## 2 Maximum Bubble Pressure Method
### Analysis of Surface Tension Using the Maximum Bubble Pressure Method  
The method of measuring the surface tension of solders at high temperatures consists of measuring the changes in the pressure of the gas supplied to the capillary immersed in liquid metal to a specified depth \(h\). The experiment is carried out at high temperatures with protective gases. The experiment begins the initial process of stabilising the thermodynamic conditions in the furnace chamber. At this stage, the chamber is heated in the presence of a gaseous protective atmosphere [14]. The actual measurement experiment begins with the immersion of the capillary in liquid solder to a preset depth \(h\). During the  
Figure 1: Bubble formation process (capillary immersion diameter \(\phi\) = 2 [mm] in glycerine to a depth of \(h\) = 30 [mm]) (own research). experiment, the pressure inside the gas bubble is recorded under a pressure-controlled gas supply outside. Between points A and B in Figure 2, there is an increase in gas pressure corresponding to bubble formation. The shape of the gas bubble surface and the value of the recorded pressure depend on the wettability of the capillary material by the liquid metal. At point C, there is a maximum pressure \(P_{max}\), and it is assumed that at the same time the bubble has the shape of a hemisphere with a radius equaling that of the capillary. Between D and E, a decrease in pressure inside the capillary is observed while the radius of curvature of the bubble is increasing. Point E corresponds to the detachment of the follicle from the capillary [11]. At equilibrium, the pressure in the bubble \(P_{p}\) balances the sum of pressures limiting the increase in the bubble's volume. For each point A of the bubble surface, the following occurs (2):  
\[P_{p}=P_{r}+P_{h}(A)+P_{\sigma}(A), \tag{2}\]  
where:  
* \(P_{p}\)--Pressure in the bubble (for practical purposes, this does not depend on point A). * \(P_{r}\)--Pressure in the retort above the bath surface (does not depend on point A). * \(P_{h}(A)\)--Hydrostatic pressure (a function of the depth \(h\) relative to the bath surface). In the experiment, the differential pressure is measured; the relation (2) can be simplified (3):  
\[\Delta P=P_{h}(A)+P_{\sigma}(A). \tag{3}\]  
The \(P_{h}(A)\) value is expressed by Equation (4):  
\[P_{h}(A)=\rho gh(A), \tag{4}\]  
where:  
* \(\rho\)--Density of the bath material. * \(g\)--Acceleration due to gravity. * \(h(A)\)--Depth of immersion of point A in the bath. The value of \(P_{\sigma}(A)\) depends on the surface tension of \(\sigma\) and the local curvature of the surface at point A. In the general case, this is expressed by the relation (5):  
\[P_{\sigma}=\sigma(\frac{1}{R_{1A}}+\frac{1}{R_{2A}}) \tag{5}\]  
Figure 2: An example showing the measurement of pressure changes in an argon bubble generated in a capillary submerged in a water bath to a depth of 30 mm at room temperature, with marked characteristic points (own research). where \(R_{1A}\) and \(R_{2A}\) are the radii of curvature at point A on two orthogonal lines passing through point A. Thus, the relation (3) takes the following form (6):  
\[\Delta P=\rho gh(A)+\sigma(\frac{1}{R_{1A}}+\frac{1}{R_{2A}}). \tag{6}\]  
In the case of a spherical surface, both radii of curvature are equal to each other and equal to the radius of the sphere \(R\); thus, the following equation holds for the sphere (7):  
\[P_{\sigma}=\frac{2\sigma}{R=const}. \tag{7}\]  
From Equation (6), it follows that the bubble does not have the shape of a spherical bowl, as the value of \(P_{h}\) increases at a constant value of \(\Delta P\) with the depth of immersion of point A; thus, the value of the second component of the sum must decrease, leading to the conclusion that the deeper the local radii of curvature, the larger the bubble. The bubble is somewhat flattened, and takes on a shape similar to a rotary ellipsoid; this is identical to the lying droplet method, where hydrostatic pressure pushes the droplet to the sides and into a flattened hemisphere [11; 13; 15].