# X-ray Particle Tracking Velocimetry in an Overflowing Foam
## 4 Discussion
### Tracer Particles  
This experimental study investigates the liquid foam flow over a horizontal weir, where the velocity field was measured by means of X-PTV using custom-tailored tracer particles. Their ability to follow the overflowing foam as well as their visibility in the X-ray images were the key for reliable flow measurements.  
Each tracer particle was composed of a polymer tetrahedron with 4.0 mm edge length and four metal beads, 0.5 mm in diameter and made of the lead-tin alloy Pb90 Sn10. Because of the high mass density of the main alloying element lead, the mass of one metal bead equal approximately 0.7 mg. In combination with the given diameter and the spherical shape, the ratio of the weight to the projected area yielded approximately 35 N m\({}^{-2}\). The yield stress of a liquid foam at 10% liquid fraction is typically in a similar range, i.e., around 40 N m\({}^{-2}\) as reported in [53], and reduces with increasing liquid fraction. As a result, an individual metal bead is likely to fall through the liquid foam [54] and, depending on its contact angle with the foam film, may cause film rupture [55]. In contrast, the total mass of one tracer particle, as shown in Figure 4b, equalled \((10.0\pm 0.4)\) mg on average, meaning that the mass fractions of the four metal beads and the polymer tetrahedron including the glue were less than 30% and more than 70%, respectively. The weight-to-area ratio of such a tracer particle was approximately 14 N m\({}^{-2}\), thus lower than the estimated yield stress of the liquid foam, which consequently was able to carry the tracer particle well. In our previous X-PTV study [42], we used tracer particles that were 3D-printed completely from metal. They had a comparable weight-to-area ratio of 15 N m\({}^{-2}\). However, their shape was less symmetrical, and for this reason, they could have preferential directions to stick to the bubble-scale foam structure. In addition, as that was the first approach, we focused on very stable liquid foams with a surfactant concentration above the critical micelle concentration, yielding less than approximately a 5% liquid fraction.  
Figure 12: Tracer particles carried by the overflowing foam (\(c_{1}=1.0\) g L\({}^{-1}\)) in the central measuring plane (\(z_{0}=0\) mm) at the weir: along their motion paths (as already shown on the left in Figure 8), the angular velocity \(\omega\) of the intrinsic rotation is plotted over the horizontal position \(x\). The same tracer particle is indicated by the same symbol and colour in both plots.  
In addition to the tracer particles used for the X-PTV measurements reported here, Table 2 lists three more combinations regarding the sizes of the polymer tetrahedron and the metal beads as well as the resulting weight-to-area ratio of the tracer particle.  
Shortening the edge length of the polymer tetrahedra from 4 mm to 2 mm while keeping the diameter of the metal beads constant at 0.5 mm has no beneficial effect on the weight-to-area ratio. Using smaller metal beads of 0.3 mm in combination with a polymer tetrahedron of 4 mm or 2 mm edge length yields a weight-to-area ratio slightly above or below 10 N m\({}^{-2}\). However, apart from the more challenging preparation and handling of smaller tracer particles, Figure 5 illustrates that metal beads smaller than 0.5 mm would attenuate the X-rays to a lesser extent, thus reducing the contrast between the projections of each metal bead and the surrounding liquid foam in the X-ray images. In addition, in the case of tracer particles having an edge length of 2 mm, the distances between the projections of the metal beads in the X-ray image would be smaller, and a partial overlap of the projections becomes more likely. As a further consequence, measurement uncertainties may increase.