On the Determination of Surface Tension, Volume, and Area of a Curvilinear Surface from the Shape of Sessile Bubbles or Pendant Drops

V. I. Melik-Gaikazyan

(Presented by the scientific-methodological seminar of the CTF)

Contactless determination of the surface tension $\sigma$, volume $V$, and area of the curvilinear surface $\Pi$ of a sessile bubble or pendant drop is necessary in the study of the mechanism of various phenomena, for example adsorption, spreading, adhesion, etc., which are of interest, respectively, for foam flotation and electrochemistry at a dropping electrode.

The simplest and at the same time sufficiently accurate method for determining $\sigma$ from the shape of a pendant drop is the method of Andreas, Hauser, and Tucker [1], according to which this shape, if it has a vertical axis of symmetry, is expressed by the ratio of the diameters of two horizontal sections: the equatorial diameter $d_e$ and the diameter $d_s$ of a section related to it, drawn from the apex $O$ (Fig. 1) at a distance equal to $d_e$. To simplify the computations, these authors compiled a special table, subsequently corrected by Fordham [2] on the basis of additional numerical solutions carried out by him of the Laplace equation for 11 values of $\beta < 0$, intermediate to the 5 previously used from the corresponding tables of Bashforth and Adams [3].

It should be noted that the computed tables [1, 2], which proved very useful in determining $\sigma$ of molten metals, salts, and viscous solutions, cover the range of forms $\beta$ from $-0.25$ to $-0.6$, i.e., they cannot be used in the case of gas bubbles smaller than 3 mm in diameter, or mercury drops smaller than 2 mm, which are of greatest interest for flotation and electrochemistry. In addition, in those cases where the bases of bubbles (drops) are not bounded, their heights may be smaller than the equatorial diameter, and this also precludes application to them of the tables computed for this method [1, 2].

In the present article, tables are given for calculating $\sigma$, $V$, and $\Pi$ from measurements made on the contour of the meridional section of any sessile bubbles or pendant drops having an equator and a height lying in the interval from 0.7 to 1.2 times the diameter of that equator. To compile these tables it was necessary to carry out about one hundred numerical solutions of the Laplace equation for values of $\beta$ from $-0.0025$ to $-0.6$, chiefly for values of $\beta$ small in absolute magnitude [4].

Method for determining $\sigma$ from the tables. On the negative of a photograph of the image of a drop or bubble, there are measured, for example on a UIM-21 measuring microscope, the equatorial diameter $d_e$ and the diameter of the section

of the section \(d_s\), drawn at a distance \(h=\kappa \cdot d_e\) from the apex of the shape at point \(O\) (Fig. 1), where the values of \(\kappa\) may be, respectively: 0.7; 0.8; 0.9; 1.0; 1.1, and 1.2. From the value of the ratio of these diameters \(S=d_s/d_e\), Table III gives, for the chosen \(\kappa\), the value \(\beta\) characterizing the given shape. To carry out the subsequent calculations it is necessary to determine the value \(b\), equal to the radius of curvature of the surface at point \(O\) and serving as the unit of scale in all the tables presented in the article. In Table II, for the value of \(\beta\) found, one finds the quantity \(\left(\dfrac{x}{b}\right)_{90^\circ}\). Dividing the actual value of the equatorial diameter \(d_e\) by twice the value \(\left(\dfrac{x}{b}\right)_{90^\circ}\) gives the value of \(b\), in the units chosen for \(d_e\).

The calculation of \(\sigma\) is performed from the following relation

\[ \sigma=\frac{g(D_1-D_2)b^2}{\beta}, \]

where \(D_1\) and \(D_2\) are the densities of the adjoining phases, and \(g\) is the acceleration due to gravity.

Fig. 1. Profile of a sessile bubble (pendant drop) with shape \(\beta=-0.2\), showing the adopted notation and the levels \(K\) for which the tables were calculated.

The accuracy of the determination of \(\sigma\) depends on the accuracy of the measurements made under the microscope, the accuracy of estimating the magnification during photography, and the magnitude of the distortions during photography.

A certain criterion of the reliability of the results obtained may be the agreement of the values of \(\beta\) determined for different levels \(\kappa\).

Method for determining the volume \(V\). The volume of the bubble or drop can be easily determined from the product \(b^3\) and the quantity

\[ \left(\frac{V}{b^3}\right)_{\kappa}, \]

found from Tables IV for the given \(\beta\) and \(\kappa\).

TABLE II

\[ \frac{x}{b};\quad \varphi = 90^\circ \]

TABLE II

\[ \frac{x}{b},\quad \varphi = 90^\circ \]

TABLE III

$S;\quad h = 0.70\,d_e$

TABLE III

\(S;\quad h = 0.70\,d_e\)

TABLE III

$S;\quad h = 0{,}80\,d_e$

TABLE III

$S;\ h = 0.80\,d_e$

TABLE III

\(S;\quad h = 0{,}90\,d_e\)

TABLE III

$S;\quad h = 0{,}90\,d_e$

TABLE III

\(S;\quad h = 1{,}00\,d_e\)

TABLE III

\(S;\ h = 1.00\,d_e\)

TABLE III

$S;\ h = 1{,}10\,d_e$

TABLE III

\(S;\quad h = 1{,}10\,d_e\)

TABLE III

\(S;\quad h = 1.20\,d_e\)

TABLE III

$S:\ h = 1{,}20\,d_e$

TABLE IV

TABLE IV

\[ \frac{V}{b^3}; \qquad h = 0{,}70\,d_e \]

TABLE IV

\[ \frac{V}{b^3}, \qquad h = 1.00\, d_e \]

TABLE IV

\[ \frac{V}{b^3}, \qquad h - 1.00\,d_e \]

TABLE IV

\[ \frac{V}{b^3}, \qquad h = 1.20\,d_e \]

TABLE IV

\[ \frac{V}{b^{3}};\qquad h=1.20\,d_e \]

TABLE V

\[ \frac{n}{b^2}; \qquad h = 0.70\, d_e \]