ON THE DETERMINATION OF THE SURFACE TENSION OF LIQUIDS FROM THE SHAPE OF SESSILE DROPS (BUBBLES)

V. I. Melik-Gaikazyan, V. V. Voronikhina

(Presented by the Scientific-Methodological Seminar of the Chemical-Technological Faculty)

For the contactless determination of the surface tension $\sigma$ of various viscous liquids—for example, molten metals, alloys, silicates, as well as resins, oils, and solutions of surface-active substances—the use of the equilibrium shapes of the large sessile drops or cavities (bubbles) formed by them, possessing axial symmetry, is usually recommended. This recommendation, found in most publications [1, 2], was theoretically justified [3] by the fact that, in the case of large drops whose equatorial diameter exceeds the ordinate of the equator by more than a factor of four, the latter quantity can be measured considerably more accurately and thereby the accuracy of determining $\sigma$ can be increased. The use of large drops, however, is associated with a number of experimental difficulties and with almost inevitable distortions of their shape during photography [4]. In this connection it seemed reasonable to use small drops and bubbles for measurements, and to evaluate their shape by the method of Andreas, Hauser, and Tucker [5], proposed earlier for the case of pendant drops. To compile the corresponding Tables VII and VIII$^{1)}$, the results of the numerical solution of the Laplace equation [7], carried out by Bashforth and Adams [8], were used.

The method for determining $\sigma$ is explained by Fig. 1, which schematically shows the shape of the meridional section of a small drop. By measuring on the negative of the photograph the equatorial diameter $d_e$ and the diameter of the section $d_s$ drawn at the chosen level K, one can, from the ratio $S=d_s/d_e$ and the corresponding curve $(S,\beta)$ in Fig. 2, or much more accurately from Table VIII, estimate from the value of $\beta$ the shape of the given drop [7, 8].

The value of $\sigma$ is calculated from the relation

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

where $D_1$ and $D_2$ are the densities of the bordering phases, $g$ is the acceleration due to gravity, and $b$ is the radius of curvature of the surface of the drop at its apex. The value of $b$ is found as the quotient obtained by dividing the true value of the equatorial diameter $d_e$ by twice the value

\[ \left(\frac{x}{b}\right)_{90^\circ}, \]

given in Table VII for the given $\beta$.

$^{1)}$ Tables VII and VIII are a continuation of Tables II and III published earlier for pendant drops [6]. In the calculations connected with the compilation of Tables VII and VIII, Engineer O. M. Konovalova took part.

Fig. 1.

Fig. 2.

Table VIII

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

Table VIII

$S:\quad h = 0.30\,d_e$

Table VIII

\(S:\quad h = 50\, d_e\)

References

1. B. S. Ellefson, N. W. Taylor. J. Am. Ceram. Soc., 21, No. 6, 193, 1938.
2. D. W. White. Trans. Am. Soc. Met., 55, p. III, 757, 1962.
3. E. W. Dismuks. J. Phys. Chem., 63, No. 2, 312, 1959.
4. C. Kemball. Trans. Faraday Soc., 42, p. 6, 7, 526, 1946.
5. J. M. Andreas, E. A. Hauser, W. B. Tucker. J. Phys. Chem., 42, 1001, 1938.
6. V. I. Melik-Gaikazyan. Izv. TPI, vol. 136, 1965.
7. N. K. Adam. The Physics and Chemistry of Surfaces. OGIZ, Moscow, 1947.
8. F. Bashforth and I. C. Adams. Capillary Action, Cambridge, University Press, 1883.