Colloids and Surfaces A: Physicochemical and Engineering Aspects
Obtaining surface tension from pendant drop volume and radius of curvature at the apex
Introduction
In a typical pendant drop tensiometer of today a CCD camera is used to capture the images of droplets formed at the tip of a capillary and these are stored directly on computer in real time. Image processing software are then used to convert these images into droplet profiles, i.e., discrete pairs of droplet radius versus vertical height, (rM, zM), data points. Superscript M is used here to distinguish the measured profile from the computed profile, which will be denoted by (rC, zC). Such computer-based tensiometers generate a large number of closely spaced data points compared to the tensiometers of just a few years back in which only a small number of points are normally reported. A large number of computational schemes have been developed to convert these large data sets into surface or interfacial tension. For example, in a currently widely adopted scheme, the governing equation of the pendant drop, the Laplace–Young equation [1]:is solved numerically to yield a computed profile (rC, zC). In Eq. (1), r1 and r2 are the local principal radii of curvature of the surface. At the apex of the droplet, z = 0 and r1 = r2 = rApex. Δρg the known specific gravitational body force acting on the droplet and γ is the unknown surface tension to be determined. This is determined by iteratively adjusting the parameter Δρg/γ (and rApex) so that the computed (rC, zC) closely matches its measured counterpart (rM, zM). This route to surface tension requires repeated integration of Eq. (1) and the least-squares minimization of the difference between (rM, zM) and (rC, zC) over the large number of data points. This is a non-trivial computational problem and a number of specialized numerical schemes have been developed to perform this task [2], [3], [4], [5], [6], [7], [8], [9], [10]. In the most recent review Hoorfar and Neumann [10] examined in great detail many of the hardware, software and computational techniques used to obtain, compute and match pendant drop profiles. They came to the conclusion that, even with the latest techniques and instruments, for certain type of drop profiles, the resulting surface tension is still not as reliable as desired. This is a reflection of the difficult nature of the pendant drop tensiometry problem.
In an alternative approach to handling pendant drop tensiometry data, spline curves are fitted locally to selected subsets of (rM, zM) [11]. These curves are then differentiated to obtain the radii of curvature r1 and r2, according to the following standard results of differential geometry:as functions of z. According to Eq. (1), 2H(z) ≡ 1/r1 + 1/r2 is linearly related to z with a slope of −Δρg/γ. Thus γ can be deduced from the best-fit straight line through the set of computed 2H(z) versus z data points. H(z) is the mean Gaussian curvature of the droplet and is a fundamental property of the surface of the pendant drop. However, it is well known that differentiation of fitted spline curves often does not lead to reliable first and second derivatives. Practical experience confirms that plot of 2H(z) versus z generally does not yield a well-defined straight line and consequently introduces considerable error into the surface tension deduced from its slope [11]. This is because differentiation of experimental data, either directly or via fitted curves, is an ill-posed problem in that the small and unavoidable noise in the data will be amplified by the differentiation process. These amplified noise are further amplified when the derivatives are substituted into Eq. (2) to compute 1/r1 and 1/r2. Tikhonov regularization is a specialized method that can be used to obtain the derivatives of experimental data. Through the built-in regularization parameter λ, this method is able to keep noise amplification under control [12], [13]. Recently Yeow et al. [14] applied this method to compute the derivatives d2r/dz2 and dr/dz of a number of pendant drop data sets and used them to obtain the mean curvature H(z). They were able to obtain reliable H(z) for general points on the droplet surface. However, because of the singularity of the derivative dr/dz at the apex, they were unable to obtain the radius of curvature rApex at this point and for points close to the apex and were forced to exclude data points from this neighbourhood in their evaluation of surface tension.
The present investigation examines the nature of the (rM, zM) data in the neighbourhood of the apex. The role of z and r is reversed with the vertical height z now regarded as a function of the droplet radius r, i.e., z(r). At the apex z(r) is a well-behaved function with a finite d2z/dr2 and vanishing odd derivatives. A modified Tikhonov regularization scheme is applied to obtain the third derivatives d3z/dr3 as a function of r which is then used to back-calculate the lower order derivatives d2z/dr2 and dz/dr in the neighbourhood of the apex. In terms of these derivatives, the equivalent of Eqs. (2a) and (2b) for r1 and r2 becomeSince all these derivatives are well behaved in the neighbourhood of the apex, a reliable Gaussian mean curvature can be obtained. The radius of curvature at the apex takes on a particularly simple form, rApex = 1/(d2z/dr2)r=0 and can be obtained directly from the results of Tikhonov regularization computation. This key property of the pendant droplet profile together with the volume of the droplet is then used to obtain γ. This way of obtaining γ makes use of the general relationship between droplet volume , rApex and Bond number, . rCap is the radius of the capillary to which the droplet is attached and Bo is a measure of the relative importance of the gravitational body force compared against surface tension. This relationship will be presented in the form of a family of curves relating the dimensionless apex curvature RApex = rApex/rCap to the dimensionless volume with Bo as an independent parameter that is held constant on each curve. The reliability of this way of determining surface tension will be demonstrated by applying it to several sets of data.
Section snippets
Integral equation for the third derivative
To simplify the development of the relevant equations, the second derivative d2z/dr2 and the third derivative d3z/dr3 of (rM, zM) will be denoted by f(r) and g(r), respectively. These derivatives are related to the droplet profile z(r) by:where f0 = f(0) is the second derivative at the apex. Eq. (3) is exact and is based on a two-term Taylor series expansion of z(r) about r = 0 with the remainder term expressed in integral form [15]. Symmetry of the drop profile
Apex radius of curvature, bond number and volume
Since the original work of Bashforth and Adams [16] on pendant drop tensiometry the relationship between surface tension and the key geometrical features of a droplet have been investigated and tabulated or plotted in various forms. The geometrical features investigated include droplet volume, the radius of curvature at the apex, droplet height and radius at the equatorial plane. The relationships between these geometrical parameters and surface tension are obtained by solving the following set
Simulated water droplet
The discrete points in Fig. 2(a) are the simulated droplet profile for water (Δρ = 998.2 kg m−3, assumed γ = 72.79 mN m−1) used by Yeow et al. [14] to test their method of processing pendant drop tensiometry data. These have been re-plotted with r as the independent variable instead of z. There are 41 points in this data set but only the 16 points in the neighbourhood of the apex, shown as filled grey squares, are used in the determination of d3z/dr3 by Tikhonov regularization. The resulting third
Discussion
The curvature of a droplet at its apex rApex appears explicitly in the Laplace–Young equation of pendant drop tensiometry. For an experimentally measured profile this parameter is not known and has to be evaluated by differentiating the experimental data or via fitted splines through the experimental data points. Apart from the general and well-known difficulties associated with differentiation of experimental data, there is an added problem. If the (rM, zM) data are treated as a function of z,
Conclusions
Treating drop height as a function of drop radius allows the use of Tikhonov regularization to obtain an estimate of the radius of curvature at the apex. This key property of a pendant drop together with its volume can be used to obtain the Bond number of the droplet and hence its surface tension from the computer-generated constant Bo curves in the dimensionless V–RApex plane. For this method to work reliably, particularly for small droplet volume, it is necessary to obtain the RApex to an
Acknowledgements
The authors gratefully acknowledge the pendant droplet profile data provided by Ms. Nicole Dingle and Prof. Michael Harris and by Prof. Shi-Yow Lin.
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