# Effects of Tube Radius and Surface Tension on Capillary Rise Dynamics of Water/Butanol Mixtures
## 2 Materials and Methods
### Methods  
Figure 1 shows the experimental setup designed for measuring the surface tension (\(\sigma\)) and capillary rise dynamics. The measurement system comprises of three subsystems: a PC/monitoring part, capillary rising part, and high-speed imaging part, as shown in Figure 1a-c, respectively. In the experiments, we raised the height of the liquid using a lab jack to connect the bottom of the capillaries with the liquid, hence facilitating the entry of liquid into the capillaries. The height of rise (\(h\)) of the liquid was measured as the point where the capillary tube just touches the liquid to the point where the meniscus stops at the equilibrium height (\(h_{\text{e}}\)). These rising processes were recorded using a high-speed camera (FASTCAM SA4, Photron Ltd., Tokyo, Japan) connected to a PC. The \(\sigma\) first measured using dual capillary tubes based on differential capillary-rise-height method [22], and the capillary dynamics using a single capillary tube was then analyzed by measuring the height of rise and its velocity [13; 14; 23]. Figure 2 shows the principle of the differential capillary rise method and the captured image for the difference in the rising height. When the liquid contacts the bottom of the capillary tube, the contact forces with the wall of the capillaries aid in the capillary rise (\(h_{\text{e}}\)) for each of the capillary tubes. The height of the meniscus is a function of the capillary  
Figure 1: Schematic of the experimental setup. (**a**) PC and monitor, (**b**) capillary tube and liquid, and (**c**) high-speed camera and lens. radius. \(\sigma\) can be calculated from the difference in the height of the capillary tubes. \(\sigma\) in the captured image on the right side of Figure 2 is calculated by the following equation [24]:  
\[\sigma=\frac{\rho g\Delta h_{\mathrm{e}}}{2}\bigg{(}\frac{r_{1}r_{2}}{r_{1}-r_{2 }}\bigg{)}, \tag{1}\]  
studies where \(\sigma\) is the surface tension (mN/m), \(g\) is the standard gravity (m/s\({}^{2}\)), \(\rho\) is the density of the liquid (kg/m\({}^{3}\)), \(r\) is the capillary radius (mm), and \(\Delta h_{\mathrm{e}}\) is the difference in equilibrium height (mm), which is experimentally measured at the rise height (\(h\)) (\(t\rightarrow\infty\)); \(r_{1}\) is larger than \(r_{2}\). The \(\sigma\) for DW was measured with three capillary tube combinations (\(r_{1}\)/ \(r_{2}\) = 0.8/1.2, 0.8/1.7, and 1.2 mm/1.7 mm). The measurements were conducted at room temperature (23 \(\pm\) 1 \({}^{\circ}\)C), and the results are presented in Figure 3. The mean value of each test case was 72.17, 72.27, and 71.99 mN/m at \(r_{1}\)/ \(r_{2}\) = 0.8/1.2, 0.8/1.7, and 1.2 mm/1.7 mm, respectively. Thus, the mean value of \(\sigma\) for DW was 72.13 mN/m at 23 \(\pm\) 1 \({}^{\circ}\)C, which was consistent with a known value of 72.29 mN/m at 23 \({}^{\circ}\)C [25]. To measure the height of rise and velocity, individual images of the liquid invasion inside a single capillary tube were captured at a frame rate of 1000 frames per second (fps) and a pixel resolution of 640 \(\times\) 640 using a high-speed camera in conjunction with a 105 mm lens (F2.8 EX DG OS HSM MACRO, SIGMA, Japan), as shown in Figure 1b. The pixel size (220 \(\upmu\)m/pixel) calibration was based on the known size of the tapeline. Figure 3: Surface tension of DW (distilled water) with different tube radius combinations in the differential capillary-rise-height method. Figure 2: Principle of the measurement method and instantaneous image of the capillary rise height. Every image for the height of rise of the liquid was obtained with a time resolution of 1 ms (1000 fps), and then the rise velocity was calculated using the frame-to-frame method (\(\Delta h/\Delta t\)). All capillary rise measurements were carried out at room temperature (23 \(\pm\) 1 \({}^{\circ}\)C). Each experiment was repeated five times, and the mean value was calculated, along with standard deviation to measure the error.