MEASUREMENT OF INTERFACIAL TENSION AT LIQUID INTERFACES FROM THE SHAPE OF A PENDANT DROP

1. MERCURY IN VACUUM

V. I. MELIK-GAIKAZYAN, V. V. VORONIKHINA, E. A. ZAKHAROVA

(Presented by the scientific seminar of the Department of Physical and Colloid Chemistry)

Measurement of the surface tension $\sigma$ of a liquid from the shape of its resting sessile or pendant drops, or of gas bubbles formed in a liquid, possessing axial symmetry, makes it possible under static conditions to determine, without contact, the absolute values of this quantity and to record possible changes in it over time. In this case the determination of $\sigma$ is based on evaluating the change in curvature of the surface of the drop* with height, which is carried out by means of various measurements, simple in principle [1—4], performed on the contour of the meridional section of the drop under consideration. From the results of these measurements and with the aid of specially compiled tables, relations, or graphs, the values of $\sigma$ of the liquid and the coefficient $\beta$, whose sign and magnitude uniquely characterize the shape of the surface under consideration, are calculated. The accuracy of determining $\sigma$ depends, other conditions being equal, on the rationality of the chosen method for evaluating the shape of the drop, on the height of the portion of its contour whose point coordinates are used in the measurements, and on the errors allowed in the measurements. In order to reduce the relative magnitude of the latter, measurements are preferably carried out on comparatively large drops. Thus, in the case of mercury, measurements in vacuum are usually performed on sessile drops with an equatorial diameter of $4\ \mathrm{cm}$. The technique of the measurements carried out in this way has been worked out in studies [5—8] so thoroughly that the values of $\sigma$ presently obtained by many authors differ among themselves and differ from data obtained by other methods by no more than one dyne [9—11], i.e., by approximately $0.2\%$ of the absolute value of the measured quantity ($485\ \mathrm{dyn}/\mathrm{cm}$ at $25^\circ\mathrm{C}$). At the same time, the “useful” height of the surface of a mercury drop, ordinarily used in the measurements and to some extent determining the accuracy of the obtained values of $\sigma$, is $0.28\ \mathrm{cm}$ for a four-centimeter equatorial diameter of a sessile drop. It should be noted that, in the case of pendant drops, the same height can easily be realized with a drop diameter of only $0.25\ \mathrm{cm}$ (Fig. 1). In addition, when measurements are carried out on pendant drops, certain experimental difficulties that are inevitable when working with sessile drops are automatically eliminated; for example, it is easier to give the drop a symmetrical shape, and the contour of the drop is obtained more sharply because of the greater curvature of its surface—

* In what follows only drops will be mentioned, although all of this applies equally to bubbles.

Fig. 1. Shapes of pendant mercury drops in vacuum at 25°C, calculated from the Laplace equation.

…ness [6, 8], the results of the measurements become independent of the possible wedge shape of the optical glass in the cell window through which the drop is photographed or its dimensions measured [7]); the more rounded shape of the drop reduces the negative influence of distortion, etc.

As we have convinced ourselves, it is best to evaluate the shape of pendant drops by the method of [3], the essence of which consists in determining the value of the ratio \(S=d_s/d_e\), where \(d_e\) and \(d_s\) are, respectively, the diameters of the equatorial section and of the section drawn, according to [3], at the level for which \(K=1\) (Fig. 2). By means of a special table, from the found values of \(S\), \(d_e\), and the known density difference between the drop and the medium, the value of \(\sigma\) is determined. The table [3], covering the interval of values of \(\beta\) from \(-0.26\) to \(-0.60\), was subsequently corrected [12], and its lower limit was extended to \(\beta=-0.150\) [13] and later to \(\beta=-0.037\) [14].

The method [3] was developed for many fixed levels (Fig. 2), and the values of \(S\) found for each of them were recalculated, by means of special five-digit tables [15], to the value of \(\beta\), which, naturally, must be constant for a given shape, regardless of the level at which it is determined. This makes it possible to check the correctness of the values of \(\beta\) obtained in this way, to determine how close to equilibrium the form under consideration is, to estimate possible errors made during the measurements, and to carry them out both on elongated and on flattened drops. The curves in Fig. 3 graphically illustrate the range of shapes and the character of the dependence \((S,\beta)\) contained in the tables [15].

