# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 2 Materials and Methods
### X-ray Radiography  
The X-ray radiographic measurements were performed using a high-power X-ray tube (ISOVOLT 450M1/25-55, GE Sensing & Inspection Technologies), operated at 200 kV X-ray tube voltage and 22.5 mA tube current. Its 5.5 mm focal spot provides a continuous and divergent beam of polychromatic X-rays at a 40\({}^{\circ}\) emergent beam angle. The X-ray beam penetrated the foam flow channel in the region of interest around the weir, highlighted by the yellow rectangle in Figure 1. The transmitted X-rays impinged on a scintillation screen (SecureX HB, Applied Scintillation Technologies), which was positioned at a distance of approximately 55 cm from the X-ray source. In order to reduce undesirable effects arising from the two-dimensional X-ray tube focal spot and the beam divergence, the scintillation screen was mounted as close as possible to the back wall of the flow channel. However, a minimum distance of 1 cm was necessary because of the rear electrodes and their wiring. Upon X-ray impingement, the scintillation screen emitted visible light, which was further deflected by a mirror through a lens system (custom build, TSO Thalheim Spezialoptik) and finally captured by means of a sCMOS camera (pco.edge 5.5, PCO). Their field of view was set to approximately 7.8 cm \(\times\) 13.8 cm. Discretised by 1438 px \(\times\) 2560 px, this yielded an image pixel size of approximately 0.05 mm for objects located in the centre plane (\(z=0\)) of the flow channel. The X-ray image sequences were acquired with an imaging frame rate of 150 fps. The image exposure time was 3 ms for the purpose of achieving a suitable contrast-to-noise ratio [49] but without causing significant motion blurring of the tracer particles in the overflowing foam. X-ray radiography is based on the measurement principle of transmission imaging, which can be described by [PERSON]'s law in the first-order approximation:  
\[I=I_{0}\cdot e^{-\mu_{X}\cdot d_{z}} \tag{1}\]  
with the linear X-ray attenuation coefficient \(\mu_{X}\), the thickness \(d_{z}\), i.e., the distance along the X-ray beam direction, and the intensities \(I_{0}\) and \(I\) of the incident and the transmitted X-rays, respectively. While passing through matter, the X-ray intensity decreases exponentially. The attenuation coefficient is element- or material-specific and also depends on the X-ray photon energy. Given the 200 kV X-ray tube voltage, the energy could not exceed 200 keV. However, according to [PERSON]'s law [50], the intensity maximum of the continuous X-ray bremsstrahlung spectrum is assumed in the range from 130 keV to 150 keV. As this is a rough estimation, we consider a larger range 100 keV \(<E_{X}<\) 200 keV in the following. In liquid foam, the X-ray attenuation due to the gas phase can be neglected, because the attenuation coefficient of air is three orders of magnitude smaller than the coefficient of water [48]. As the concentration of the dissolved surfactant is low (Table 1), its contribution to the effective attenuation coefficient of the liquid phase can be neglected, too. According to [PERSON]'s law (Equation (1)) and its graphic representation in Figure 5, the liquid foam in the channel (\(d_{z}=100\) mm), here at a 10% liquid fraction (Figure 3a), reduces the X-ray transmittance by less than 20%. The left and right side walls of the foam channel, made of acrylic glass (PMMA, \(d_{z}=2\cdot 10\) mm \(=20\) mm), additionally reduce the transmittance by approximately 30%. For both the liquid foam and the PMMA channel walls, the blue and green graphs in Figure 5 suggest that the dependency on the X-ray photon energy is negligible in the considered range 100 keV \(<E_{X}<\) 200 keV. The X-ray images in Figure 4c-e show only the four metal beads belonging to each tracer particle, but the polymer tetrahedron is not visible.