# The Analysis of Periodic Order in Monolayers of Colloidal Superballs
## 4 Results and Discussion
Figure 4e depicts the vertical cross sections taken at \(q_{x}=6.12\,\upmu\text{m}^{-1}\), which contain the \(h1\) peaks. Here, it can be seen that there is a clear evolution of the peaks. Specifically, the 01 peak becomes visible only at \(\phi=50^{\circ}\). By contrast, the 11 and 21 peaks are both visible at up to \(\phi=20^{\circ}\) and disappear at higher angles with only the 11 peak appearing again at \(\phi=50^{\circ}\). The observation that both the 10 and the 01 peaks only appear upon a considerable rotation indicates that these peaks must be close to a minimum in the form factor. Furthermore, it can be seen that the \(h2\) and \(h3\) peaks appear to be broadened along the vertical \(y\)-direction, as indicated in Figure 4a-c. The biggest influence of the particle form factor was observed for monolayers prepared from hollow superballs with \(m=3.63\pm 12\%\) and \(D=1.01\,\upmu\text{m}\pm 6.6\%\) that have a distinct cube-like shape. Figure 5a-c show the 2D SAXS patterns imaged at different rotations. Strikingly, here not even a single Bragg peak is significantly visible in the pattern at a rotation of \(\phi=0^{\circ}\) as seen in Figure 5a. Upon rotating the sample to \(\phi=20\) and \(40^{\circ}\), some of the diffraction peaks can be made visible (Figure 5b,c). Again, we examine the vertical cross sections of the diffraction patterns at \(q_{x}=0\,\upmu\text{m}^{-1}\) (Figure 5d). From the evolution of the peaks, it is clear that the 10 peak appears in the patterns at \(\phi=30^{\circ}\) and higher rotations, while the 20 and 30 peaks are only visible up to \(\phi=30^{\circ}\). Furthermore, in the scattering patterns depicted in Figure 5a-c, some intensity can  
Figure 5: (**a–c**) 2D SAXS patterns of a monolayer prepared from hollow colloidal silica superballs with an average shape parameter of \(m=3.63\pm 12\%\) and diameter of \(D=1.01\,\upmu\text{m}\pm 6.6\%\) imaged at different rotations around the \(x\)-axis of \(\phi\). The Bragg peaks are labelled on their left. (**d**,**e**) Vertical cross sections of the diffraction patterns at (**d**) \(q_{x}=0\,\upmu\text{m}^{-1}\) and (**e**) \(q_{x}=5.76\,\upmu\text{m}^{-1}\). The vertical lines indicate the \(q_{y}\text{cos}(\phi)\) positions of the identified peaks. be seen where the \(h1\) and \(h2\) peaks are expected. These regions are highlighted in Figure 5a. In Figure 5e the cross sections of the diffraction patterns imaged at the various rotations \(\phi\) taken at \(q_{x}=5.76\,\mathrm{\SIUnitSymbolMicro m}^{-1}\) are depicted, corresponding to the intensity ascribed to the \(h1\) peaks. Due to their broadness, however, no clear Bragg peaks can be assigned here as the \(h1\) peaks appear to form a continuous band. To further understand the appearance and disappearance of the peaks, we examine the different form factor contribution of the superball colloids used here, with spherical, semi-cubic and cube-like shapes. The form factor of a hollow superball is a function of the shape and size as well as the thickness of the shell for which there is no analytical solution. Therefore, we have calculated their form factors numerically. Figure 6a-c show the resulting calculated 2D form factors at a rotation of \(\phi=0^{\circ}\), in which the semi-cubic and cube-like superballs lie flat on the PDMS-coated substrate, and an effectively 2D projection is obtained. Although the form factor for spherical particles has an analytical solution, as seen in Figure 3e, we included it here as a numerical calculation for a direct comparison. It can be seen that upon increasing the shape parameter \(m\) of the colloidal superballs, the isotropic rings visible for spheres (\(m=2\)) began to show oscillations with maxima in the horizontal and vertical directions, i.e., in the directions of the flat sides of hollow superballs, and minima in the diagonal directions, i.e., the corners of the hollow superballs. These changes in the form factor explain why the structure factor peaks visible for the spherical particles disappear for the hollow superballs.