From measurements carried out earlier on the image of sessile drops it was known that an inaccuracy, allowed when aligning the axis of symmetry of the drop image with the reference axis of the measuring microscope by one degree, leads to an error in the value of \(\sigma\) of 1% [4]. Calculations carried out by us for the example of a pendant drop with \(\beta=-0.2\) for the levels 0.7, 1.0, and 1.2 showed (Fig. 4) that even errors visible to the eye in aligning the axis of symmetry of the drop with the reference axis of the measuring microscope do not, in the case of the method [3], have a noticeable effect on the accuracy of determining the surface shape, and therefore…

Fig. 2. Diagram illustrating the method for estimating the shape of the surface of a drop from the ratio of the diameters of two horizontal sections: the equatorial one \((d_e)\) and any one \((d_s)\) of those checked at fixed levels \(K\).

Fig. 3. Curves for finding the value of \(\beta\) from the values of \(S\) found from measurements made at various levels of a pendant drop (Fig. 2).

Fig. 4. Errors permitted in determining the shape of a pendant drop by the method of [3] and caused by the noncoincidence of the symmetry axis of the drop and the \(oz'\) axis of the measuring microscope.

Consequently, they also affect the accuracy of the determination of \(\sigma\). This conclusion was fully confirmed in the results obtained for mercury drops with \(\beta\) from \(-0.15\) to \(-0.25\), the images of which, during measurements under the microscope, were rotated through various angles. The vertical dashed line in the graph of Fig. 4 bounds the region of maximum errors allowed by us in carrying out the operation under consideration, namely the alignment of the axes.

In Fig. 5, on the right, two schemes are presented that correspond to the two most unfavorable combinations of errors allowed in measuring the quantities whose values are necessary for calculating the shape of the surfac-

... according to the method [3]. The lengths of the arrows under the diameters \(d_e\) and \(d_s\) schematically indicate in which direction from the true value their possible deviation is assumed in the given case. The same is indicated by the signs before \(\Delta_0\), \(\Delta_1\), \(\Delta_2\), which express the absolute error, each of which in the case under consideration is equal either to zero or to \(0.001\,b\), which, according to the experimental data, corresponds to their actually possible value. The quantity \(b\), chosen in the tables [1,

Fig. 5. Errors allowed in determining the shape of a pendant drop by the method [3] and caused by errors in determining the coordinates of points on its contour. The numbers marking the curves \((\beta, K)\) correspond to the indices of the errors taken into account in the calculations.

12, 13, 15] as the unit of scale, is the radius of curvature of the surface at the point of its intersection with the axis of symmetry. To the left of each of the diagrams in Fig. 5 are given the calculated curves expressing the deviation of the quantity \(\beta\) from its true value, introduced by the errors \(\Delta_0\), \(\Delta_1\), and \(\Delta_2\) at different levels \(K\). For the calculation a shape with \(\beta=-0.2\) was chosen. Since all the values used in the calculation are dimensionless [15], the results obtained are to a considerable extent applicable also to other shapes close to the one considered. The numbers marking the curves in Fig. 5 correspond to the index of the error, or to the combination of errors, that produced the given deviation of \(\beta\). For example, the curve marked “0” corresponds to the case where all measurements have been carried out correctly, except for the operation of finding point 0, to which, during the measurements, a position corresponding to points \(0'\) or \(0''\) was erroneously assigned. The curves “012” correspond to cases in which all measurements are erroneous, and in the most unfavorable combinations. From the curves in Fig. 5 it follows:

a) all other conditions being equal, measurements should be carried out at the greatest possible value of \(K\);

b) from the character of the dependence \((\beta, K)\) obtained from the measurements, one can judge the degree of correctness of the calculated values of \(\beta\), or determine which of the values of \(\beta\) is closest to the true one;

c) simple averaging of the values of \(\beta\) obtained from measurements made at different levels, as apparently is sometimes done [4], is obviously impermissible.

