Dynamic properties of interfaces have attracted attention for many years because they help in understanding the behaviour of polymer, surfactant or mixed adsorption layers.6 In particular, interfacial rheology (dilational properties) is crucial for many technological processes (emulsions, flotation, foaming, etc).1 The present work deals with the adsorption of MeC at the air-water interface. Because of its amphiphilic character MeC is able to adsorb at the liquid interface thus lowering the surface tension. Our aim is to quantify how surface active this polymer is, and to determine the rheological properties of the layer. A qualitative and quantitative evaluation of the adsorption process and the dilata-tional surface properties have been realised by dynamic interface tension measurements using a drop tensiometer and an axisymmetric drop shape analysis. 

The Gauss-Laplace equation describing liquid menisci in general was discussed in detail by Padday Russel (1960) and Padday et al. (1975). The profile of an axisymmetric drop can be calculated in dimensionless co-ordinates from the following equation (Rotenberg et al. 1983),... 

Kwok, D. Y., Cheung, L. K., Park, C. B., and Neumann, A. W., Study on the surface tensions of polymer melts using axisymmetric drop shape analysis, Polym. Eng. Sci., 38, 757-764 (1998). 

Additional information on interfacial layers can be gained from rheological and ellipsometry experiments. There is quite a number of different experimental setups used to determine surface rheological parameters (27). New possibilities to determine surface dilational parameters arise from oscillating-drop experiments. Using axisymmetric drop shape analysis (ADSA) the change in interfacial ten-... 

C. Axisymmetric Drop Shape Analysis and Spinning Drop Technique... 

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. 

Contact Angle Measurements by Axisymmetric Drop Shape Analysis... 

For even better results, drop profile analysis can be applied instead of measuring the contact angle directly (axisymmetric drop shape analysis, ASDA Fig. 4.20). This technique extracts experimental drop profiles from video images while slowly increasing or decreasing the droplet volume [42, 43]. The best fit of experimental data with theoretical assumptions based on the Laplace equation of capillarity allows one to calculate the surface/inter-facial tension and subsequently the contact angle. Also droplet radius, droplet volume, and the contact area are computed. ADSA can therefore reveal... 

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