If, when more careful repeated measurements are made, the course of the dependence \((\beta, K)\) does not change, i.e., remains decreasing or increasing,

which leads to the conclusion that the shape under consideration is either nonequilibrium or distorted. Thus, carrying out measurements not at one [3, 12–14], but at many levels makes it possible to check the results obtained and may serve as a guarantee of their reliability and accuracy.

Experimental procedure and results obtained

Figure 6 shows a diagram of the apparatus we used in measuring the surface tension of mercury in vacuum from the shape of a pendant drop. On the right are shown two diagrams illustrating the possibility (upper) and impossibility (lower) of holding a liquid drop at the tip of a capillary, the surface of which is, respectively, wetted and not wetted by this liquid, while a liquid meniscus may form in the capillary. It follows from the diagram that, in order for a mercury drop to hang from the tip of the capillary, the latter must be made of a material wetted by mercury, for example, amalgamated platinum, or of glass, but with the obligatory exclusion of a meniscus in the tube with mercury directly connected to the capillary at whose end the drop hangs. This was achieved by means of a carefully ground vacuum stopcock 1, which breaks the column of mercury in the tube after the drop has been suspended at the end of the capillary. The lubricant for the stopcock is the mercury itself. To prevent the stopcock plug from being drawn into the sleeve, it is made hollow and has, in its upper part, an opening through which the mercury in the plug is connected with the mercury in the reservoir. The stopcock is provided with a ground cap, beneath which, above the mercury reservoir, a vacuum is created equal to that at the level of this reservoir in the mercury tubes connected to stopcock 1. In this case the stopcock plug can rotate freely, and no appreciable flow of mercury from the reservoir into the system and back occurs even when the experiment lasts several days.

Before the experiment the apparatus was thoroughly washed with concentrated sulfuric acid and hot distilled water, practically free of organic impurities [17], sealed off, and dried—

Fig. 6. Diagram of the apparatus for determining the surface tension of mercury in vacuum from the shape of a pendant drop.

Table 1

Values of surface tension in vacuum at 25°C, found from the shape of a pendant drop of high-viscosity potassium in the apparatus diagrammed in Fig. 6

Mean value:

was carried out in vacuum. After this, thoroughly purified and vacuum-distilled mercury was poured into the apparatus; following the example of [6–8], it was distilled in vacuum for several days in a closed circuit, i.e., through condenser 2, stopcock 1, and capillary 3, from which, dripping continuously, it again entered distillation flask 4. The vacuum in the system was produced by a three-stage mercury diffusion pump 5. When it was necessary to stop pump 5, the measuring part of the setup was separated from the pump, stopcocks, and the rest of the system by mercury seal 6.

The pendant mercury drops were photographed in transmitted parallel light by means of a horizontal microscope and a Zenit camera on RF-3 motion-picture film through optical glass plates ground to the glass flanges of the cell and attached to them along the edges with epoxy resin.

Measurements were made on images both of stationary pendant drops and of slowly growing ones (the exposure during photography was \(1/125\) sec). From the results given in Table 1 it follows that, irrespective of the lifetime of the drop, the differences in the values of \(\sigma\) obtained lie within the limits of the measurement accuracy and are in good agreement with data reported for analogous conditions by other authors [7–12].

The mean value of \(\sigma\) for mercury in vacuum obtained in our work at \(25^\circ\mathrm{C}\) proved to be \(484.6 \pm 1.3\) dyn/cm, where \(1.3\) dyn/cm is the root-mean-square error of an individual measurement.

It has been shown that when the grounding of the mercury is disconnected, the value of \(\sigma\) rapidly decreases by 30 or more dyn/cm, but becomes normal again when the grounding is restored. It should be noted that the necessity of grounding mercury when measuring \(\sigma\) in vacuum had been noted earlier [6–8]; however, such a strong influence of the charging of the mercury surface on the measurement results was not found, which is apparently connected with the considerably smaller size of the drops used by us for the measurements. Thus, when large sessile drops were used [8], disconnecting the grounding led to a decrease in \(\sigma\) of only 6 dyn/cm.

Application of the method [3], developed for many levels, makes it possible to record electrocapillary curves simultaneously with curves of the differential capacitance of the double layer in electrolyte solutions, both pure and containing surface-active impurities and, in some cases, an oil emulsion. These results will be presented in subsequent publications.

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